2. Preliminaries

Let $ M(p,q,\mathbb{R})$ and $ M(p,q,\mathbb{C})$ be, respectively, the spaces of $ \mathbb{R}$ - and $ \mathbb{C}$ -valued $ p \times q$ matrices, $ p,q \in \mathbb{N}$ , and let $ M(p,\mathbb{R}) = M(p,p,\mathbb{R})$ and $ M(p,\mathbb{C}) = M(p,p,\mathbb{C})$ . Also, let $ GL(p,\mathbb{R})$ and $ GL(p,\mathbb{C})$ denote the corresponding groups of nonsingular matrices on the fields $ \mathbb{R}$ and $ \mathbb{C}$ , respectively. For $ M \in M(p,\mathbb{C})$ , $ \text{eig}(M)$ denotes the set of possibly repeated eigenvalues (characteristic roots) of M, and $ \Re \text{eig}(M)$ denotes the set of their (possibly repeated) real parts. Whenever convenient, given $ M\in M(p,\mathbb{C})$ , we write $ \lambda_i(M)$ , $ i=1,\ldots, p$ , for the (possibly repeated) eigenvalues of M, indexed by the ordering $ \Re \lambda_1(M)\leq\ldots\leq \Re \lambda_p(M)$ . The symbol I denotes the identity matrix, and $ \text{diag}(d_1,\ldots,d_p)$ represents a diagonal matrix with main diagonal entries $ d_1,\ldots,d_p \in \mathbb{C}$ . The symbol $ \|{\cdot}\|$ denotes the Euclidean norm of a vector or the corresponding operator norm for a matrix. In the latter case, for a square matrix M, $ \|M\|^2$ is given by the largest eigenvalue of $ M^*M$ or $ MM^*$ .

2.1. Stochastic integrals

In this section, we use compensated Poisson random measures associated with finite-second-moment Lévy measures to describe a framework for stochastic integration. This framework provides a multivariate generalization of the ones in Benassi et al. [Reference Benassi, Cohen and Istas8, Reference Benassi, Cohen and Istas9] and Marquardt [Reference Marquardt44] (see also Marquardt [Reference Marquardt45]). The ultimate goal is to construct moving average and harmonizable classes of fractional stochastic processes (Section 3), so we consider stochastic integration in both frequency (Fourier) and time domains. In this section, we provide the definitions and expressions that are essential in the construction of ofLm. The Poisson random measures considered herein can be viewed as stemming from the jump measure of a Lévy process (see, e.g., Sato [Reference Sato62], Chapter 4). More properties of stochastic integrals can be found in Section A.

In the proposed framework, the differences between integration in the Fourier and time domains lie in the Poisson random measure domain ( $ \mathbb{C}^p$ or $ \mathbb{R}^p$ , respectively) and in the classes of integrands considered. In the former case, we mainly consider Hermitian integrands, so as to ensure $ \mathbb{R}^p$ -valued stochastic integrals.

We first consider the Fourier domain. So, let $ \mu_{\mathbb{C}^p}(d{\mathbf z}) \equiv \mu(d{\mathbf z})$ be a Lévy measure on $ {\mathcal B}(\mathbb{C}^p)$ satisfying

(7) \begin{equation}\int_{\mathbb{C}^{\,p}} {\mathbf z}^* {\mathbf z} \mu(d{\mathbf z}) < \infty, \quad \mu(\{{\mathbf 0}\}) = 0.\end{equation}

Example 2.1. Let $ \mu(d{\mathbf z})=c\nu(d{\mathbf z})$ , where $ c>0$ and $ \nu(d{\mathbf z})$ is any probability measure on $ \mathbb{C}^p$ with finite second moments satisfying $ \nu(\{\mathbf 0\}) = 0$ . Then (7) is satisfied.

Example 2.2. Let $ \alpha\in(0,2)$ . For some $ c>0$ , define $ \mu(d\mathbf z)= e^{-c\|\mathbf z\|}\big/\|\mathbf z\|^{1+\alpha} d\mathbf z$ . Then (7) is satisfied (the measure $ \mu$ is an instance of a tempered stable distribution; see Rosiński [Reference Rosiński57] or Grabchak [Reference Grabchak27]).

Now, for $ \mu(d{\mathbf z})$ as in (7), let

(8) \begin{equation}\widetilde{N}(dx,d{\mathbf z}) = N(dx,d{\mathbf z})- \mathbb{E} N(dx,d{\mathbf z})= N(dx,d{\mathbf z}) - dx \mu(d{\mathbf z}) \in \mathbb{R}\end{equation}

be a compensated Poisson random measure on $ {\mathcal B}(\mathbb{R} \times \mathbb{C}^p)$ (see, for example, Sato [Reference Sato62], Section 19, or Applebaum [Reference Applebaum3], Section 2.3). We define the space of integration kernels

\begin{align*}{\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}& \equiv {\mathcal L}^2_{dx\otimes \mu_{\mathbb{C}^p}(d{\mathbf z})}\equiv {\mathcal L}^2\big(\mathbb{R} \times \mathbb{C}^p, {\mathcal B}\big(\mathbb{R} \times \mathbb{C}^p\big),dx\otimes \mu_{\mathbb{C}^p}(d{\mathbf z})\big)\\& = \Big\{\text{measurable }\varphi\,:\,\mathbb{R} \times \mathbb{C}^p \rightarrow \mathbb{C}^p \,:\, \|\varphi\|_{{\mathcal L}^2_{dx\otimes \mu_{\mathbb{C}^p}(d{\mathbf z})}} < \infty\Big\},\end{align*}

where

(9) \begin{equation}\|\varphi\|^2_{{\mathcal L}^2_{dx\otimes \mu_{\mathbb{C}^p}(d{\mathbf z})}} \equiv \|\varphi\|^2_{{\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}} \,:\!=\, \int_{\mathbb{R}}\int_{\mathbb{C}^p} \varphi(x,{\mathbf z})^* \varphi(x,{\mathbf z}) \mu(d{\mathbf z})dx.\end{equation}

Fix the sets $ B_{1,i} \times B_{2,i} \in {\mathcal B}(\mathbb{R}) \otimes {\mathcal B}(\mathbb{C}^p)$ , as well as the vectors $ \varphi_i \in \mathbb{C}^p$ , $ i = 1,\ldots, I$ . Consider the elementary function $ \varphi(x,{\mathbf z}) = \sum^{I}_{i=1} \varphi_{i}1_{B_{1,i} \times B_{2,i}}(x,{\mathbf z})$ . We define the stochastic integral of the elementary function $ \varphi$ with respect to the random measure $ \widetilde{N}$ by means of the expression

\begin{equation*}\int_{\mathbb{R} \times \mathbb{C}^p} \varphi(x,{\mathbf z}) \widetilde{N}(dx,d{\mathbf z}) \,:\!=\, \sum^{I}_{i=1} \varphi_{i} \widetilde{N}\big(B_{1,i},B_{2,i}\big).\end{equation*}

Next, fix $ \varphi \in {\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}$ , and let $ \{\varphi_{n}\}_{n \in \mathbb{N}} \subseteq {\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}$ be elementary functions converging to $ \varphi$ in the norm $ \|{\cdot}\|_{{\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}}$ . Then

(10) \begin{equation}\mathbb{C}^p \ni \int_{\mathbb{R} \times \mathbb{C}^p} \varphi(x,{\mathbf z}) \widetilde{N}(dx,d{\mathbf z}) = L^2(\mathbb{P})\text{--}\lim_{n \rightarrow \infty} \int_{\mathbb{R} \times \mathbb{C}^p}\varphi_{n}(x,{\mathbf z})\widetilde{N}(dx,d{\mathbf z})\end{equation}

is well defined as the stochastic integral of the function $ \varphi$ with respect to the compensated Poisson random measure $ \widetilde{N}$ (see Section A for details). In particular, the limit random vector does not depend on the chosen sequence of elementary functions. Now let

(11) \begin{equation}\widetilde{g} \in L^2\big(\mathbb{R},M(p,\mathbb{C})\big)= \bigg\{ \text{measurable } {\mathfrak g}\,:\,\mathbb{R} \rightarrow M(p,\mathbb{C}): \int_{\mathbb{R}} \text{tr}\big( {\mathfrak g}(x) {\mathfrak g}(x)^* \big) dx < \infty \bigg\}.\end{equation}

We also define the random measure $ \widetilde{{\mathcal M}}(dx)$ on $ {\mathcal B}(\mathbb{R})$ by means of the relation

(12) \begin{equation}\int_{\mathbb{R}} \widetilde{g}(x) \widetilde{{\mathcal M}}(dx) \,:\!=\, \int_{\mathbb{R} \times \mathbb{C}^p} \big\{\widetilde{g}(x){\mathbf z} + \widetilde{g}({-}x)\overline{{\mathbf z}}\big\}\widetilde{N}(dx,d{\mathbf z}) \in \mathbb{C}^p.\end{equation}

In particular, for

(13) \begin{equation}\widetilde{g} \in L^2_{\text{Herm}}(\mathbb{R}) = \Big\{{\mathfrak g} \in L^2\big(\mathbb{R},M(p,\mathbb{C})\big)\,:\, {\mathfrak g}({-}x) = \overline{{\mathfrak g}(x)}\Big\},\end{equation}

the expression (12) reduces to

(14) \begin{equation}\int_{\mathbb{R}} \widetilde{g}(x) \widetilde{{\mathcal M}}(dx) = \int_{\mathbb{R} \times \mathbb{C}^p} 2 \Re\big(\widetilde{g}(x){\mathbf z}\big) \widetilde{N}(dx,d{\mathbf z}) = 2 \Re\bigg( \int_{\mathbb{R} \times \mathbb{C}^p} \widetilde{g}(x) {\mathbf z} \widetilde{N}(dx,d{\mathbf z}) \bigg)\! \in \mathbb{R}^p.\end{equation}

So, let $ \big\{\widetilde{g}_t\big\}_{t \in \mathbb{R}} \subseteq L^2_{\text{Herm}}(\mathbb{R})$ . We can define the stochastic process $ \widetilde{X} =\big\{\widetilde{X}(t)\big\}_{t \in \mathbb{R}}$ by means of the stochastic integral

(15) \begin{equation}\widetilde{X}(t) = \int_{\mathbb{R}}\widetilde{g}_{t}(x)\widetilde{{\mathcal M}}(dx), \quad t \in \mathbb{R}.\end{equation}

Equivalently, based on the relation (14), we can re-express $ \widetilde{X}$ as

(16) \begin{equation}\widetilde{X}(t) = \int_{\mathbb{R} \times \mathbb{C}^p} 2 \Re\big(\widetilde{g}_{t}(x){\mathbf z}\big)\widetilde{N}(dx,d{\mathbf z})= 2 \Re\bigg(\int_{\mathbb{R} \times \mathbb{C}^p} \widetilde{g}_{t}(x){\mathbf z}\widetilde{N}(dx,d{\mathbf z}) \bigg).\end{equation}

To construct the analogous time-domain framework, we start with the following definition. As in (8), we consider the compensated Poisson random measure

(17) \begin{equation}\widetilde{N}(ds,d{\mathbf z}) = N(ds,d{\mathbf z})- \mathbb{E} N(ds,d{\mathbf z})= N(ds,d{\mathbf z}) - ds\mu(d{\mathbf z}) \in \mathbb{R}\end{equation}

on $ {\mathcal B}\big(\mathbb{R}^{p+1}\big)$ , where $ \mu(d{\mathbf z})$ is a Lévy measure on $ {\mathcal B}\big(\mathbb{R}^p\big)$ satisfying

(18) \begin{equation}\int_{\mathbb{R}^p} {\mathbf z}^* {\mathbf z}\, \mu(d{\mathbf z}) < \infty, \quad \mu(\{{\mathbf 0}\}) = 0.\end{equation}

We naturally define the space of integration kernels $ {\mathcal L}^2_{ds\otimes \mu(d{\mathbf z})}$ as in (9), where $ \mathbb{R}^{p+1}$ replaces $ \mathbb{R} \times \mathbb{C}^p$ , $ \mu(d{\mathbf z})$ is a Lévy measure on $ {\mathcal B}(\mathbb{R}^{p})$ , and

\begin{equation*}\|\varphi\|^2_{{\mathcal L}^2_{ds\otimes \mu(d{\mathbf z})}}\end{equation*}

is defined as in (9) with $ \mathbb{R}^{p+1}$ replacing $ \mathbb{R} \times \mathbb{C}^p$ . Let $ \{\varphi_{n}\}_{n \in \mathbb{N}} \subseteq {\mathcal L}^2_{ds\otimes \mu(d{\mathbf z})}$ be elementary functions converging to $ \varphi$ in the norm $ \|{\cdot}\|_{{\mathcal L}^2_{ds\otimes \mu(d{\mathbf z})}}$ . The stochastic integral

(19) \begin{equation}\int_{\mathbb{R}^{p+1}} \varphi(s,{\mathbf z}) \widetilde{N}(ds,d{\mathbf z}) \in \mathbb{R}^p, \quad \varphi \in {\mathcal L}^2_{ds \otimes \mu(d{\mathbf z})},\end{equation}

is then naturally defined as in (10).

We further define the random measure $ {\mathcal M}(ds)$ by means of the relation

(20) \begin{equation}\int_{\mathbb{R}} g(s) {\mathcal M}(ds) \,:\!=\, \int_{\mathbb{R}\times \mathbb{R}^p} g(s){\mathbf z} \widetilde{N}(ds,d{\mathbf z}) \in \mathbb{R}^p,\end{equation}

where

(21) \begin{equation}g \in L^2\big(\mathbb{R},M(p,\mathbb{R})\big)= \bigg\{ \text{measurable } {\mathfrak g}\,:\,\mathbb{R} \rightarrow M(p,\mathbb{R}): \int_{\mathbb{R}} \text{tr}\big( {\mathfrak g}(s) {\mathfrak g}(s)^* \big) ds < \infty \bigg\}\end{equation}

$ \big($ in particular, $ g(s){\mathbf z} \in {\mathcal L}^2_{ds \otimes \mu(d{\mathbf z})}\big)$ . Note that the space of integrands for the random measure $ {\mathcal M}(ds)$ (i.e., (21)) is different from that for $ \widetilde{{\mathcal M}}(d x)$ (i.e., (11)).

Analogously to the expression (15), for $ \{g_t(s)\}_{t \in \mathbb{R}} \subseteq L^2\big(\mathbb{R},M(p,\mathbb{R})\big)$ , we can define the stochastic process $ X = \{X(t)\}_{t \in \mathbb{R}}$ by means of the stochastic integral

\begin{equation*}X(t) = \int_{\mathbb{R}} g_t(s) {\mathcal M}(ds).\end{equation*}

Equivalently, based on the relation (20), we can re-express X as

(22) \begin{equation}\{X(t)\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \bigg\{\!\int_{\mathbb{R}\times \mathbb{R}^p} g_t(s) {\mathbf z} \widetilde{N}(ds,d{\mathbf z})\bigg\}_{t \in \mathbb{R}}.\end{equation}

2.2. On integral representations of operator fractional Brownian motion

Recall that an ofBm is a Gaussian, o.s.s., stationary-increment stochastic process. Harmonizable representations are the natural starting point for the study of ofBm. This is so because, as briefly recalled in the introduction, almost every instance of ofBm admits the representation (3), where $ \widetilde{{\mathcal M}}(dx)=\widetilde B(dx)$ is a p-variate complex Gaussian random measure satisfying $ \widetilde B({-}dx)= \overline{\widetilde B(dx)}$ almost surely (a.s.) and $ \mathbb{E}\widetilde B(dx)\widetilde B(dx)^* = dx \times I$ (see Example A.1). So, for notational simplicity, define

(23) \begin{equation}D = H - (1/2) I.\end{equation}

Let

(24) \begin{equation}L^2_{\text{Herm}}(\mathbb{R}) \ni \widetilde{g}_t(x) = \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x} \Big\{x^{-D}_{+}A + x^{-D}_{-}\overline{A} \Big\}, \quad x \neq 0,\end{equation}

be the integrand of the harmonizable representation of ofBm. Defining Fourier transforms entrywise, for

(25) \begin{equation}g_t(s) \,:\!=\, {\mathcal F}^{-1}(\widetilde{g}_t)(s) \in L^2\big(\mathbb{R},M(p,\mathbb{R}) \big),\end{equation}

ofBm also admits a moving average representation of the form (4), where the Gaussian random measure $ {\mathcal M}(ds) = B(ds)$ satisfies $ \mathbb{E} B(ds)B(ds)^* = ds \times I$ . For most cases of interest, we can explicitly recast the moving average representation of ofBm. In fact, we can set

(26) \begin{equation}g_t(s) = \left\{\begin{array}{c@{\quad}c}\Big\{(t-s)^{D}_+ - ({-}s)^{D}_+ \Big\}M_+ + \Big\{(t-s)^{D}_- - ({-}s)^{D}_- \Big\}M_-, & \text{if }\Re\text{eig}(H) \subseteq (0,1)\setminus\{1/2\};\\ & \\\{\text{sign}(t-s) - \text{sign}({-}s)\big\}M + \log \Big( \frac{|t-s|}{|s|}\Big)N, & \text{if }H = (1/2)I,\end{array}\right.\end{equation}

for some matrix constants $ M_+, M_- \in M(p,\mathbb{R})$ or $ M, N \in M(p,\mathbb{R})$ (Didier and Pipiras [Reference Didier and Pipiras23], Theorem 3.2). For the instances

(27) \begin{equation}\Re\text{eig}(H) \subseteq (0,1)\setminus\{1/2\},\end{equation}

the expression (26) can be extracted based on the fact that

\begin{equation*}{\mathcal F}\Big( (t-s)^{D}_{\pm}-({-}s)^{D}_{\pm} \Big)(x) \,:\!=\, \int_{\mathbb{R}}e^{{\mathbf i} s x}\Big\{ (t-s)^{D}_{\pm}-({-}s)^{D}_{\pm} \Big\} ds\end{equation*}

(28) \begin{equation}= \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x} |x|^{-D}\Gamma\big(D+I\big) e^{\mp \text{sign}(x) {\mathbf i} \pi D/2}\end{equation}

(see Proposition 3.1 and Theorem 3.2 in Didier and Pipiras [Reference Didier and Pipiras23], in particular the expressions (3.20), (3.24), and (3.25)). In (28), $ \Gamma\big(D+I\big)$ is interpreted as a primary matrix function (Horn and Johnson [Reference Horn and Johnson28], Sections 6.1 and 6.2). Further note that, when

\begin{equation*}\text{eig}(H) \cap \big\{ z \in \mathbb{C}: \Re z = 1/2 \big\} \neq \emptyset \quad \text{and} \quad H \neq (1/2)I,\end{equation*}

moving average representations can be quite intricate (see Example 3.1 in Didier and Pipiras [Reference Didier and Pipiras23]).

3. Operator fractional Lévy motion

We are now in a position to define the ofLm class. For the sake of simplicity, hereinafter we focus on purely non-Gaussian constructs. We first define ofLm in the Fourier and time domains, and then establish its fundamental properties.

Definition 3.1. Let $ H \in M(p,\mathbb{R})$ be a (Hurst) matrix whose eigenvalues satisfy

(29) \begin{equation}\Re \text{eig}(H) \subseteq (0,1).\end{equation}

Let $ \widetilde{{\mathcal M}}(ds)$ be a $ \mathbb{C}^p$ -valued random measure as in (14) whose Lévy measure $ \mu(d{\mathbf z})\equiv \mu_{\mathbb{C}^p}(d{\mathbf z})$ satisfies the condition (7). A real harmonizable operator fractional Lévy motion (rhofLm) without Gaussian component $ \widetilde{X}_H = \big\{\widetilde{X}_H(t)\big\}_{t \in \mathbb{R}}$ is a stochastic process such that

1. (i)

   (30) \begin{equation}\text{the distribution of $\widetilde{X}_H(t)$ is proper}, \quad t \neq 0;\end{equation}

2. (ii) it satisfies the relation

   (31) \begin{equation}\big\{\widetilde{X}_H(t)\big\}_{t \in \mathbb{R}} = \bigg\{ \int_{\mathbb{R}}\widetilde{g}_t(x) \widetilde{{\mathcal M}}(dx) \bigg\}_{t \in \mathbb{R}}.\end{equation}

In (31), the integrand is given by $ \widetilde{g}_t(x)$ as in (24) for some matrix constant $ A \in M(p,\mathbb{C})$ .

We can now turn to the time domain.

Example 3.1. From (33) and (26), when $ \Re \text{eig}(H)\subseteq (0,1)\setminus\{1/2\}$ , and $ M_+ = M_- =:\, M$ , maofLm admits the well-balanced representation

(34) \begin{equation}\big\{X_H(t)\big\}_{t \in \mathbb{R}} = \bigg\{\!\int_{\mathbb{R}} \big\{|t-s|^{D} - |-s|^{D}\big\} M {\mathcal M}(ds)\bigg\}_{t \in \mathbb{R}}\end{equation}

(cf. Benassi et al. [Reference Benassi, Cohen and Istas9], Definition 2.1).

In the following proposition, we establish fundamental properties of both rhofLm and maofLm. Statement (vii) pertains to sample path properties, whereas all remaining statements pertain to existence, continuity, and distributional properties.

Example 3.2. A simple example of a Lévy measure satisfying (40) is given by

\begin{equation*}\mu_{\mathbb{C}^p}(d{\mathbf z}) = \sum^{p}_{k=1}\delta_{(1+{\mathbf i})e_k}(d{\mathbf z}),\end{equation*}

where $ e_k \in \mathbb{R}^p$ , $ k = 1,\ldots,p$ , are the first p canonical vectors.

Recall that, in the Gaussian case (ofBm), harmonizable and moving average stochastic integrals are representations of the same stochastic process (see Section 2.2). Equivalently, they have the same covariance structure. As established in Theorem 3.1, under assumptions on the Lévy measure, rhofLm and maofLm share the covariance structure of ofBm. Nevertheless, they are rather distinct from ofBm. We shed light on such differences in the next three propositions. In Proposition 3.1, we provide natural alternative stochastic integral representations of rhofLm and maofLm in the time and Fourier domains, respectively. The representations (i.e., (42) and (43)) are formally similar to (31) and (33), respectively. However, the random measures involved in each expression do not satisfy the conditions stated in Definitions 3.1 and 3.2. In particular, the random measures generally display orthogonal but dependent increments. On the other hand, even though ofLm is never o.s.s., in Proposition 3.2 we establish that (rescaled) maofLm and rhofLm converge to an ofBm over different time ranges, i.e., in the large and small scaling limits, respectively. Remarkably, in Proposition 3.3 we further show that maofLm and rhofLm may display operator self-similarity in the other limit directions; namely, maofLm can be o.s.s. in the small-scale limit, whereas rhofLm can be o.s.s. in the large-scale limit. However, such limits may display heavy-tailed marginal distributions; i.e., they are not ofBms.

We begin by establishing natural alternative stochastic integral representations of rhofLm and maofLm. These representations are based on random measures with uncorrelated increments; for the reader’s convenience they are given explicitly in Proposition E.1.

Example 3.3. Suppose $ M_- = 0$ and

(44) \begin{equation}\Re \text{eig}(H) \cap \{1/2\} = \emptyset.\end{equation}

Let $ A = \Gamma(D+I)e^{-i\pi D/2}$ . Then, by the expression (28), we can recast the representation (42) in the form

\begin{equation*}\big\{X_H(t)\big\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \bigg\{\!\int_\mathbb{R} \frac{e^{{\mathbf i} tx}-1}{{\mathbf i} x} \Big\{ x^{-D}_+ A + x^{-D}_- \overline A \Big\}\Phi_{\mathcal M}(dx) \bigg\}_{t \in \mathbb{R}}.\end{equation*}

On the other hand, for this same choice of the parameter A and still assuming the condition (44) holds, again by the expression (28) we can rewrite (43) as

\begin{equation*}\Big\{\widetilde{X}_H(t)\Big\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \bigg\{\!\int_\mathbb{R} \Big\{(t-s)_+^D - ({-}s)_+^D \Big\} \Phi_{\widetilde {\mathcal M}}(ds)\bigg\}_{t\in\mathbb{R}}.\end{equation*}

In Proposition 3.2, we establish the large- and small-scale behaviors of maofLm and rhofLm, respectively. In the statement of the proposition, $ \stackrel{\text{f.d.d.}}{\to}$ denotes the convergence of finite-dimensional distributions.

In Proposition 3.3, we show that some maofLm and rhofLm instances are o.s.s. in the small- and large-scale limits, respectively—in both cases, with a different matrix scaling exponent. This occurs when the associated random measures $ \mathcal M$ and $ \widetilde{ \mathcal M}$ are chosen to be ‘locally’ operator-stable, in the sense that their Lévy measures around $ {\mathbf 0}$ behave like that of an operator-stable Lévy process. These limiting processes, in turn, are instances of operator-stable o.s.s. processes recently studied in Kremer and Scheffler [Reference Kremer and Scheffler36]. For the reader’s convenience, the precise definition and more details about such measures and associated independently scattered random measures are provided in Section F.

Recall that for any $ M\in M(p,\mathbb{R})$ , $ \lambda_i(M)$ denotes the ith eigenvalue of M in the ordering $ \Re \lambda_1(M)\leq \ldots \leq \Re \lambda_p(M)$ , where an arbitrary ordering of eigenvalues is adopted in case real parts are equal.

4. Time-reversibility

Recall that a stochastic process $ X = \{X(t)\}_{t \in \mathbb{R}}$ is said to be time-reversible if $ \{X({-}t)\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \{X(t)\}_{t \in \mathbb{R}}$ . In this section, we provide characterizations of time-reversibility for maofLm and rhofLm under mild assumptions. In the characterizations, the true difficulty lies in establishing necessary conditions, i.e., in what the assumption of time-reversibility implies about the parametric representations of maofLm and rhofLm. The proofs require results on the uniqueness of multivariate stochastic integral representations, which are developed in Section D. To provide these uniqueness results, we adapt the fundamental framework constructed in Sections 2.1 and 2.2 of Kabluchko and Stoev [Reference Kabluchko and Stoev32] (see also Maruyama [Reference Maruyama47], Chapter 3 of Samorodnitsky [Reference Samorodnitsky60], and Rosiński [Reference Rosiński58]).

To investigate time-reversibility in the framework of ofLm, it is convenient to slightly generalize the notation. Simply put, the new argument $ \omega$ stands for either the Fourier or the time argument x or s. In turn, the vector $ {\boldsymbol \varpi} = (\omega,{\mathbf z})$ includes both $ \omega \in \mathbb{R}$ and the Lévy measure argument $ {\mathbf z} \in \mathbb{R}^q$ , where either $ q = p$ or $ q = 2p$ . So, more precisely, let $ \overline{\Omega} = \mathbb{R} \times \mathbb{R}^q$ , $ {\mathcal B}= {\mathcal B}(\overline{\Omega})$ (cf. the expression (136)). Let

(51) \begin{equation}\kappa(d {\boldsymbol \varpi}) = d\omega\otimes \mu(d{\mathbf z}),\end{equation}

where $ \mu(d{\mathbf z})$ is a Lévy measure satisfying (18). Whenever convenient, we write $ \eta(d\omega) \equiv d \omega$ . Also define

(52) \begin{align}{\mathcal L}^2_{\kappa(d {\boldsymbol \varpi})}\big(\overline{\Omega}\big) & = {\mathcal L}^2_{d\omega\otimes \mu(d{\mathbf z})}\big(\mathbb{R} \times \mathbb{R}^q\big)\nonumber\\& = \bigg\{\varphi\,:\,\mathbb{R} \times \mathbb{R}^q \rightarrow \mathbb{R}^p\,:\, \int_{\mathbb{R}}\int_{\mathbb{R}^q} \varphi(\omega,{\mathbf z})^*\varphi(\omega,{\mathbf z}) \mu(d{\mathbf z})d\omega< \infty\bigg\}\end{align}

(cf. the expression (9)). We then express the compensated Poisson random measure on $ {\mathcal B}(\overline{\Omega})$ as

(53) \begin{equation}\mathbb{R} \ni \widetilde{N}(d{\boldsymbol \varpi}) \equiv N(d{\boldsymbol \varpi}) - \kappa(d{\boldsymbol \varpi}) \equiv \widetilde{N}(d\omega,d {\mathbf z}),\end{equation}

where $ N(d{\boldsymbol \varpi}) \equiv N(d\omega,d {\mathbf z})$ is a Poisson random measure (cf. (8) and (17)). Let $ f_t({\boldsymbol \varpi})$ and $ \mathfrak{g}_t(\omega)$ , $ t \in \mathbb{R}$ , be two families of $ \mathbb{R}^p$ - and $ M(p,q,\mathbb{R})$ -valued functions, respectively, where

(54) \begin{equation}\{f_t({\boldsymbol \varpi})\}_{t \in \mathbb{R}} \,:\!=\, \{\mathfrak{g}_t(\omega){\mathbf z} \}_{t \in \mathbb{R}} \subseteq {\mathcal L}^2_{\kappa(d{\boldsymbol \varpi})}(\overline{\Omega}).\end{equation}

Example 4.2. For (38), we can write (54) with

\begin{equation*}\omega = s, \quad {\mathfrak g}_t(s) = g_t(s) \in M(p,\mathbb{R}), \quad q = p, \quad \text{and} \quad \mu(d{\mathbf z})\text{ as in (38)}.\end{equation*}

For (36), we can re-express (54) with

\begin{equation*}\omega = x, \quad {\mathfrak g}_t(x) = \big(\Re \widetilde{g}_t(x),\Im \widetilde{g}_t(x)\big) \in M(p,2p,\mathbb{R}), \quad q = 2p, \quad \text{and} \quad \mu(d{\mathbf z}) = \big(\mu_{\mathbb{R}^{2p}} \circ \varsigma^{-1}\big)(d{\mathbf z}),\end{equation*}

where $ \varsigma({\mathbf z}) = (2 \Re{\mathbf z},-2 \Im{\mathbf z})$ .

The main results in this section require some notion of minimal (stochastic integral) representation. In the following definition, we revisit the notion of minimality as put forward in Kabluchko and Stoev [Reference Kabluchko and Stoev32].

Definition 4.1. Let $ T\subseteq \mathbb{R}$ , and consider the $ \mathbb{R}^p$ -valued stochastic process $ X = \{X(t)\}_{t \in T}$ given by the stochastic integral representation

(55) \begin{equation}X(t) = \int_{\overline{\Omega}}f_t({\boldsymbol \varpi}) \widetilde{N}(d{\boldsymbol \varpi}) = \int_{\mathbb{R} \times \mathbb{R}^q} {\mathfrak g}_t(\omega) {\mathbf z} \widetilde{N}(d\omega,d{\mathbf z}), \quad t \in T.\end{equation}

We say $ \{f_t\}_{t \in T}$ is a minimal representation of the ID stochastic process X with respect to $ {\mathcal B} \mod \kappa$ if the following two conditions hold:

1. (i) $ \sigma(\{f_t\}_{t \in T}) = {\mathcal B} \mod \kappa$ , i.e., for every $ B \in {\mathcal B}$ , there exists $ A \in \sigma(\{f_t\}_{t \in T})$ such that $ \kappa(A \Delta B) = 0$ ; and

2. (ii) there is no $ B \in {\mathcal B}$ such that $ \kappa(B) > 0$ and, for every $ t \in \mathbb{R}$ , $ f_t \equiv 0$ a.e. on B.

In the following theorem, we characterize time-reversibility for maofLm. For comments on the minimality assumption, see Remark 4.1.

Theorem 4.1. Let H be a (Hurst) matrix whose eigenvalues satisfy (27). Let $ X_H = \{X_H(t)\}_{t \in \mathbb{R}}$ be a maofLm with Hurst matrix H. Further assume that

(56) \begin{equation}M_+,M_- \in GL(p,\mathbb{R})\end{equation}

and $ \{f_t({\boldsymbol \varpi}),t \in \mathbb{R}\}= \{g_t(s){\mathbf z},t \in \mathbb{R}\}$ is a minimal representation of $ X_H$ with respect to $ {\mathcal B}\big(\mathbb{R}^{p+1}\big) \mod \kappa$ , where $ \kappa(d{\boldsymbol \varpi}) = ds \otimes \mu(d{\mathbf z})$ and $ \mu(d{\mathbf z})$ is as in (38). Then the following conditions are equivalent:

1. (i) $ X_H$ is time-reversible.

2. (ii) The following two conditions hold:

3. (a) $ (M^{-1}_{-}M_{+})|_{\text{supp}(\mu)}$ is an involution; i.e.,

   (57) \begin{equation}M^{-1}_- M_+ {\mathbf z} = M^{-1}_+ M_- {\mathbf z} \quad \mu(d{\mathbf z})\text{-a.e.}\end{equation}

4. (b) The map $ \mathbf z \mapsto M^{-1}_+ M_-\mathbf z$ preserves the measure $ \mu$ ; i.e.,

   (58) \begin{equation}\mu\big((M^{-1}_- M_+) d{\mathbf z} \big) = \mu(d{\mathbf z}).\end{equation}

In (ii), the condition (a) can be replaced by the following:

1. \begin{align*}&(a^{\prime})\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\& \qquad \qquad\qquad\quad g_{-t}(s)\mathbf z=g_t({-}s)M_-^{-1}M_+ \mathbf z=g_t({-}s)M_+^{-1}M_- \mathbf z \quad \kappa(ds,d\mathbf z)\text{-a.e.}\end{align*}

Example 4.3. It is illustrative to compare the conditions for time-reversibility for ofBm and maofLm. In the case of the former, time-reversibility (i.e., the expression (6)) is equivalent, in the time domain, to the parametric condition

(59) \begin{align}& \cos\bigg(\frac{\pi D}{2}\bigg)\big(M_+ + M_-\big)\big(M^*_+ - M^*_-\big)\sin\bigg(\frac{\pi D^*}{2}\bigg)\nonumber\\& \quad = \sin\bigg(\frac{\pi D}{2}\bigg)\big(M_+ - M_-\big)\big(M^*_+ + M^*_-\big)\cos\bigg(\frac{\pi D^*}{2}\bigg)\end{align}

(Didier and Pipiras [Reference Didier and Pipiras23], Corollary 5.1), where $ D = H-(1/2)I$ and the cosine and sine of matrices are interpreted in the sense of primary matrix functions (Horn and Johnson [Reference Horn and Johnson28]).

So, suppose the conditions used in Theorem 4.1 hold; namely, suppose that (27) and (56) are satisfied for a minimal representation, and also that $ X_H$ is time-reversible. In addition, assume the Lévy measure $ \mu$ satisfies the second moment condition (39)—otherwise, ofBm and maofLm have incompatible parametrizations (cf. Theorem 3(iv)). Based on a change of variable $ {\mathbf w} = M^{-1}_+ M_- {\mathbf z}$ , using (58), we have

\begin{equation*}\big(M^{-1}_+ M_-\big) \bigg(\int_{\mathbb{R}^p}{\mathbf z}{\mathbf z}^* \mu(d{\mathbf z})\bigg)\Big(M^*_- \big(M^*_+\big)^{-1}\Big) =\int_{\mathbb{R}^p}{\mathbf w}{\mathbf w}^* \mu(d{\mathbf w}) = I.\end{equation*}

Hence, $ M_+ M^*_+ = M_- M^*_-$ . Also, under (39), (57) implies $ M^{-1}_- M_+= M^{-1}_+ M_- $ . As a consequence,

\begin{equation*}\big(M_+ + M_-\big)\big(M^*_+ - M^*_-\big) = \mathbf 0= -\big(M_- + M_+\big)\big(M^*_+ - M^*_-\big),\end{equation*}

which in turn implies the condition (59). Conversely, we may pick $ M_+,M_-$ satisfying (59) but for which $ \big(M_+ + M_-\big)\big(M^*_+ - M^*_-\big) \neq {\mathbf 0}$ . In other words, among the instances of maofLm satisfying (18), the conditions for time-reversibility of maofLm as established in Theorem 4.1 are more stringent than those for the time-reversibility of ofBm.

Example 4.4. Let $ X_H$ be a time-reversible maofLm. Assume $ \Sigma \,:\!=\, \int_{\mathbb{R}^p} \mathbf z \mathbf z^* \mu(d\mathbf z)$ has full rank. A simple calculation shows that there exists a symmetric orthogonal matrix O such that

(60) \begin{equation}M_+^{-1} M_-=M_-^{-1} M_+ = \Sigma^{1/2} O\Sigma^{-1/2}\end{equation}

(Lemma E.2). In other words, (60) is a necessary condition for time-reversibility. In light of Theorem 4.1, this implies the following:

1. (i) If $ p=1$ , $ X_H$ is time-reversible if and only if either (a) $ M_+=M_-$ ; or (b) $ M_+=-M_-$ and $ \mu$ is symmetric (i.e., $ \mu({-}dz) = \mu(dz)$ ). (Indeed, the relation (60) implies necessity, whereas sufficiency follows from both conditions (58) and (57)).

2. (ii) If $ \mu$ is a full, zero-mean Gaussian measure on $ \mathbb{R}^p$ , then it is characterized by the matrix $ \Sigma$ . Therefore, $ X_H$ is time-reversible if and only if (60) holds.

Turning to the Fourier domain, let

(61) \begin{equation}\widetilde{X}_H = \big\{\widetilde{X}_H(t) \big\}_{t \in \mathbb{R}}\end{equation}

be a rhofLm with kernel $ f_t({\boldsymbol \varpi}) = \widetilde{g}_{t}(x){\mathbf z}$ and measure (51) given by $ \kappa(d{\boldsymbol \varpi}) = dx \otimes \mu(d{\mathbf z}) \equiv \mu(d{\mathbf z}) \otimes dx$ . Note that $ \widetilde{X}_H$ can also be represented based on the measure

(62) \begin{equation}\widetilde \kappa(d\boldsymbol \varpi) = dx \otimes \widetilde \mu (d\mathbf z),\end{equation}

where

(63) \begin{equation}\widetilde \mu(d\mathbf z) = \frac{\mu(d\mathbf z) +\mu(\overline{d\mathbf z}) }{2}\end{equation}

(see Lemma E.3). This fact is used in the following theorem, where we characterize time-reversibility for rhofLm. For comments on the minimality assumption, see Remark 4.1.

Theorem 4.2. Let H be a (Hurst) matrix satisfying (29). Let $ \widetilde{X}_H = \big\{\widetilde{X}_H(t)\big\}_{t \in \mathbb{R}}$ be a rhofLm with Hurst matrix H. Assume

\begin{equation*}\{f_t({\boldsymbol \varpi})\}_{t \in \mathbb{R}}= \big\{\widetilde{g}_t(x){\mathbf z}\big\}_{t \in \mathbb{R}}\text{ is a minimal representation}\end{equation*}

(64) \begin{equation}\text{of $\big\{\widetilde{X}_H(t)\big\}_{t \in \mathbb{R}}$ with respect to }{\mathcal B}\big(\mathbb{R}\times \mathbb{R}^{2p}\big) \mod {\widetilde \kappa},\end{equation}

where $ {\widetilde \kappa}$ is a measure of the form (62) and we identify $ \mu_{\mathbb{C}^p}\equiv\mu_{\mathbb{R}^{2p}}$ (see (37)). Further assume

(65) \begin{equation} A,\overline A \in GL(p,\mathbb{C}).\end{equation}

Then $ \widetilde X_H$ is time-reversible if and only if the map $ \mathbf z\mapsto - A^{-1} \overline A \mathbf z$ preserves the measure $ \widetilde \mu$ as in (62), i.e.,

(66) \begin{equation}\widetilde{\mu}\Big( -\overline A^{-1} A d \mathbf z\Big)= \widetilde{\mu}(d\mathbf z).\end{equation}

Example 4.6. As in Example 4.3, we now compare the conditions for time-reversibility of ofBm to those for rhofLm. For the former, time-reversibility (i.e., the expression (6)) is equivalent, in the Fourier domain, to the parametric condition

(67) \begin{equation}AA^* = \overline{AA^*}\end{equation}

(Didier and Pipiras [Reference Didier and Pipiras23], Theorem 5.1).

So, suppose the conditions used in Theorem 4.2 hold; namely, suppose the conditions (29), (65), and (64) are satisfied. In addition, assume the Lévy measure $ \mu$ satisfies the second moment condition (40), so as to ensure that ofBm and rhofLm have the same covariance structure and compatible parametrizations (cf. Theorem 3.1(iv)). Then, for $ {\mathbf z} = {\mathbf z}_1 + {\mathbf i} {\mathbf z}_2$ ,

(68) \begin{equation}\int_{\mathbb{C}^p} {\mathbf z}{\mathbf z}^* \widetilde \mu(d{\mathbf z}) = (1/2)I + {\mathbf i} \int_{\mathbb{C}^p} \big({\mathbf z}_2{\mathbf z}^*_1 - {\mathbf z}_1{\mathbf z}^*_2\big) \widetilde \mu(d{\mathbf z}) = (1/2)I,\end{equation}

where the last equality is a consequence of the general property $ \widetilde \mu(d\mathbf z) = \widetilde\mu(\overline{d\mathbf z})$ . Thus, assuming time-reversibility, based on a change of variable $ \mathbf w= -\overline A^{-1} A \mathbf z$ , the condition (66) implies that

(69) \begin{equation}\Big( \overline A^{-1} A \Big) \int_{\mathbb{C}^p} {\mathbf z}{\mathbf z}^* \widetilde{\mu}(d{\mathbf z})\Big( \overline A^{-1} A \Big)^* =\int_{\mathbb{C}^p}{\mathbf w}{\mathbf w}^* \widetilde{\mu}(d{\mathbf w}) = (1/2)I.\end{equation}

Hence, $ \Big( \overline A^{-1} A \Big) \Big( \overline A^{-1} A \Big)^*=I$ , which implies the condition (67). In regard to the converse, however, by choosing $ A,A^*$ satisfying (67), we may easily find a Lévy measure $ \mu$ under the condition (40) such that (66) is not satisfied. In other words, among the instances of rhofLm satisfying (40), the conditions for time-reversibility of rhofLm as established in Theorem 4.2 are stronger than those for time-reversibility of ofBm.

Example 4.7. Write $ \Sigma= \int_{\mathbb{C}^p} {\mathbf z}{\mathbf z}^* \widetilde \mu(d{\mathbf z})$ , and suppose $ \Sigma$ has full rank. By reasoning similar to that of Example 4.4, a necessary condition for the time-reversibility of $ \widetilde X_H$ is that

(70) \begin{equation}A^{-1}\overline A = \Sigma^{1/2}U\Sigma^{-1/2},\end{equation}

where $ U\in M(p,\mathbb{C})$ is some unitary matrix. This implies the following:

1. (i) If $ p=1$ , $ \widetilde X_H$ is time-reversible if and only if $ \widetilde \mu({-}dz)=\widetilde \mu\big(e^{{\mathbf i} 2\theta} dz\big)$ , where $ \theta = \text{arg} A$ .

2. (ii) If $ \mu$ is a zero-mean Gaussian measure on $ \mathbb{C}^p$ satisfying $ \mu(d\mathbf z) = \mu (e^{{\mathbf i} \theta} d\mathbf z)$ , $ \theta\in({-}\pi,\pi]$ , then $ \widetilde X_H$ is time-reversible if and only if (70) holds. (Indeed, in this case $ \widetilde \mu = \mu$ , and $ \mu$ is completely determined by $ \Sigma$ , showing that (70) holds if and only if (66) holds by taking second moments.)

Appendix A. Properties of stochastic integrals

Before establishing the results in Section 3, we lay out a few basic facts about the general stochastic integrals defined in Section 2.1. We start off with the Fourier domain. By construction, for $ \varphi_1, \varphi_2 \in {\mathcal L}^2_{ds\otimes \mu(d{\mathbf z})}$ , the $ \mathbb{C}^p$ -valued stochastic integrals of the form (10) satisfies the isometry-type property

\begin{align*} & \mathbb{E} \bigg(\int_{\mathbb{R} \times \mathbb{C}^p} \varphi_1(x,{\mathbf z} ) \widetilde{N}(dx,d{\mathbf z} ) \bigg) \bigg(\int_{\mathbb{R} \times \mathbb{C}^p} \varphi_2\big(x^{\prime},{\mathbf z}^{\prime} \big) \widetilde{N}\big(dx^{\prime},d{\mathbf z}^{\prime} \big)\bigg)^*\\& = \int_{\mathbb{R}} \int_{ \mathbb{C}^p} \varphi_1(x,{\mathbf z} )\varphi_2(x,{\mathbf z} )^* \mu(d{\mathbf z} )dx.\end{align*}

In particular, consider the functions

(71) \begin{equation}\varphi_i(x,{\mathbf z} )= 2 \Re\big(\widetilde{g}_i(x){\mathbf z}\big), \quad \widetilde{g}_i \in L^2_{\text{Herm}}(\mathbb{R}), \quad i = 1,2.\end{equation}

Then, since $ \widetilde{g}_i$ , $ i=1,2$ , is Hermitian,

(72) \begin{align}& \mathbb{E} \bigg(\int_{\mathbb{R}} \widetilde{g}_1(x)\widetilde{{\mathcal M}}(dx)\bigg)\bigg(\int_{\mathbb{R}} \widetilde{g}_2(x^{\prime})\widetilde{{\mathcal M}}\big(dx^{\prime}\big)\bigg)^*\nonumber\\& \quad = 4 \int_{\mathbb{R}} \Re \widetilde{g}_1(x) \bigg(\int_{\mathbb{C}^p} \Re {\mathbf z} \Re {\mathbf z}^* \mu(d{\mathbf z}) \bigg)\Re \widetilde{g}_2(x)^* d x + 4 \int_{\mathbb{R}} \Im \widetilde{g}_1(x) \bigg(\int_{\mathbb{C}^p} \Im {\mathbf z} \Im {\mathbf z}^* \mu(d{\mathbf z}) \bigg)\Im \widetilde{g}_2(x)^* d x.\end{align}

Moreover, the joint characteristic function of the real and imaginary parts of the $ \mathbb{C}^p$ -valued stochastic integral $ \int \varphi d\widetilde{N}$ , $ \varphi \in {\mathcal L}^2_{dx\otimes \mu(d{\mathbf z})}$ , is given by

(73) \begin{align}& \mathbb{E} \exp \bigg\{ {\mathbf i} \bigg({\mathbf u}^*_1 \int \Re(\varphi)d\widetilde{N}+ {\mathbf u}^*_2 \int \Im(\varphi)d\widetilde{N}\bigg)\bigg\}\nonumber\\& \quad = \exp \bigg\{ \int_{\mathbb{R}}\int_{ \mathbb{C}^p} \Big[ e^{{\mathbf i}\big({\mathbf u}^*_1 \Re(\varphi)+{\mathbf u}^*_2 \Im(\varphi)\big)}-1 -{\mathbf i}\big({\mathbf u}^*_1 \Re(\varphi)+{\mathbf u}^*_2 \Im(\varphi)\big)\Big]\mu(d{\mathbf z})dx\bigg\},\end{align}

for $ {\mathbf u}_1,{\mathbf u}_2 \in \mathbb{R}^p$ (Sato [Reference Sato63]). Note that, under the condition (7), the integral on the right-hand side of (73) is finite in view of the inequality

(74) \begin{equation}\big|e^{{\mathbf i} y}-1 - {\mathbf i} y\big| \leq |y|^2, \quad y \in \mathbb{R}.\end{equation}

In particular, for a function $ \varphi(x,{\mathbf z})$ of the form (71),

(75) \begin{equation}\mathbb{E} \exp \bigg\{ {\mathbf i} \bigg({\mathbf u}^* \int_{\mathbb{R} \times \mathbb{C}^p} \varphi(x,{\mathbf z}) \widetilde{N}(d x, d {\mathbf z})\bigg) \bigg\}= \exp \bigg\{ \int_{\mathbb{R} } \widetilde{\psi}(\widetilde{g}(x)^*{\mathbf u}) dx\bigg\}, \quad {\mathbf u} \in \mathbb{R}^p,\end{equation}

where

(76) \begin{equation}\widetilde{\psi}({\mathbf v}) = \int_{\mathbb{C}^p} \Big( e^{{\mathbf i} 2 \Re \langle {\mathbf v},{\mathbf z}\rangle } -1 -{\mathbf i} 2 \Re \langle {\mathbf v},{\mathbf z}\rangle \Big) \mu(d{\mathbf z}) , \quad {\mathbf v} \in \mathbb{C}^p,\end{equation}

and $ \langle {\mathbf v},{\mathbf z}\rangle \,:\!=\, {\mathbf v}^*{\mathbf z}$ . Equivalently, if we regard $ \mu(d\mathbf z)=\mu_{\mathbb{R}^{2p}}(d\mathbf z)$ as a measure on $ {\mathcal B}\big(\mathbb{R}^{2p}\big)$ , identifying each $ \mathbf z_1 + {\mathbf i} \mathbf z_2 \in \mathbb{C}^p$ with $ (\mathbf z_1,\mathbf z_2)\in \mathbb{R}^{2p}$ , then for $ \mathbf z, \mathbf v \in \mathbb{R}^{2p}$ , we may write

(77) \begin{equation}\widetilde{\psi}({\mathbf v}) = \int_{\mathbb{R}^{2p}} \Big( e^{{\mathbf i} 2 \langle \mathbf v, \mathbf z\rangle } -1 -{\mathbf i} 2 \langle \mathbf v, \mathbf z\rangle )\Big) \mu_{\mathbb{R}^{2p}}(d{\mathbf z}), \quad {\mathbf v}\in \mathbb{R}^{2p}.\end{equation}

Given the stochastic integral (16), for every n, any $ (t_1,\ldots,t_n) \in \mathbb{R}^n$ , and any $ {\mathbf u}_1,\ldots,{\mathbf u}_n \in \mathbb{R}^p$ , the characteristic function of the finite-dimensional distributions of the $ \mathbb{R}^p$ -valued stochastic process $ \widetilde{X}$ is given by

(78) \begin{align}& \mathbb{E} \exp \Bigg\{{\mathbf i} \sum^{n}_{k=1}\langle {\mathbf u}_k, \widetilde{X}(t_k) \rangle\Bigg\}\nonumber\\& \quad = \exp\Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \Big( \sum^{n}_{k=1}{\mathbf u}^*_k \widetilde{g}_{t_k}(x){\mathbf z} \Big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k \widetilde{g}_{t_k}(x){\mathbf z} \Bigg)\Bigg]\mu(d{\mathbf z}) dx \Bigg\}.\end{align}

In particular, the random vectors $ \widetilde{X}(t)$ , $ t \in \mathbb{R}$ , are ID (cf. Samorodnitsky [Reference Samorodnitsky60], Theorem 3.3.2(ii)).

Turning to the time domain, let $ \varphi_i(s,{\mathbf z})= g_i(s){\mathbf z}$ , $ i = 1,2$ . By construction, the stochastic integral (20) satisfies the isometry property

(79) \begin{equation}\mathbb{E} \bigg(\int_{\mathbb{R}} g_1(s){\mathcal M}(ds)\bigg)\bigg(\int_{\mathbb{R}} g_2\big(s^{\prime}\big){\mathcal M}\big(ds^{\prime}\big)\bigg)^* = \int_{\mathbb{R}} g_1(s)\bigg(\int_{\mathbb{R}^{p}} {\mathbf z}{\mathbf z}^* \mu(d {\mathbf z}) \bigg)g_2(s)^* ds,\end{equation}

where $ \text{tr}\big(\int_{\mathbb{R}^{p}} {\mathbf z}{\mathbf z}^* \mu(d {\mathbf z})\big) < \infty$ . Moreover, for $ \varphi(s,{\mathbf z}) \in {\mathcal L}^2_{ds \otimes \mu(d{\mathbf z})}$ , the characteristic function of the $ \mathbb{R}^p$ -valued stochastic integral $ \int \varphi(s,{\mathbf z}) \widetilde{N}(ds,d{\mathbf z})$ is given by

(80) \begin{align}& \mathbb{E} e^{{\mathbf i} {\mathbf u}^* \int_{\mathbb{R}^{p+1}} \varphi(s,{\mathbf z}) \widetilde{N}(ds,d{\mathbf z}) }\nonumber\\& = \exp \bigg\{\!\int_{\mathbb{R}} \int_{\mathbb{R}^p} \bigg(e^{{\mathbf i} \langle {\mathbf u}, \varphi(s,{\mathbf z})\rangle }-1-{\mathbf i} \langle {\mathbf u}, \varphi(s,{\mathbf z})\rangle\bigg)\mu(d{\mathbf z})ds \bigg\}, \quad {\mathbf u} \in \mathbb{R}^p.\end{align}

Under the condition (7) (restricted to $ \mathbb{R}^p$ ), the integral on the right-hand side of (80) is convergent in view of the inequality (74).

Given the expression (22), for any n, the joint characteristic function of the stochastic process $ X = \{X(t)\}_{t \in \mathbb{R}}$ at the time points $ t_1,\ldots,t_n$ is given by

(81) \begin{align} \mathbb{E} e^{{\mathbf i} \sum^{n}_{k=1}\langle{\mathbf u}_k, X(t_k)\rangle} & = \exp \Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^p} \Bigg(e^{{\mathbf i} \sum^{n}_{k=1} \langle{\mathbf u}_k, g_{t_k}(s){\mathbf z}\rangle }-1-{\mathbf i} \sum^{n}_{k=1}\langle{\mathbf u}_k, g_{t_k}(s){\mathbf z}\rangle\Bigg)\mu(d{\mathbf z})ds \Bigg\}\nonumber\\& = \exp \Bigg\{\!\int_{\mathbb{R}} \psi\Bigg(\sum^{n}_{k=1} g_{t_k}(s)^*{\mathbf u}_k\Bigg) ds \Bigg\}, \quad {\mathbf u}_1,\ldots,{\mathbf u}_n \in \mathbb{R}^p.\end{align}

In (81), the Lévy symbol $ \psi$ can be expressed as

(82) \begin{equation}\psi({\mathbf u}) = \int_{\mathbb{R}^p} \big(e^{{\mathbf i} \langle{\mathbf u}, {\mathbf z}\rangle}-1-{\mathbf i} \langle{\mathbf u}, {\mathbf z}\rangle \big)\mu(d {\mathbf z}),\quad {\mathbf u} \in \mathbb{R}^p.\end{equation}

In particular, the random vectors X(t), $ t \in \mathbb{R}$ , are ID (cf. Samorodnitsky [Reference Samorodnitsky60], Theorem 3.3.2(ii)).

Example A.1. For the reader’s convenience, we explicitly show how $ \widetilde{\mathcal M}(dx)$ and $ {\mathcal M}(ds)$ can be constructed to encompass both Gaussian and purely non-Gaussian instances of ofLm. For rhofLm, consider a $ \mathbb{C}^{p}$ -valued independently scattered ID random measure on $ \mathbb{R}$ given by

\begin{equation*}{\mathfrak M(dx) \,:\!=\, \mathfrak B(dx) + \int_{\mathbb{C}^{p}} \mathbf z \widetilde N(dx, d\mathbf z)} = \Re(\mathfrak M(dx)) + {\mathbf i} \Im(\mathfrak M(dx))=:\,{\mathcal M}^{(1)}(dx) + {\mathbf i} {\mathcal M}^{(2)}(dx),\end{equation*}

where $ \mathfrak B(dx)\,:\!=\,\mathfrak B^{(1)}(dx)+ {\mathbf i} \mathfrak B^{(2)}(dx)$ is independent of $ \widetilde N( dx, d\mathbf z)$ and $ \mathfrak B^{(\ell)}(dx)$ , $ \ell=1,2$ , are $ \mathbb{R}^p$ -valued independently scattered Gaussian random measures on $ \mathbb{R}$ . We may then define

\begin{equation*}\widetilde {\mathcal M}(dx) = \Big(\widetilde{\mathcal M}^{(1)}(dx) + \widetilde{\mathcal M}^{(1)}({-}dx)\Big) + {\mathbf i} \Big(\widetilde{\mathcal M}^{(2)}(dx) - \widetilde{\mathcal M}^{(2)}({-}dx)\Big)\end{equation*}

(n.b.: $ \widetilde{\mathcal M}(dx)$ is not independently scattered). When $ \widetilde N(ds,d\mathbf z)$ is absent ( $ \mathfrak M(dx)\equiv \mathfrak B(dx)$ ), the resulting measure $ \widetilde{\mathcal M}(dx) =:\widetilde B(dx)$ is Gaussian. Assuming properness, the corresponding process (31) is an ofBm. Without loss of generality, we may further assume $ \mathfrak B^{(1)}(dx),\mathfrak B^{(2)}(dx)$ are independent and take $ \mathbb{E} \widetilde B(dx) \widetilde B(dx)^* = dx\times I$ (note that a slightly different construction of $ \widetilde{ B}(dx)$ —but one that is equivalent for representing ofBm—is given in Didier and Pipiras [Reference Didier and Pipiras23]; see also Samorodnitsky and Taqqu [Reference Samorodnitsky and Taqqu61], Section 7.2.2). When $ \mathfrak M(dx)$ is purely non-Gaussian $ (\mathfrak B(dx)\equiv \mathbf 0)$ , one recovers rhofLm as in Definition 3.1.

To encompass both Gaussian and non-Gaussian instances of maofLm, one can take $ \mathcal M(ds) = B(ds) + \int_{\mathbb{R}^p} \mathbf z \widetilde N(ds,d\mathbf z)$ , where B(ds) is an $ \mathbb{R}^p$ -valued independently scattered Gaussian random measure on $ \mathbb{R}$ independent of $ N(ds,d\mathbf z)$ , without loss of generality satisfying $ \mathbb{E} B(ds) B(ds)^* = ds\times I$ . In this case, when $ \widetilde N$ is absent from $ \mathcal M$ , assuming properness, the corresponding process (33) is an ofBm.

Appendix B. Proofs: Section 3

Proof of Theorem 3.1: Statement (i) is a consequence of the facts that $ \widetilde{g}_t \in L^2_{\text{Herm}}(\mathbb{R})$ , $ g_t \in L^2(\mathbb{R},M(p,\mathbb{R}))$ , $ t \in \mathbb{R}$ (cf. Didier and Pipiras [Reference Didier and Pipiras23]).

In regard to (ii), it follows from the dominated convergence theorem that the covariance functions of both $ \widetilde{X}_H$ and $ X_H$ are continuous. Therefore, both processes are stochastically continuous.

As for (iii), the relation (35) is a consequence of the expression (78) for the characteristic function of stochastic integrals of the form (15). The expression (36) now follows from the fact that, for $ {\mathbf z} = {\mathbf z}_1 + {\mathbf i} {\mathbf z}_2 \in \mathbb{C}^p$ , and $ j = 1,\ldots,m$ ,

\begin{align*} \Re\Big\{\Big(\Re g_{t_j}(x)+ {\mathbf i} \Im g_{t_j}(x)\Big){\mathbf z} \Big\} & = \Re\Big\{\Big[\Re g_{t_j}(x) {\mathbf z}_1 - \Im g_{t_j}(x) {\mathbf z}_2 + {\mathbf i} \Big(\Re g_{t_j}(x){\mathbf z}_2 + \Im g_{t_j}(x){\mathbf z}_1\Big)\Big]\Big\}\\& = \Re g_{t_j}(x){\mathbf z}_1 - \Im g_{t_j}(x){\mathbf z}_2 .\end{align*}

In turn, the relation (38) is a consequence of (81). Moreover, $ \widetilde{X}_H$ and $ X_H$ are cov.o.s.s. as a consequence of the isometry relations (72) and (79), as well as of the scaling properties

(83) \begin{equation}\widetilde{g}_{ct}(x) = c^{D+I} \widetilde{g}_t(cx) \quad \text{a.e.}, \end{equation}

(84) \begin{equation}g_{ct}(cs) = {\mathcal F}^{-1} \Big(c^{-1}\widetilde{g}_{ct}\big(c^{-1} \cdot \big)\Big)(s) = c^{D}{\mathcal F}^{-1} \big(\widetilde{g}_{t}\big)(s)= c^{D} g_{t}(s) \quad \text{a.e.},\end{equation}

for any $ c > 0$ . This establishes (iii).

We now turn to (iv). First consider $ X_H$ , and let $ H= P J_H P^{-1}$ be the Jordan decomposition of H. Define $ H^{\prime} = Q H Q^{-1}$ , and observe that $ \Re \text{eig}(H^{\prime})\in(0,1)$ . The expression (79) and Parseval–Plancherel imply $ \mathbb{E} X_H(s) X_H(t)^* = \int_{\mathbb{R}} g_s(u) Q Q^* g_t(u)^* du= \int_{\mathbb{R}} \widetilde g_s(x)Q Q^* \widetilde g_t(x)^* dx$ , which is the covariance function of an ofBm with parameters H ^′ and QA in its harmonizable representation. This establishes the claim. The claim under (27) follows, since in this case $ Q=I$ . The statements for $ \widetilde{X}_H$ follow similarly based on the expression (72).

In regard to (v), for any $ t,h \in \mathbb{R}$ ,

(85) \begin{align}X_{H}(t+h)- X_{H}(h) & =\int_{\mathbb{R}} \big( g_{t+h}(s) - g_{h}(s)\big) {\mathcal M}(ds)\nonumber\\ & = \int_{\mathbb{R}}\! \Big( \Big\{(t+h-s)^{D}_+ -(h-s)^{D}_+ \Big\}M_+ + \Big\{(t+h-s)^{D}_- -(h-s)^{D}_-\Big\} M_-\Big) {\mathcal M}(ds)\nonumber\\&\stackrel{\text{f.d.d.}}= \int_{\mathbb{R}} \Big( \Big\{\big(t-s^{\prime}\big)^{D}_+ -\big({-}s^{\prime}\big)^{D}_+ \Big\}M_+ + \Big\{\big(t-s^{\prime}\big)^{D}_- -\big({-}s^{\prime}\big)^{D}_-\Big\} M_-\Big) {\mathcal M}(ds).\end{align}

In (85), the equality of finite-dimensional distributions follows from a change of variable $ h-s = s^{\prime}$ in the characteristic function for

\begin{equation*}\big( X_{H}(t+h_1)- X_{H}(h_1),\ldots,X_{H}(t+h_m)- X_{H}(h_m) \big), \quad h_1,\ldots,h_m \in \mathbb{R},\end{equation*}

which in turn stems from the expression (81). This shows that the maofLm $ X_{H}$ has strict-sense stationary increments. That $ \widetilde X_H$ has wide-sense stationary increments is a direct consequence of (72) and (79). Moreover, under the condition (41), for any $ t,h \in \mathbb{R}$ ,

\begin{align*}\widetilde{X}_{H}(t+h)- \widetilde{X}_{H}(h) & =\int_{\mathbb{R}} \big( \widetilde{g}_{t+h}(x) - \widetilde{g}_{h}(x)\big) \widetilde{{\mathcal M}}(dx)\\ & = \int_{\mathbb{R}} \bigg( \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x}\bigg)\big\{x^{-D}_+ A+x^{-D}_- \overline{A}\big\} e^{{\mathbf i} h x}\widetilde{{\mathcal M}}(dx)\\ & \stackrel{\text{f.d.d.}}= \int_{\mathbb{R}} \bigg( \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x}\bigg)\big\{x^{-D}_+ A+x^{-D}_- \overline{A}\big\} \widetilde{{\mathcal M}}(dx) = \widetilde{X}_{H}(t).\end{align*}

This shows that the rhofLm $ \widetilde{X}_{H}$ has strict-sense stationary increments. This establishes (v).

We now show (vi). For the sake of contradiction, suppose $ X_H$ is o.s.s. Then the relation (2) holds for $ X_H$ and some matrix $ \mathcal H$ . So, fix $ c > 0$ and $ t \neq 0$ . Then

(86) \begin{align}& \exp\bigg\{ \int_{\mathbb{R}}\int_{\mathbb{R}^p} \Big( e^{{\mathbf i} {\mathbf u}^* g_{ct}(s){\mathbf z} }-1 -{\mathbf i} {\mathbf u}^* g_{ct}(s){\mathbf z} \Big) \mu(d{\mathbf z}) ds \bigg\}\nonumber\\& = \exp\bigg\{ \int_{\mathbb{R}}\int_{\mathbb{R}^p} \Big( e^{{\mathbf i} {\mathbf u}^* c^{\mathcal H} g_{t}(s){\mathbf z} }-1 -{\mathbf i} {\mathbf u}^* c^{\mathcal H} g_{t}(s){\mathbf z} \Big) \mu(d{\mathbf z}) ds \bigg\}, \quad {\mathbf u} \in \mathbb{R}^p.\end{align}

Define the measurable functions $ H_1(s,{\mathbf z}) = g_{ct}(s){\mathbf z}$ and $ H_2(s,{\mathbf z}) = c^{\mathcal H} g_{t}(s){\mathbf z}$ , and consider the product measure $ \mu \otimes \eta$ . Now define the induced measures on $ \mathbb{R}^p$ via

\begin{equation*}\nu_i(d{\mathbf y})= [ H_i (\mu \otimes \eta)](d{\mathbf y})= (\mu \otimes \eta)\big[H^{-1}_i (d{\mathbf y})\big], \quad i=1,2.\end{equation*}

By a change of measures, we can rewrite (86) as

\begin{equation*}\exp\bigg\{ \int_{\mathbb{R}^{p}} \Big( e^{{\mathbf i} \langle{\mathbf u},{\mathbf y}\rangle }-1 -{\mathbf i} \langle{\mathbf u}, {\mathbf y}\rangle \Big) \nu_1(d{\mathbf y}) \bigg\}= \exp\bigg\{ \int_{\mathbb{R}^{p}}\Big( e^{{\mathbf i} \langle{\mathbf u},{\mathbf y}\rangle }-1 -{\mathbf i} \langle{\mathbf u}, {\mathbf y} \rangle\Big) \nu_2(d{\mathbf y}) \bigg\}, \quad {\mathbf u} \in \mathbb{R}^p.\end{equation*}

By the measure-theoretic convention $ 0 \times \infty = 0$ , we arrive at

\begin{align*}& \exp\bigg\{ \int_{\mathbb{R}^{p}\setminus\{{\mathbf 0}\}} \Big( e^{{\mathbf i} \langle{\mathbf u},{\mathbf y}\rangle }-1 -{\mathbf i} \langle{\mathbf u}, {\mathbf y}\rangle \Big) \nu_1(d{\mathbf y}) \bigg\}\\& \quad = \exp\bigg\{ \int_{\mathbb{R}^{p}\setminus\{{\mathbf 0}\}}\Big( e^{{\mathbf i} \langle{\mathbf u},{\mathbf y}\rangle }-1 -{\mathbf i} \langle{\mathbf u}, {\mathbf y} \rangle\Big) \nu_2(d{\mathbf y}) \bigg\}, \quad {\mathbf u} \in \mathbb{R}^p.\end{align*}

By the uniqueness of the Lévy measure, $ \nu_1(B)= \nu_2(B)$ for all $ B \in \mathcal B(\mathbb{R}^p\setminus\{\mathbf 0\})$ . Equivalently,

(87) \begin{equation}\int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{B}\big(g_{ct}(s){\mathbf z}\big) \mu(d{\mathbf z})ds = \int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{B}\big(c^{\mathcal H} g_{t}(s){\mathbf z}\big) \mu(d{\mathbf z})ds, \quad B \in {\mathcal B}(\mathbb{R}^p \setminus\{{\mathbf 0}\}).\end{equation}

Note that the kernel $ g_{t}$ satisfies the scaling relation (84). By a change of variable $ s = c w$ and (84) applied to the integral $ \int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{B}\big(g_{ct}(s){\mathbf z}\big) \mu(d{\mathbf z})ds$ , we can rewrite the relation (87) as

(88) \begin{equation}c \int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{c^{(1/2)I-H}B}\big(g_{t}(w){\mathbf z}\big) \mu(d{\mathbf z})dw = \int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{c^{-\mathcal H}B}\big(g_{t}(s){\mathbf z} \big) \mu(d{\mathbf z})ds.\end{equation}

Let

(89) \begin{equation}\nu_*(B) =\int_{\mathbb{R}}\int_{\mathbb{R}^p} 1_{B}\Big(g_{t}(s){\mathbf z} \Big) \mu(d{\mathbf z})ds, \quad B \in {\mathcal B}(\mathbb{R}^{p}\setminus\{\mathbf 0\}),\end{equation}

and set $ \nu_*(\{\mathbf 0\})\,:\!=\,0$ . Note that

\begin{equation*}\int_{\mathbb{R}^p} \|\mathbf y\|^2 \nu_*(d\mathbf y) = \int_{\mathbb{R}}\int_{\mathbb{R}^p} \|g_t(s) \mathbf z\|^2 \mu(d\mathbf z)ds\leq \int_{\mathbb{R}}\|g_t(s)\|^2 ds \cdot \int_{\mathbb{R}^p} \| \mathbf z\|^2 \mu(d\mathbf z)<\infty.\end{equation*}

Based on $ \nu_*$ , we can rewrite (88) as

(90) \begin{equation}c \nu_*\big(c^{(1/2)I-H}B\big) = \nu_*\big(c^{-\mathcal H} B\big).\end{equation}

Fix $ B_0 \in {\mathcal B}(\mathbb{R}^p)$ . Then, for $ B(c)\,:\!=\,c^{H-(1/2)I}B_0 \in {\mathcal B}(\mathbb{R}^{p})$ , (90) implies that $ c \nu_*(B_0) = \nu_*\big(c^{-\mathcal H} c^{ H -(1/2)I}B_0\big)$ , i.e.,

(91) \begin{equation}c \nu_*(d{\mathbf y}) = \nu_*\big(c^{-\mathcal H} c^{ H -(1/2)I}d{\mathbf y}\big).\end{equation}

Starting from the Lévy measure $ \nu_*$ , we can define a Lévy process $ L_*$ by means of the characteristic function

(92) \begin{equation}\mathbb{E} e^{{\mathbf i} \langle\mathbf{u}, L_*(t)\rangle} = \exp\bigg\{t \int_{\mathbb{R}^{p}}\Big( e^{{\mathbf i} \langle{\mathbf u},{\mathbf y}\rangle }-1 -{\mathbf i} \langle{\mathbf u}, {\mathbf y}\rangle \Big) \nu_*(d{\mathbf y}) \bigg\}.\end{equation}

In particular, $ L_*(0) = 0$ , $ L_*$ has finite second moment, and $ L_*(t)$ is proper for $ t \neq 0$ . By (91),

(93) \begin{equation}\mathbb{E} e^{{\mathbf i} \langle\mathbf{u}, L_*(ct)\rangle} = \mathbb{E} e^{{\mathbf i} \big\langle\mathbf{u}, c^{\mathcal H} c^{(1/2)I-H} L_*(t)\big\rangle}, \quad c > 0.\end{equation}

Since $ L_*$ is a Lévy process, the relation (93) implies that, for every $ c > 0$ , there exist a linear operator B(c) and a vector b(c)—namely, $ B(c) = c^{\mathcal H} c^{(1/2)I-H}$ and $ b(c) = {\mathbf 0}$ —such that $ \{ L_*(ct)\}_{t\in\mathbb{R}} \stackrel {\text{f.d.d.}} = \{ B(c) L_*(t)+ b(c)\}_{t\in\mathbb{R}}$ . Therefore, since $ L_*$ is proper, Theorem 1 in Hudson and Mason [Reference Hudson and Mason29] shows that there exist a matrix H ^′ and a nonrandom function $ d\,:\,[0,\infty)\to \mathbb{R}^p$ such that $ \{ L_*(ct)\}_{t\in\mathbb{R}} \stackrel {\text{f.d.d.}} = \big\{ c^{H^{\prime}}L_*(t)+d(c)\big\}_{t\in\mathbb{R}}$ . Furthermore, since $ L_*$ is stochastically continuous, Theorem 7 in Hudson and Mason [Reference Hudson and Mason29] implies that $ d\equiv 0$ , and that $ L_*(1)$ is operator-stable (with exponent H ^′). However, $ L_*$ has finite second moment, which in turn implies $ L_*$ must be Gaussian. This contradicts the fact that, by (92), $ L_*$ has no Gaussian component. Hence, maofLm $ X_H$ is not o.s.s., as claimed. The case for rhofLm $ \widetilde X_H$ is analogous.

We now turn to (vii). Without loss of generality, we can rewrite $ \Re h_1 \leq \ldots \leq \Re h_p$ . Let $ d_1 = \Re h_1-\tfrac{1}{2}$ . Again without loss of generality, let $ 0\leq s<t\leq 1$ , and write $ r=t-s$ . Recall that the Frobenius inner product $ \langle \cdot,\cdot\rangle_F$ of two matrices $ A,B\in M(p,\mathbb{C})$ is given by $ \langle A ,B\rangle_F=\text{tr}(A^*B)$ , and write $ \|\cdot \|_F$ for the corresponding norm. Also, write $ H=P J_H P^{-1}$ for the Jordan decomposition of H, and recall that for any Jordan block $ J_{h}$ corresponding to $ h\in \text{eig}(H)$ of size k,

\begin{equation*}r^{J_h} = \begin{pmatrix}r^h & \quad 0 & \quad 0 & \quad \dots & \quad 0\\[3pt]({\log} r) r^h & \quad r^h & \quad 0 & \quad \dots & \quad 0\\[3pt]\frac{(\log r)^2}{2!}r^h & \quad (\log r) r^h & \quad r^h & &\\[3pt]\vdots & \quad \vdots & &\quad \ddots &\\[3pt]\frac{(\log r)^{k-1}}{(k-1)!}r^h& \quad \frac{(\log r)^{k-2}}{(k-2)!}r^h & \quad \ldots & \quad (\log r)r^h & \quad r^h\end{pmatrix}, \quad r > 0.\end{equation*}

By stationarity of the increments of maofLm,

\begin{align*}\mathbb{E} \| X_H(t)-X_H(s)\|^2 & = \mathbb{E} \| X_H(r)\|^2 = \text{tr} \big( \mathbb{E} X_H(r) X_H(r)^*\big)\\ & =\text{tr} \big( r^H\mathbb{E} X_H(1) X_H(1)^* r^{H^*}\big)= \text{tr} \big( r^{H^*}r^H \mathbb{E} X_H(1) X_H(1)^*\big)\\ & \leq \big\| r^{H^*}r^H\big\|_F \big\| \mathbb{E} X_H(1) X_H(1)^*\big\|_F \leq C \big\| r^{H}\big\|_F^2 \leq C^{\prime} \big\| r^{J_H}\big\|_F^2\\ & \leq C^{\prime\prime}\max_{h\in \text{eig}(H)}\big\| r^{J_h}\big\|_F^2 \leq C^{\prime\prime\prime} \big({|\log r|^{2(p-1)}} \vee 1\big) r^{2\min \big\{\Re h_1,\ldots,\Re h_p\big\}}\\ & {\leq C^{\prime\prime\prime\prime} r^{2 d_1 +1-\varepsilon}},\end{align*}

for every $ \varepsilon>0$ . Hence, by the Kolmogorov– $ \breve{\text{C}}$ entsov theorem (e.g., Kallenberg [Reference Kallenberg33], Theorem 2.23), X has a modification that is a.s. locally $ \gamma$ -Hölder continuous for each $ \gamma\in (0, d_1)$ , as claimed. The statement also holds for rhofLm in view of its wide-sense stationary increments (see Statement (v)).

Proof of Proposition 3.1: Both statements are a consequence of the Parseval-type relations (142), (145) and Proposition E.1.

Proof of Proposition 3.2: We begin with (i). For any $ t \neq 0$ , let $ g_{t}$ be as in (25), and recall the scaling relation (84) that the kernel $ g_{t}$ satisfies. For $ n \in \mathbb{N}$ , let $ t_1,\ldots,t_n\in \mathbb{R}$ . By Theorem 3.1(iii), the joint characteristic function of $ X_H(t_1),\ldots,X_H(t_n)$ is given by

\begin{equation*}\mathbb{E} \exp \Bigg \{{\mathbf i} \sum_{j=1}^n\langle {\mathbf u}_j, X_H(t_j)\rangle \Bigg\}= \exp \Bigg\{\!\int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{t_j}(s)^*{\mathbf u}_j\Bigg)ds\Bigg\},\end{equation*}

where the Lévy symbol $ \psi$ is as in (82). Now consider the collection of rescaled vectors

\begin{equation*}c^{-H}X_H(ct_1),c^{-H}X_H(ct_2),\ldots,c^{-H}X_H(ct_n).\end{equation*}

Then their joint characteristic function is given by

\begin{align}& \mathbb{E} e^{ {\mathbf i}\sum_{j=1}^n\big\langle {\mathbf u}_j,{c^{-H}X_H(ct_j)}\big\rangle}= \mathbb{E} e^{ {\mathbf i}\sum_{j=1}^n\big\langle c^{-H^*}{\mathbf u}_j,X_H(ct_j)\big\rangle}= \exp \Bigg\{ \int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{ct_j}(s)^*c^{-H^*}{\mathbf u}_j\Bigg)ds \Bigg\}\nonumber\\& \stackrel{cv=s}{=} \exp \Bigg\{ \int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{ct_j}(cv)^*c^{-H^*}{\mathbf u}_j\Bigg)cdv \Bigg\} = \exp \Bigg\{\int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{t_j}(v)^*c^{-\frac{1}{2}I}{\mathbf u}_j\Bigg)c dv \Bigg\} \nonumber\end{align}

(94) \begin{align}& =\exp \Bigg\{\int_\mathbb{R}\int_{\mathbb{R}^p}\Bigg(\exp \Bigg\{{\mathbf i}\Bigg\langle \sum_{j=1}^ng_{t_j} (v)^*c^{-\frac{1}{2}I}{\mathbf u}_j,{\mathbf z}\Bigg\rangle \Bigg\} - 1 - {\mathbf i}\Bigg\langle{ \sum_{j=1}^n g_{t_j}(v)^*c^{-\frac{1}{2}I}{\mathbf u}_j,{\mathbf z}}\Bigg\rangle \Bigg)c\mu(d{\mathbf z})dv \Bigg\}\nonumber\\& =\exp \Bigg\{ \int_\mathbb{R}\int_{\mathbb{R}^p}\Bigg(\exp \Bigg\{ c^{-1/2}{\mathbf i}\Bigg\langle \sum_{j=1}^n g_{t_j} (v)^*{\mathbf u}_j,{\mathbf z}\Bigg\rangle \Bigg\} - 1 - {\mathbf i}\Bigg\langle{ c^{-1/2}\sum_{j=1}^n g_{t_j}(v)^*{\mathbf u}_j,{\mathbf z}}\Bigg\rangle \Bigg)c \mu(d{\mathbf z})dv \Bigg\}.\end{align}

Note that

\begin{equation*}h_c(y)\,:\!=\, c\Big(e^{{\mathbf i} c^{-1/2}y}-1- {\mathbf i} c^{-1/2}y\Big) \sim -\frac{1}{2}y^2\end{equation*}

as $ c\to\infty$ and that $ |h_c(y)|\leq y^2$ for all $ y\in\mathbb{R}$ . If we write $ \xi(v,{\mathbf z})=\big\langle{ \sum_{j=1}^ng_{t_j}(v)^*{\mathbf u}_j,{\mathbf z}}\big\rangle$ , the integrand $ h_c(\xi(v,{\mathbf z}))$ in the expression (94) satisfies

\begin{equation*}|h_c(\xi(v,{\mathbf z}))|\leq \xi(v,{\mathbf z})^2 =\sum_{j,k=1}^n {{\mathbf u}^*_jg_{t_j}(v){\mathbf z}{\mathbf z}^*g_{t_k}^*(v)u_k},\end{equation*}

which is integrable with respect to $ \mu(d{\mathbf z})dv$ since each $ g_{t_j}(\cdot)\in L^2(\mathbb{R},M(p,\mathbb{R}))$ and $ \big\|\int_{\mathbb{R}^p} {\mathbf z}{\mathbf z}^*\mu(d{\mathbf z})\big\|=1$ . Thus, by the dominated convergence theorem, as $ c\to\infty$ , (94) converges to

(95) \begin{align}& \exp \Bigg \{-\frac{1}{2} \int_\mathbb{R}\int_{\mathbb{R}^p}\Bigg(\sum_{j=1}^n\big\langle{g_{t_j}(v)^*{\mathbf u}_j,{\mathbf z}}\big\rangle\Bigg)^2 \mu(d{\mathbf z})dv\Bigg\}\nonumber\\& \quad =\exp \Bigg \{-\frac{1}{2} \int_\mathbb{R}\int_{\mathbb{R}^p}\sum_{j,k=1}^n{{\mathbf u}^*_jg_{t_j}(v) {\mathbf z}{\mathbf z}^* g_{t_k}^*(v){\mathbf u}_k} \mu(d{\mathbf z})dv\Bigg\}\nonumber\\& \quad =\exp \Bigg \{-\frac{1}{2} \int_\mathbb{R}\sum_{j,k=1}^n{{\mathbf u}^*_jg_{t_j}(v)g_{t_k}^*(v){\mathbf u}_k} dv\Bigg\},\end{align}

where we use the condition (39). Note that (95) is equal to $ \exp\!\big\{{-}\frac{1}{2} {\mathbf u}^*\Sigma_{B_H}{\mathbf u}\big\}$ , where $ \Sigma_{ B_H}$ is the $ pn \times pn$ block matrix

\begin{equation*} \Sigma_{B_H} = \bigg( \int_{\mathbb{R}} g_{t_i}(s) g_{t_j}(s)^*ds \bigg)_{i,j=1,\ldots,n}. \end{equation*}

Hence, (95) is the characteristic function of an ofBm at times $ t_1,\ldots,t_n$ .

For (ii), for any $ t \neq 0$ , let $ \widetilde{g}_{t}$ be as in (24). For $ s \in \mathbb{R}$ and $ \varepsilon > 0$ , Theorem 3.1 (iv) implies that

\begin{equation*}\widetilde{X}_{H}(s + \varepsilon t)- \widetilde{X}_{H}(s) \stackrel{\text{f.d.d.}}= \widetilde{X}_{H}(\varepsilon t).\end{equation*}

So, fix $ n \in \mathbb{N}$ and let $ {\mathbf u}_1,\ldots,{\mathbf u}_n \in\mathbb{R}^p$ , $ t_1,\ldots,t_n \in\mathbb{R}$ . Note that $ \widetilde{g}_{\varepsilon t}\big(\varepsilon^{-1}x\big) = x^{D+I}\widetilde{g}_{t}(x)$ . By Theorem 3.1(iii),

(96) \begin{align}\mathbb{E} e^{{\mathbf i} \sum^{N}_{j=1}{\mathbf u}^*_j \varepsilon^{-H}\widetilde{X}_H(t_j)} & = \exp \Bigg\{\int_\mathbb{R} \widetilde\psi\Bigg( \sum_{j=1}^n \widetilde{g}_{\varepsilon t_j}(x)^*\varepsilon^{-H^*}{\mathbf u}_j\Bigg)dx\Bigg\}\nonumber\\& \stackrel{y = \varepsilon x}{=} \exp \Bigg\{ \int_\mathbb{R} \widetilde \psi\Bigg( \sum_{j=1}^n \widetilde{g}_{ t_j}(y \varepsilon^{-1})^*\varepsilon^{-H^*}{\mathbf u}_j\Bigg)\frac{dy}{\varepsilon} \Bigg\}\nonumber\\& =\exp \Bigg\{ \int_\mathbb{R} \widetilde \psi\Bigg(\sum_{j=1}^n \widetilde{g}_{ t_j}(y)^*\varepsilon^{\frac{1}{2}I}{\mathbf u}_j\Bigg)\frac{dy}{\varepsilon} \Bigg\},\end{align}

where $ \widetilde \psi$ is given by (76). Recast $ {\mathbf z} = {\mathbf z}_1 + {\mathbf i} {\mathbf z}_2$ . By a dominated convergence argument similar to that of (i), and based on the fact that $ \widetilde{g}_{t_j} \in L^{2}_{\text{Herm}}(\mathbb{R})$ , $ j=1,\ldots,n$ , as $ \varepsilon \to 0$ the expression (96) converges to

\begin{equation*}\exp\Bigg\{-2 \Bigg(\int_{\mathbb{R} \times \mathbb{C}^p} \sum_{j,k=1}^n {\mathbf u}^*_j\widetilde{g}_{t_j}(y){\mathbf z}_1 {\mathbf z}^*_1 \widetilde{g}_{t_k}^*(y){\mathbf u}_k +\sum_{j,k=1}^n {\mathbf u}^*_j\widetilde{g}_{t_j}(y){\mathbf z}_2 {\mathbf z}^*_2 \widetilde{g}_{t_k}^*(y){\mathbf u}_k \Bigg)\mu(d{\mathbf z})dy \Bigg\}.\end{equation*}

By using the condition (40), we arrive at

\begin{equation*}\exp \Bigg\{-\frac{1}{2} \int_\mathbb{R} \sum_{j,k=1}^n{{\mathbf u}^*_j\widetilde{g}_{t_j}(y) \widetilde{g}_{t_k}^*(y){\mathbf u}_k}\ dy\Bigg\}.\end{equation*}

This establishes (ii).

Proof of Proposition 3.3: Whenever convenient, we write $ D=H-(1/2)I$ (see (23)). First note that, since $ \Re \lambda_p(B)<1$ , the expression (147) is well defined and corresponds to the Lévy symbol of a full operator-stable distribution in $ \mathbb{R}^p$ (see Section 11). Furthermore, the process (47) is well defined by Theorem 5.4 in Maejima and Mason [Reference Maejima and Mason41] (see also Theorem 4.2 in Kremer and Scheffler [Reference Kremer and Scheffler36]), since

\begin{equation*}\Re\lambda_1\big(\widetilde H - B\big) + \Re\lambda_1(B) = \Re\lambda_1(D)+\Re\lambda_1(B) >0\end{equation*}

and

\begin{equation*}\Re\lambda_p\big(\widetilde H - B -I\big) + \Re\lambda_p(B) = \Re \lambda_p(D)-1+\Re\lambda_p(B)<0.\end{equation*}

(N.b.: there is a typo in the original statement of Theorem 5.4 in Maejima and Mason [Reference Maejima and Mason41]; in the notation of that paper, their assumption should read $ \Lambda_{D-B-I}+\Lambda_B<0$ .) We now proceed as in the proof of Proposition 3.2(i). In fact, fix $ n \in \mathbb{N}$ and $ t_1,\ldots,t_n \in \mathbb{R}$ , as well as the vectors $ {\mathbf u}_1,\ldots,{\mathbf u}_n \in \mathbb{R}^p$ . Let $ \varepsilon > 0$ . Consider $ g_{t_j}$ as in (25), $ j = 1,\ldots,n$ , and recall the scaling relation (84). Then, by stationary increments and (45),

\begin{align}& \mathbb{E} e^{ {\mathbf i}\sum_{j=1}^n\big\langle {\mathbf u}_j,{\varepsilon^{-\widetilde H_1} \big( X_H(s+\varepsilon t_j)-X_H(s)\big)}\big\rangle}= \mathbb{E} e^{ {\mathbf i}\sum_{j=1}^n\big\langle {\mathbf u}_j,{\varepsilon^{-\widetilde H_1} X_H\big(\varepsilon t_j\big)}\big\rangle}\nonumber\\[3pt]& \quad = \exp\Bigg\{\int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{\varepsilon t_j}(s)^*\varepsilon^{-\widetilde H_1^*}{\mathbf u}_j\Bigg)ds\Bigg\}\stackrel{\varepsilon v=s}{=} \exp \Bigg\{ \int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{\varepsilon t_j}(\varepsilon v)^*\varepsilon ^{-\widetilde H_1^*}{\mathbf u}_j\Bigg)\varepsilon dv\Bigg\}\nonumber\\[3pt]& \quad =\exp \Bigg\{\int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n g_{t_j}( v)^*\varepsilon ^{(D-\widetilde H_1)^*}{\mathbf u}_j\Bigg)\varepsilon dv \Bigg\}=\exp \Bigg\{\int_\mathbb{R} \psi\Bigg( \sum_{j=1}^n \varepsilon ^{-B^*}g_{t_j}( v)^*{\mathbf u}_j\Bigg)\varepsilon dv\Bigg\}\nonumber\\[3pt]& \quad =\exp \Bigg\{\int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}\Bigg(\exp\Bigg\{{\mathbf i}\Bigg\langle \sum_{j=1}^n g_{t_j} (v)^*{\mathbf u}_j,(r/\varepsilon)^B\boldsymbol\theta\Bigg\rangle\Bigg\}\nonumber\\[3pt]& \qquad - 1 - {\mathbf i}\Bigg\langle{ \sum_{j=1}^n g_{t_j}(v)^*{\mathbf u}_j,(r/\varepsilon)^B\boldsymbol\theta}\Bigg\rangle \Bigg)\varepsilon q(r,\boldsymbol\theta)\frac{dr}{r^2}\lambda(d\boldsymbol\theta)dv \Bigg\}\nonumber\end{align}

(97) \begin{align}& \quad \stackrel{\xi=r/\varepsilon}{=}\exp \Bigg\{\int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}\Bigg(\exp\Bigg\{{\mathbf i}\Bigg\langle \sum_{j=1}^n g_{t_j} (v)^*{\mathbf u}_j,\xi^B\boldsymbol\theta\Bigg\rangle\Bigg\}\nonumber\\[3pt]& \qquad - 1 - {\mathbf i}\Bigg\langle{ \sum_{j=1}^n g_{t_j}(v)^*{\mathbf u}_j,\xi^B\boldsymbol\theta}\Bigg\rangle \Bigg)q(\varepsilon \xi,\boldsymbol\theta)\frac{d\xi}{\xi^2}\lambda(d\boldsymbol\theta)dv \Bigg\}\nonumber\\[3pt]& \quad \to\exp \Bigg\{\int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}\Bigg(\exp\Bigg\{{\mathbf i}\Bigg\langle \sum_{j=1}^n g_{t_j} (v)^*{\mathbf u}_j,\xi^B\boldsymbol\theta\Bigg\rangle\Bigg\}\nonumber\\[3pt]& \qquad - 1 - {\mathbf i}\Bigg\langle{ \sum_{j=1}^n g_{t_j}(v)^*{\mathbf u}_j,\xi^B\boldsymbol\theta}\Bigg\rangle \Bigg)\frac{d\xi}{\xi^2}\lambda(d\boldsymbol\theta)dv \Bigg\},\end{align}

as $ \varepsilon\to 0^+$ . The limit in (97) is a consequence of the dominated convergence theorem and of the relation (150), since $ |q(r,\boldsymbol\theta)|\leq 1$ . By Proposition 2.17 in Sato [Reference Sato63], (97) is the characteristic function of (47). Therefore, the limit (46) of the rescaled finite-dimensional distributions of $ X_H$ holds.

For (ii), we first verify the existence of the limiting process (50) by applying Theorem 2.5 in Kremer and Scheffler [Reference Kremer and Scheffler36]. To use the proposition, we view (50) as a process in $ \mathbb{R}^{2p}$ , where we make the identification

\begin{equation*}\mathbb{R}^{2p\times 2p}\ni \widetilde g_t(x) \equiv \begin{pmatrix}{\Re \widetilde{g}_t(x)} & \quad {-\Im \widetilde{g}_t(x)}\\[4pt]0 & \quad 0\end{pmatrix}\end{equation*}

(see Proposition 5.10 in Kremer and Scheffler [Reference Kremer and Scheffler35]). Fix $ t\neq 0$ . To apply Theorem 2.5 in Kremer and Scheffler [Reference Kremer and Scheffler36], we need to verify

(98) \begin{equation}\int_{\{x:\|\widetilde g_t(x)\|<R\}} \|\widetilde g_t(x)\|^{{\frac{1}{\Re\lambda_p(B)} - \delta_1}}dx + \int_{\{x:\|\widetilde g_t(x)\|>R\}} \|\widetilde g_t(x)\|^{{\frac{1}{\Re\lambda_1(B)} + \delta_2}} dx <\infty\end{equation}

for some $ R>0$ and appropriate

\begin{equation*}\delta_1\in \left(0,\frac{1}{\Re \lambda_p(B)}\right), \qquad \delta_2>0.\end{equation*}

Recall that $ D = H - (1/2)I$ . Since $ \|\widetilde g_t(x)\|\to0$ as $ |x|\to\infty$ , and $ \|\widetilde g_t(\cdot)\|$ is continuous on $ \mathbb{R}\setminus\{0\}$ , for R large enough there exists an $ \varepsilon>0$ such that the set $ \{x:\|\widetilde g_t(x)\|>R\}\subseteq\{x:|x|<\varepsilon\}$ . Further note that for each $ \delta>0$ there exists $ C>0$ such that

(99) \begin{align} \max\big\{\big\|\Re\widetilde{g}_{t}(x)\big\|,\big\|\Im\widetilde{g}_{t}(x)\big\|\big\}\mathbf 1_{\big\{|x|>\varepsilon\big\}} \leq C |x|^{\delta-\Re{\lambda_1 (D)}-1},\nonumber\\ \text{max}\big\{\big\|\Re\widetilde{g}_{t}(x)\big\|,\big\|\Im\widetilde{g}_{t}(x)\big\|\big\}\mathbf 1_{\big\{|x|\leq \varepsilon\big\}} \leq C |x|^{-\delta-\Re{\lambda_p (D)}},\end{align}

(see Theorem 2.2.4 in Meerschaert and Scheffler [Reference Meerschaert and Scheffler50]), where in the second inequality we used the fact that $ \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x}$ is bounded for all small $ |x|$ .

So, take $ \delta>0$ small enough so that

(100) \begin{equation}\Re\lambda_1(D)+ 1 -\Re \lambda_p(B) = \Re\lambda_1(H)+\Bigg( \frac{1}{2} - \Re\lambda_p(B)\Bigg) > \delta\end{equation}

and

(101) \begin{equation}\Re\lambda_p(D) +1 -\Re \lambda_1(B)=\Re\lambda_p(H)+\Bigg(\frac{1}{2}-\Re \lambda_1(B)\Bigg)<1-\delta.\end{equation}

Let $ C>0$ be the constant satisfying both inequalities (99). For notational simplicity, write $ \rho_1=-(\delta - \Re{\lambda_1 (D)}-1)$ and $ \rho_2 = \delta+\Re\lambda_p (D)$ . Now, for any $ \delta_1>0$ ,

(102) \begin{equation}\int_{\big\{|x|>\varepsilon \big\}} \max\Big\{\big\|\Re\widetilde{g}_{t}(x)\big\|,\big\|\Im\widetilde{g}_{t}(x)\big\|\Big\}^{\frac{1}{\Re\lambda_p(B)} - \delta_1}dx\leq C \int_{\big\{|x|> \varepsilon \big\}} |x|^{\frac{-\rho_1}{\Re\lambda_p(B)} + \delta_1\rho_1}dx.\end{equation}

By (100),

\begin{equation*}\frac{-\rho_1}{\Re\lambda_p(B)}= \frac{\delta-\Re\lambda_1 (D)-1}{\Re\lambda_p(B)}<-1.\end{equation*}

By choosing $ \delta_1$ so $ \rho_1\delta_1$ is small enough, we obtain

\begin{equation*}{\frac{-\rho_1}{\Re\lambda_p(B)} + \delta_1\rho_1}<-1,\end{equation*}

implying that the integral (102) is finite. This shows that the first summand in (98) is finite, since

\begin{equation*}\int_{\big\{x:\|\widetilde g_t(x)\|<R\big\}} \big\|\widetilde g_t(x)\big\|^{{\frac{1}{\Re\lambda_p(B)} + \delta_1}}\big(\mathbf 1_{\{|x|\leq \varepsilon\}}+\mathbf 1_{\{|x|>\varepsilon\}}\big)dx \leq C^{\prime}+ \int_{\{|x|> \varepsilon \}} |x|^{\frac{-\rho_1}{\Re\lambda_p(B)} + \delta_1\rho_1}dx\end{equation*}

by (102). Now, for any $ \delta_2>0$ ,

(103) \begin{equation}\int_{\big\{|x|\leq \varepsilon\big\}} \max\Big\{\big\|\Re\widetilde{g}_{t}(x)\big\|,\big\|\Im\widetilde{g}_{t}(x)\big\|\Big\}^{\frac{1}{\Re\lambda_1(B)} + \delta_2}dx\leq C \int_{\{|x|\leq \varepsilon\}}|x|^{\frac{-\rho_2}{\Re\lambda_1(B)} - \delta_2\rho_2}dx.\end{equation}

By (101), we see that

\begin{equation*}\frac{-\rho_2}{\Re\lambda_1(B)}=\frac{-\delta-\Re \lambda_p(D)}{\Re\lambda_1(B)} >-1.\end{equation*}

Hence, by choosing $ \delta_2$ small enough we have

\begin{equation*}\frac{\rho_2}{\Re\lambda_1(B)} + \delta_2\rho_2>-1,\end{equation*}

and the integral (103) is also finite. If we write $ \|{\cdot}\|_{q\times q}$ for the operator norm in $ \mathbb{R}^{q\times q}$ , then clearly $ \| \widetilde g_t(x)\|_{2p\times 2p} \leq2\max\big\{\big\|\Re\widetilde{g}_{t}(x)\big\|_{p\times p},\big\|\Im\widetilde{g}_{t}(x)\big\|_{p\times p}\big\}$ . Hence, the conditions of Theorem 2.5 in Kremer and Scheffler [Reference Kremer and Scheffler36] are satisfied, which implies that the process (50) exists by Proposition 5.10 in Kremer and Scheffler [Reference Kremer and Scheffler35]. Moreover, by Corollary 5.11(b) in Kremer and Scheffler [Reference Kremer and Scheffler35], the characteristic function of the candidate limiting process (50) at times $ t_1,\ldots,t_n$ is given by

(104) \begin{align} & \exp \int_\mathbb{R}\int_{\mathbb{R}^{2p}}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{t_k}(y) \mathbf z_1 - \Im \widetilde{g}_{t_k}(y) \mathbf z_2 \big) \Bigg)\Bigg)\mu_{\widetilde B}(d\mathbf z) dy\nonumber\\& \quad =\exp \int_\mathbb{R}\int_{ S_0}\int_{\mathbb{R}_+}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta - \Im \widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta \big) \Bigg)\Bigg)\frac{d\xi}{\xi^2}\lambda(d\boldsymbol \theta)dy\end{align}

(see (146)), where for notational simplicity we used the expression $ W(y)=e^{{\mathbf i} y}-1-{\mathbf i} y$ , $ y\in\mathbb{R}$ . Now, to establish the convergence (49), observe that the scaling relation

\begin{equation*}\widetilde{g}_{ct}\big(x c^{-1}\big)c^{-\widetilde H_2} = \widetilde g_t(x) c^{H+\frac{1}{2}I-\widetilde H_2} = \widetilde g_t(x) c^{B}\end{equation*}

holds. So the characteristic function of the rescaled vector $ \left(c^{-\widetilde H_2}\widetilde X_H(c t_1),\ldots,c^{-\widetilde H_2}\widetilde X_H(c t_1)\right)^*$ is given by

\begin{align*} & \exp\Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \big( \sum^{n}_{k=1}{\mathbf u}^*_k c^{-\widetilde H_2}\widetilde{g}_{ct_k}(x){\mathbf z}\big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k c^{-\widetilde H_2}\widetilde{g}_{ct_k}(x){\mathbf z}\Bigg)\Bigg]\mu(d{\mathbf z}) dx \Bigg\}\\ & \qquad = \exp\Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^{2p}}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_kc^{-\widetilde H_2} \big(\Re\widetilde{g}_{ct_k}(x) {\mathbf z}_1 - \Im \widetilde{g}_{ct_k}(x) {\mathbf z}_2\big) \Bigg)\mu_{\mathbb{R}^{2p}}(d{\mathbf z}) dx \Bigg\},\\ & \quad \stackrel{y=cx}= \exp\Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^{2p}}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_kc^{-\widetilde H_2} \big(\Re\widetilde{g}_{ct_k}(yc^{-1}) {\mathbf z}_1 - \Im \widetilde{g}_{ct_k}\big(yc^{-1}\big) {\mathbf z}_2\big) \Bigg)\mu_{\mathbb{R}^{2p}}(d{\mathbf z}) \frac{dy}c \Bigg\},\\ & \qquad = \exp\Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^{2p}}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{ct_k}(y)c^B{\mathbf z}_1- \Im \widetilde{g}_{ct_k}(y) c^B{\mathbf z}_2\big) \Bigg)\mu_{\mathbb{R}^{2p}}(d{\mathbf z}) \frac{dy}c \Bigg\},\\ & \quad \stackrel{(148)}=\exp \int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{t_k}(y) (cr)^B\boldsymbol \theta - \Im \widetilde{g}_{t_k}(y) (cr)^B\boldsymbol \theta \big) \Bigg)q(r,\boldsymbol \theta)\frac{dr}{r^2}\lambda(d\boldsymbol \theta)\frac{dy}{c}\\ & \quad \stackrel{\xi = cr}=\exp \int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta - \Im \widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta \big) \Bigg)q\big(c^{-1}\xi,\boldsymbol \theta\big)\frac{d\xi}{\xi^2}\lambda(d\boldsymbol \theta)dy\\ & \quad \stackrel{c \to \infty}\to \exp \int_\mathbb{R}\int_{S_0}\int_{\mathbb{R}_+}W\Bigg(2 \sum^{n}_{k=1}{\mathbf u}^*_k\big(\Re\widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta - \Im \widetilde{g}_{t_k}(y) \xi^B\boldsymbol \theta \big) \Bigg)\Bigg)\frac{d\xi}{\xi^2}\lambda(d\boldsymbol \theta)dy,\end{align*}

where the limit is again a consequence of the dominated convergence theorem and the relation (149). The conclusion follows.

Appendix C. Proofs: Section 4

Proof of Theorem 4.1: Let $ \{Y_H(t)\}_{t \in \mathbb{R}} = \{X_H({-}t)\}_{t \in \mathbb{R}}$ be the time-reversed process. First note that, if $ \{f_t({\boldsymbol \varpi}),t \in \mathbb{R}\}= \{g_t(s){\mathbf z},t \in \mathbb{R}\}$ is a minimal representation of $ X_H$ with respect to $ {\mathcal B} \mod \kappa$ , then

(105) \begin{equation}\{f_{-t}({\boldsymbol \varpi}),t \in \mathbb{R}\} \text{ is a minimal representation of $Y_H$ with respect to } {\mathcal B} \mod \kappa.\end{equation}

We first show that (ii) $ \Rightarrow$ (i). Note that, by (57),

(106) \begin{align}g_{-t}(s){\mathbf z} & = \Big\{\Big[({-}t - s)^{D}_+ - ({-}s)^{D}_+\Big]M_+ + \Big[({-}t -s)^{D}_- - ({-}s)^{D}_-\Big]M_- \Big\}{\mathbf z}\nonumber\\[4pt]& = \Big[(t + s)^{D}_- - (s)^{D}_-\Big]M_- \Big(M^{-1}_+ M_-\Big) {\mathbf z}+ \Big[(t + s)^{D}_+ - s^{D}_+\Big] M_+ \Big(M^{-1}_+ M_-\Big){\mathbf z} \quad \mu(d{\mathbf z})\text{-a.e.}\end{align}

So, for any $ m \in \mathbb{N}$ , fix $ t_1 < \ldots < t_m$ , and pick any vectors $ {\mathbf u}_1,\ldots,{\mathbf u}_m \in \mathbb{R}^p$ . By the expressions (81) and (106), the finite-dimensional distributions of $ \{X_H({-}t)\}_{t \in \mathbb{R}}$ are given by

\begin{align*}\mathbb{E} \exp\Bigg\{ \sum^{n}_{j=1} {\mathbf u}^*_j X_H({-}t_j) \Bigg\} = \exp \Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^p} \Bigg(e^{{\mathbf i} \sum^{n}_{j=1} {\mathbf u}^*_j g_{-t_j}(s) {\mathbf z}}- 1 - {\mathbf i} \sum^{n}_{j=1} {\mathbf u}^*_j g_{-t_j}(s){\mathbf z} \Bigg) \mu(d{\mathbf z}) ds \Bigg\}\\= \exp \Bigg\{ \int_{\mathbb{R}} \int_{\mathbb{R}^p} \Bigg(e^{{\mathbf i} \sum^{n}_{j=1} {\mathbf u}^*_j g_{t_j}\big(s^{\prime}\big) {\mathbf z}^{\prime}}- 1 - {\mathbf i} \sum^{n}_{j=1} {\mathbf u}^*_j g_{t_j}\big(s^{\prime}\big){\mathbf z}^{\prime} \Bigg) \mu\big(d{\mathbf z}^{\prime}\big) ds \Bigg\}= \mathbb{E} \exp\Bigg\{ \sum^{n}_{j=1} {\mathbf u}^*_j X_H(t_j) \Bigg\},\end{align*}

where we make the change of variable $ \big(s^{\prime},{\mathbf z}^{\prime}\big) = \big({-}s,\big(M^{-1}_+ M_-\big){\mathbf z}\big)$ and apply the conditions (57) and (58). Therefore, $ X_H$ is time-reversible. This establishes (i).

Now, we establish (i) $ \Rightarrow$ (ii). So, suppose $ X_H$ is time-reversible. We first show that (57) holds. In terms of spectral representations, time-reversibility means that, for $ f_t({\boldsymbol \varpi})$ as in (54),

\begin{equation*}\bigg\{\int_{\mathbb{R}^{p+1}} f_t({\boldsymbol \varpi}) \widetilde{N}(d{\boldsymbol \varpi})\bigg\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \Big\{\int_{\mathbb{R}^{p+1}} f_{-t}({\boldsymbol \varpi}) \widetilde{N}(d{\boldsymbol \varpi})\Big\}_{t \in \mathbb{R}}.\end{equation*}

By assumption and by (105), both $ \{f_t({\boldsymbol \varpi})\}_{t \in \mathbb{R}}$ and $ \{f_{-t}({\boldsymbol \varpi})\}_{t \in \mathbb{R}}$ are minimal representations of $ X_H$ on the space $ \big(\mathbb{R}^{p+1},{\mathcal B}\big(\mathbb{R}^{p+1}\big),\kappa\big)$ . Then, Proposition D.1 implies that there is a (unique modulo $ \kappa$ -null sets) mapping

\begin{equation*}\Phi\,:\,\mathbb{R}^{p+1} \rightarrow \mathbb{R}^{p+1}, \quad {\boldsymbol \varpi} \mapsto (\Phi_1({\boldsymbol \varpi}),\Phi_2({\boldsymbol \varpi})), \quad \Phi_1({\boldsymbol \varpi}) \in \mathbb{R}, \Phi_2({\boldsymbol \varpi})\in \mathbb{R}^p,\end{equation*}

such that, for all $ t \in \mathbb{R}$ ,

(107) \begin{equation}f_{-t}({\boldsymbol \varpi}) = f_t(\Phi({\boldsymbol \varpi})) \quad \kappa(d{\boldsymbol \varpi}) \text{-a.e.}\end{equation}

So, for each $ t\in\mathbb{R}$ , let

\begin{equation*}V_t = \big\{{\boldsymbol \varpi}\,:\, (107) \text{ holds at } {\boldsymbol \varpi}\big\} \in \mathcal B(\mathbb{R}\times\mathbb{R}^p).\end{equation*}

Define $ V = \bigcap_{t\in \mathbb Q} V_{t}$ . Observe that

(108) \begin{equation}f_{-t}({\boldsymbol \varpi}) = f_{t}(\Phi({\boldsymbol \varpi})), \quad {\boldsymbol \varpi} = (s,{\mathbf z}) \in V,\quad t \in \mathbb Q.\end{equation}

Now, for notational simplicity, consider the integrand $ g_t$ with s in place of $ -s$ . Fix any

(109) \begin{equation}{\boldsymbol \varpi}_0 = (s_0,{\mathbf z}_0) \in V, \quad s_0 < 0.\end{equation}

Let $ \{t_n\}_{n\in{\mathbb N}} \subseteq \mathbb Q$ be a sequence such that $ t_n \uparrow \infty$ . For large enough n,

\begin{equation*}\Big[({-}t_n + s_0)^{D}_- - (s_0)^{D}_-\Big]M_- {\mathbf z}_0\end{equation*}

\begin{equation*}= \Big(\Big[\big(t_n+\Phi_1({\boldsymbol \varpi}_0)\big)^{D}_+ - \big(\Phi_1\big({\boldsymbol \varpi}_0\big)\big)^{D}_+\Big]M_+ - \big(\Phi_1\big({\boldsymbol \varpi}_0\big)\big)^{D}_- M_- \Big)\Phi_2\big({\boldsymbol \varpi}_0\big);\end{equation*}

i.e.,

\begin{equation*}({-}t_n + s_0)^{D}_- M_- {\mathbf z}_0 - \big(t_n+\Phi_1({\boldsymbol \varpi}_0)\big)^{D}_+ M_+ \Phi_2({\boldsymbol \varpi}_0)\end{equation*}

(110) \begin{equation}= (s_0)^{D}_- M_- {\mathbf z}_0 - \big(\Phi_1({\boldsymbol \varpi}_0)\big)^{D}_+ M_+ \Phi_2({\boldsymbol \varpi}_0) - \big(\Phi_1({\boldsymbol \varpi}_0)\big)^{D}_- M_- \Phi_2({\boldsymbol \varpi}_0).\end{equation}

Consider the Jordan decomposition $ D = PJ_DP^{-1}$ . The right-hand side of (110) is a constant with respect to n. Therefore, after pre-multiplying both sides by $ P^{-1}$ , we can recast (110) in the form

(111) \begin{equation}({-}t_n + s_0)^{J_D}_- P^{-1}M_- {\mathbf z}_0 - (t_n+\Phi_1({\boldsymbol \varpi}_0))^{J_D}_+ P^{-1}M_+ \Phi_2({\boldsymbol \varpi}_0) = C \in M(p,\mathbb{R}).\end{equation}

We want to show that

(112) \begin{equation}M_- {\mathbf z}_0 = M_+ \Phi_2({\boldsymbol \varpi}_0) \equiv M_+ \Phi_2(s_0,{\mathbf z}_0).\end{equation}

Without loss of generality, we can assume $ J_D$ is a single Jordan block. In view of the condition (27), it suffices to consider two cases, namely, when $ J_D$ is a Jordan block associated with an eigenvalue d with positive real part or with negative real part. So, first assume $ \Re(d) > 0$ and rewrite (111) as

\begin{equation*}\bigg(\frac{t_n+\Phi_1({\boldsymbol \varpi}_0)}{t_n-s_0}\bigg)^{-J_D}P^{-1}M_- {\mathbf z}_0 - P^{-1}M_+ \Phi_2({\boldsymbol \varpi}_0) = (t_n + \Phi_1({\boldsymbol \varpi}_0))^{-J_D} C .\end{equation*}

If $ C \neq {\boldsymbol 0}$ , by taking $ n \rightarrow \infty$ , we arrive at a contradiction, since

\begin{equation*}\lim_{n \rightarrow \infty}\frac{t_n+\Phi_1({\boldsymbol \varpi}_0)}{t_n - s_0}=1\end{equation*}

and $ \lim_{n \rightarrow \infty}\|(t_n - s_0)^{-J_D} C\|=\infty$ . Therefore,

(113) \begin{equation}({-}t_n + s_0)^{J_D}_- P^{-1}M_- {\mathbf z}_0 = (t_n+\Phi_1({\boldsymbol \varpi}_0))^{J_D}_+ P^{-1}M_+ \Phi_2({\boldsymbol \varpi}_0).\end{equation}

Alternatively, assume $ \Re(d) < 0$ . Rewrite (111) as

\begin{equation*}P^{-1}M_- {\mathbf z}_0 - \left(\frac{t_n+\Phi_1({\boldsymbol \varpi}_0)}{t_n - s_0}\right)^{J_D} P^{-1}M_+ \Phi_2({\boldsymbol \varpi}_0) = (t_n - s_0)^{-J_D} C .\end{equation*}

Again by taking $ n \rightarrow \infty$ , we arrive at a contradiction unless $ C = {\boldsymbol 0}$ . So (113) also holds. Thus, in any case, by taking $ n \rightarrow \infty$ we conclude that (112) holds, as we wanted to show.

Still for $ s_0 < 0$ , now let $ \{t_n\}_{n \in \mathbb{N}}\subseteq \mathbb Q$ be a sequence such that $ t_n \downarrow -\infty$ . Then, for large enough n,

\begin{equation*}\Big\{\Big[\big({-}t_n + s_0\big)^{D}_+ \Big]M_+ + \Big[ - (s_0)^{D}_-\Big]M_- \Big\}{\mathbf z}_0\end{equation*}

\begin{equation*}= \Big\{\Big[ - (\Phi_1({\boldsymbol \varpi}_0))^{D}_+\Big]M_+ + \Big[ (t_n+\Phi_1({\boldsymbol \varpi}_0)\big)^{D}_- - (\Phi_1({\boldsymbol \varpi}_0))^{D}_- \Big] M_- \Big\}\Phi_2({\boldsymbol \varpi}_0).\end{equation*}

By an argument analogous to the one leading to (112), we conclude that

(114) \begin{equation}M_+ {\mathbf z}_0 = M_- \Phi_2({\boldsymbol \varpi}_0) \equiv M_- \Phi_2(s_0,{\mathbf z}_0).\end{equation}

As a consequence of (112) and (114), for arbitrary $ (s,{\mathbf z}) \in V $ with $ s<0$ ,

(115) \begin{equation}\Phi_2(s,\mathbf z) = M_-^{-1}M_+ \mathbf z = M_+^{-1}M_- {\mathbf z}.\end{equation}

This establishes (57) for all $ (s,{\mathbf z}) \in V$ such that $ s < 0$ . Now let

(116) \begin{equation}{\boldsymbol \varpi}_0 = (s_0,{\mathbf z}_0) \in V, \quad s_0 > 0.\end{equation}

Let $ \{t_n\}_{n \in \mathbb{N}} \subseteq \mathbb{Q}$ be a sequence such that $ t_n \uparrow \infty$ as $ n \rightarrow \infty$ . By adapting the arguments for showing (113), we conclude that

(117) \begin{equation}(t_n - s_0)^{D}_{+} M_- {\mathbf z}_0 = (t_n + \Phi_1({\boldsymbol \varpi}_0))^{D}_{+} M_+ \Phi_2({\boldsymbol \varpi}_0).\end{equation}

Thus, by adapting the argument we conclude that the relation (112) holds also for $ s > 0$ . Similarly, the relation (114) holds for $ s > 0$ .

In summary, we conclude that

(118) \begin{equation}\Phi_2(s,\mathbf z) = M_-^{-1}M_+ \mathbf z = M_+^{-1}M_- {\mathbf z}, \quad (s,{\mathbf z}) \in V, \quad s\neq0.\end{equation}

In other words, (57) holds, as we wanted to show.

Next, we show that (58) holds. So, fix again $ {\boldsymbol \varpi}_0 = (s_0,\mathbf z_0)\in V$ , $ s_0 < 0$ , as in (109). Consider a sequence $ \{t_n\}_{n\in{\mathbb N}} \subseteq \mathbb{Q}$ such that $ t_n \uparrow \infty$ . Then, for some fixed large n (depending on $ (s_0,\mathbf z_0)$ ), the expressions (111) (with $ C = {\boldsymbol 0}$ ) and (115) imply that

\begin{equation*}\left(\frac{({-}t_n+s_0)_-}{t_n+\Phi_1(s_0,\mathbf z_0)}\right)^D M_- \mathbf z_0 =\left(\frac{t_n-s_0}{t_n+\Phi_1(s_0,\mathbf z_0)}\right)^D M_- \mathbf z_0 = M_+\Phi_2(s_0,\mathbf z_0) = M_-\mathbf z_0.\end{equation*}

In particular, 1 is an eigenvalue of

\begin{equation*}\left(\frac{t_n-s_0}{t_n+\Phi_1(s,\mathbf z_0)}\right)^D\end{equation*}

with corresponding eigenvector $ M_-\mathbf z_0$ . However, in view of the condition (27), $ \text{eig}(D)\cap \{0\} = \emptyset$ . Hence,

\begin{equation*}\{1\}\in \text{eig}\left( \left(\frac{t_n-s_0}{t_n+\Phi_1(s,\mathbf z)}\right)^D\right) = \Big\{ w\in \mathbb{C} \,:\, w= \left(\frac{t_n-s_0}{t_n+\Phi_1(s,\mathbf z)}\right)^d, d\in \text{eig}(D) \Big\}\end{equation*}

\begin{equation*}\Leftrightarrow \frac{t_n-s_0}{t_n+\Phi_1(s_0,\mathbf z_0)} =1.\end{equation*}

Thus,

\begin{equation*}\Phi_1(s_0,\mathbf z_0) = -s_0.\end{equation*}

Now fix again $ {\boldsymbol \varpi}_0 = (s_0,\mathbf z_0)\in V$ , $ s_0 > 0$ , as in (116), and let $ \{t_n\}_{n \in \mathbb{N}}$ be a sequence such that $ t_n \uparrow \infty$ as $ n \rightarrow \infty$ . Hence, (117) holds. Then, by the explicit expression (118) for $ \Phi_2$ , we conclude that 1 is an eigenvalue of

\begin{equation*}\left(\frac{t_n-s_0}{t_n + \Phi_1({\boldsymbol \varpi}_0)}\right)^D.\end{equation*}

Thus, once again we arrive at the relation $ t_n + s_0 = t_n - \Phi_1({\boldsymbol \varpi}_0)$ , i.e.,

\begin{equation*}\Phi_1({\boldsymbol \varpi}_0)= - s_0.\end{equation*}

Since $ (s_0,\mathbf z_0)$ was arbitrary, we conclude that

\begin{equation*}\Phi_1(s,\mathbf z)=-s\quad \text{for all }(s,\mathbf z) \in V.\end{equation*}

In particular,

\begin{equation*}\Phi(s,\mathbf z) = ({-}s, M_-^{-1}M_+ \mathbf z) \quad \kappa(ds,d{\mathbf z})\text{-a.e. }\end{equation*}

However, by Proposition D.1(ii), the mapping $ \Phi$ is a measure space isomorphism from the space $ (\mathbb{R}\times \mathbb{R}^p, \mathcal{B}(\mathbb{R}\times \mathbb{R}^p), \kappa )$ to itself. In particular, the expression (134) holds with $ \kappa_1 = \kappa = \kappa_2$ . Since, in addition, $ \eta[-1,0]=1=\eta[0,1]$ , we have

\begin{equation*}\mu(B) = \kappa([0,1]\times B) =\kappa \circ \Phi^{-1}\big([0,1]\times B)\big)= \kappa\Big([-1,0]\times \Phi^{-1}_2(B)\Big) =\mu\big(\Phi^{-1}_2(B)\big)\end{equation*}

for any Borel set $ B\in \mathcal B(\mathbb{R}^p)$ . Hence, the expression (58) holds. This shows that (i) $ \Rightarrow$ (ii). Therefore, (i) $ \Leftrightarrow$ (ii), as claimed.

We now show that (a) $ \Leftrightarrow$ (a^′). So, suppose (a) in (ii) holds. Then, for each $ s,t\in\mathbb{R}$ , $ M_+\mathbf z = M_-M_+^{-1}M_-\mathbf z$ $ \mu(d{\mathbf z})$ -a.e. Thus,

\begin{align*}g_{-t}({-}s) \mathbf z & =\Big\{\Big[({-}t +s)^{D}_+ - s^{D}_+\Big]M_+ + \Big[({-}t +s)^{D}_- - s^{D}_-\Big]M_- \Big\}{\mathbf z}\\ & = \Big\{\Big[({-}t +s)^{D}_+ - s^{D}_+\Big]M_- + \Big[({-}t +s)^{D}_- - s^{D}_-\Big]M_+ \Big\}M_+^{-1}M_-{\mathbf z}\\ & = \Big\{\Big[(t -s)^{D}_- - ({-}s)^{D}_-\Big]M_- + \Big[(t -s)^{D}_+ - ({-}s)^{D}_+\Big]M_+ \Big\}M_+^{-1}M_-{\mathbf z} = g_{t}(s)M_+^{-1}M_-{\mathbf z}.\end{align*}

Analogously, $ g_{-t}({-}s){\mathbf z}=g_{t}(s) M_-^{-1}M_+{\mathbf z}$ $ \mu(d\mathbf z)$ -a.e. Hence, (a^′) holds. In turn, assuming (a^′), for fixed s, by taking large enough t, through reasoning similar to that of the argument leading to (113), we obtain the relation

\begin{equation*}\bigg(\frac{t+s}{t-s}\bigg)^D M_+ M_-^{-1}M_+ \mathbf z = M_-\mathbf z \quad \mu(d\mathbf z)\text{-a.e.}\end{equation*}

By taking the limit $ t\to\infty$ , we obtain (57). Thus, (a) holds. In other words, (a) $ \Leftrightarrow$ (a^′), as claimed.

Proof of Theorem 4.2: As in the proof of Theorem 4.1, let $ \big\{\widetilde{Y}_H(t)\big\}_{t \in \mathbb{R}} = \big\{\widetilde{X}_H({-}t)\big\}_{t \in \mathbb{R}}$ be the time-reversed process. First note that, if $ \big\{f_t({\boldsymbol \varpi}),t \in \mathbb{R}\big\}= \big\{\widetilde{g}_t(x){\mathbf z},t \in \mathbb{R}\big\}$ is a minimal representation of $ \widetilde{X}_H$ with respect to $ {\mathcal B} \mod \kappa$ , then

(119) \begin{equation}\big\{f_{-t}({\boldsymbol \varpi}),t \in \mathbb{R}\big\} \text{ is a minimal representation of $\widetilde{Y}_H$ with respect to } {\mathcal B} \mod \kappa.\end{equation}

First, we show that the condition (66) implies time-reversibility. Observe that

\begin{align*}\Re \big(\widetilde g_t(x)\mathbf z\big) & = \Re \left( \frac{e^{{\mathbf i} t x}-1}{{\mathbf i} x} \left[x^{-D}_{+}A + x^{-D}_{-}\overline{A} \right]\mathbf z\right)\\& =\Re \left( \left(\frac{e^{{\mathbf i} t x}-1}{-{\mathbf i} x}\right) \left[x^{-D}_{+}\overline A ({-}\overline A ^{-1} A ) + x^{-D}_{-}A({-} A^{-1} \overline A) \right]\mathbf z\right)\\& =\Re \left( \left(\frac{e^{-{\mathbf i} t x^{\prime}}-1}{{\mathbf i} x^{\prime}}\right) \left[\big(x^{\prime}\big)^{-D}_{-}\overline A + \big(x^{\prime}\big)^{-D}_{+}A \right]\mathbf z^{\prime}\right) =\Re \big(\widetilde g_{-t}(x^{\prime})\mathbf z^{\prime}\big),\end{align*}

where $ (x^{\prime},\mathbf z^{\prime}) = \Big({-}x, - A^{-1} \overline A \mathbf z \mathbf 1_{\{x<0\}}- \overline A^{-1} A \mathbf z\mathbf 1_{\{x>0\}}\Big)=:\, \Psi(x,\mathbf z)$ . Then,

\begin{equation*}\Re \big(\widetilde g_t(x)\mathbf z\big) = f_t(\boldsymbol\varpi) = f_{-t}\big(\Psi^{-1}(\boldsymbol\varpi)\big).\end{equation*}

For $ \widetilde{\kappa}(d{\boldsymbol \varpi}) = dx \otimes \widetilde \mu(d\mathbf z)$ , recall that we define

\begin{equation*}\widetilde \mu(d\mathbf z) = \frac{\mu(d\mathbf z) + \mu\big(\overline{d\mathbf z}\big)}{2}.\end{equation*}

Also observe that, by the condition (66), $ \widetilde \mu(d\mathbf z) = \widetilde \mu \big({-}A^{-1}\overline A d\mathbf z\big)$ . We now show that

(120) \begin{equation}\widetilde \kappa = \widetilde \kappa \circ \Psi^{-1}.\end{equation}

Indeed, if $ I\in \mathcal B (\mathbb{R})$ , $ B \in \mathcal B(\mathbb{C}^p)$ , $ I\pm = I\cap \mathbb{R}_{\pm}$ , then

\begin{equation*}\widetilde\kappa(I_+\times B) = \eta(I_+)\widetilde \mu(B) = \eta({-}I_+)\widetilde \mu\big({-} A^{-1} \overline A B\big) = \widetilde\kappa \circ \Psi^{-1}(I_+\times B)\end{equation*}

and

\begin{equation*}\widetilde\kappa(I_-\times B) = \eta(I_-)\widetilde \mu(B) = \eta({-}I_-)\widetilde \mu\big({-} \overline A^{-1} A B\big) = \widetilde\kappa \circ \Psi^{-1}(I_-\times B).\end{equation*}

This shows that $ \widetilde\kappa(I\times B) = \widetilde\kappa \circ \Psi^{-1} (I\times B)$ , which in turn implies that the measures coincide on $ \mathcal B(\mathbb{R}\times \mathbb{C}^p)$ . This establishes (120).

Therefore, starting from (78), by a change of variables,

\begin{align*}& \mathbb{E} \exp \Bigg\{{\mathbf i} \sum^{n}_{k=1}{\mathbf u}^*_k \widetilde{X}_H(t_k) \Bigg\}\\& = \exp\Bigg\{ \int_{\mathbb{R}\times \mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \big( \sum^{n}_{k=1}{\mathbf u}^*_k f_{t_k}(\boldsymbol \varpi)\big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k f_{t_k}(\boldsymbol \varpi)\Bigg)\Bigg]\widetilde \kappa(d\boldsymbol \varpi) \Bigg\}\\& = \exp\Bigg\{ \int_{\mathbb{R}\times \mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \big( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}(\Psi^{-1}(\boldsymbol \varpi))\big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}\big(\Psi^{-1}(\boldsymbol \varpi)\big)\Bigg)\Bigg]\widetilde\kappa (d\boldsymbol \varpi) \Bigg\}\\& = \exp\Bigg\{ \int_{\mathbb{R}\times \mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \big( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}(\boldsymbol \varpi)\big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}(\boldsymbol \varpi)\Bigg)\Bigg]\big(\widetilde\kappa \circ \Psi^{-1}\big) (d\boldsymbol \varpi) \Bigg\}\\& = \exp\Bigg\{ \int_{\mathbb{R}\times \mathbb{C}^p}\Bigg[e^{{\mathbf i} 2 \Re \big( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}(\boldsymbol \varpi)\big)}- 1 -{\mathbf i} 2 \Re \Bigg( \sum^{n}_{k=1}{\mathbf u}^*_k f_{-t_k}(\boldsymbol \varpi)\Bigg)\Bigg]\widetilde\kappa(d\boldsymbol \varpi) \Bigg\}\\& = \mathbb{E} \exp \Bigg\{{\mathbf i} \sum^{n}_{k=1}{\mathbf u}^*_k \widetilde{X}_H({-}t_k) \Bigg\}.\end{align*}

This shows that $ \widetilde X_H$ is time-reversible.

Now suppose that $ \widetilde X_H$ is time-reversible, i.e., that

\begin{equation*}\Bigg\{\int_{\mathbb{R}^{p+1}} f_{-t}({\boldsymbol \varpi})\widetilde{N}(d{\boldsymbol \varpi})\Bigg\}_{t \in \mathbb{R}} \stackrel{\text{f.d.d.}}= \Bigg\{\int_{\mathbb{R}^{p+1}} f_{t}({\boldsymbol \varpi})\widetilde{N}(d{\boldsymbol \varpi})\Bigg\}_{t \in \mathbb{R}}.\end{equation*}

Recall that $ \widetilde{\kappa}(d {\boldsymbol \varpi})\equiv dx \otimes \widetilde{\mu}(d{\mathbf z})$ . Under the condition (64), Proposition D.1 implies that there exists a measurable bijection

\begin{equation*}\Phi\,:\, \mathbb{R}^{p+1} \rightarrow \mathbb{R}^{p+1}, \quad {\boldsymbol \varpi} \mapsto \big( \Phi_1({\boldsymbol \varpi}),\Phi_2({\boldsymbol \varpi})\big)\in \mathbb{R} \times \mathbb{R}^p,\end{equation*}

such that, for all $ t \in \mathbb{R}$ ,

(121) \begin{equation}\Re \big(\widetilde g_t(x)\mathbf z\big) = \Re \big(\widetilde g_{-t}\big(\Phi_1(x,\mathbf z)\big)\Phi_2(x,\mathbf z)\big) \quad dx\otimes \widetilde{\mu}(d\mathbf z)\text{-a.e.}\end{equation}

Moreover, by Proposition D.1, the mapping $ \Phi$ is a measure space isomorphism from the space $ \big(\mathbb{R}\times \mathbb{C}^p, \mathcal{B}\big(\mathbb{R}\times \mathbb{C}^p\big), \widetilde \kappa \big)$ to itself. In particular, it also preserves the measure $ \widetilde \kappa$ , the same being true of $ \Phi^{-1}$ (cf. (134)). Noting that the set [0, 1] has Lebesgue measure 1, we have

\begin{equation*}\widetilde \mu(B) = \widetilde{\kappa}([0,1] \times B) = \widetilde \kappa \circ \Phi^{-1}([0,1] \times B), \quad \widetilde{\kappa}\Big([0,1] \times \big({-} A^{-1} \overline{A B}\big)\Big)= \widetilde \mu\Big({-} A^{-1} \overline{A B}\Big).\end{equation*}

Our goal is to show that

(122) \begin{equation}\widetilde \kappa \circ \Phi^{-1}([0,1] \times B)=\widetilde{\kappa}\Big([0,1] \times \big({-} A^{-1} \overline{A B}\big)\Big),\end{equation}

whence (66) is established. For this purpose, we need to conveniently re-express $ \Phi$ .

So, for each t, let

\begin{equation*}\widetilde{V}_t = \{ {\boldsymbol \varpi}: (121) \text{ holds at }{\boldsymbol \varpi} \} \in {\mathcal B}(\mathbb{R} \times \mathbb{R}^p),\end{equation*}

and set $ \widetilde{V} = \bigcap_{t\in \mathbb Q} \widetilde{V}_t$ . In particular, $ \widetilde{\kappa}\big(\widetilde{V}_t^c\big)=0$ . Fix a vector $ (x_0,\mathbf z_0)\in \widetilde{V}$ with

(123) \begin{equation}x_0>0,\end{equation}

and note that $ A \mathbf z_0 \neq 0$ as a consequence of the condition (65). For notational simplicity, write

\begin{equation*}\Phi_1=\Phi_1(x_0,\mathbf z_0), \quad \Phi_2=\Phi_2(x_0,\mathbf z_0).\end{equation*}

By (121), for all $ t \in \mathbb Q$ ,

(124) \begin{equation} \Re \bigg( \frac{e^{{\mathbf i} t x_0}-1}{{\mathbf i} x_0} (x_0)^{-D}_+ A \mathbf z_0\bigg) = \Re \bigg( \frac{e^{-{\mathbf i} t \Phi_1}-1}{{\mathbf i} \Phi_1} \Big[(\Phi_1)^{-D}_{+}A + (\Phi_1)^{-D}_{-}\overline{A} \Big] \Phi_2\bigg).\end{equation}

By continuity in t, the relation (124) holds for all $ t\in \mathbb{R}$ . Taking derivatives in t, we find that for all integers $ k\geq 0$ ,

(125) \begin{equation}\Re \Big(({\mathbf i} x)^k e^{{\mathbf i} t x_0}(x_0)^{-D}_{+}A \mathbf z_0\Big)=\Re \Big( -({-}{\mathbf i} \Phi_1)^k e^{-{\mathbf i} t \Phi_1}\Big[(\Phi_1)^{-D}_{+}A + (\Phi_1)^{-D}_{-}\overline{A} \Big] \Phi_2\Big), \quad t \in \mathbb{R}.\end{equation}

By taking $ k=4$ and $ k=0$ , respectively, we arrive at the equalities

\begin{equation*}x_0^4 \Re \Big( e^{{\mathbf i} t x_0}(x_0)^{-D}_{+}A \mathbf z_0\Big) = \Phi_1^4 \Re \Big( - e^{-{\mathbf i} t \Phi_1}\Big[(\Phi_1)^{-D}_{+}A + (\Phi_1)^{-D}_{-}\overline{A} \Big] \Phi_2\Big)\end{equation*}

\begin{equation*}= \Phi_1^4 \Re \Big( e^{{\mathbf i} t x}(x_0)^{-D}_{+}A {\mathbf z}_{0}\Big).\end{equation*}

In particular,

(126) \begin{equation}\text{ either }\Phi_1= x_0 \text{ or }\Phi_1=-x_0.\end{equation}

In view of (126), first suppose $ \Phi_1=x_0 > 0$ . Then, from (125) with $ k=0$ , and by the invertibility of the matrix $ (x_0)_+^{-D}$ ,

(127) \begin{equation}\Re \big( e^{{\mathbf i} t x_0}A \mathbf z_0\big)=\Re \big( - e^{-{\mathbf i} t x}A \Phi_2 \big),\quad t\in \mathbb{R}.\end{equation}

Hence, by choosing $ t =0$ in (127), we see that $ \Re (A \mathbf z_0)= \Re ({-}A \Phi_2)$ . On the other hand, by choosing t such that $ e^{{\mathbf i} t x_0}={\mathbf i}$ in (127), we see that $ \Im (A \mathbf z_0)= -\Im ({-}A \Phi_2)$ . This implies that $ A \mathbf z_0 = - \overline{A \Phi_2}$ . Hence,

(128) \begin{equation}\Phi_2(x_0,\mathbf z_0)=- A^{-1}\overline{A \mathbf z_0},\quad \text{if }\Phi_1(x_0,\mathbf z_0)=x_0.\end{equation}

Alternatively, suppose $ \Phi_1=-x_0$ in (126). From (125) with $ k=0$ , we obtain

\begin{equation*}\Re \big( e^{{\mathbf i} t x_0}A \mathbf z_0\big)=\Re \Big({-} e^{{\mathbf i} t x_0}\overline A \Phi_2\Big),\quad t\in \mathbb{R}.\end{equation*}

As with the relation (127), this implies that $ \Re (A \mathbf z_0) = \Re({-}\overline A \Phi_2)$ , $ \Im (A \mathbf z_0) = \Im({-}\overline A \Phi_2)$ through an appropriate choice of t. In other words, $ A \mathbf z_0 = -\overline A \Phi_2$ . Hence,

(129) \begin{equation}\Phi_2(x_0,\mathbf z_0)= -\overline A^{-1}A \mathbf z_0,\quad \text{if }\Phi_1(x_0,\mathbf z_0)=-x_0.\end{equation}

Consequently, for each $ (x,\mathbf z) \in \widetilde{V}$ with $ x>0$ (namely, under the condition (123)), either (128) or (129) holds. An analogous argument shows that $ |\Phi_1(x,\mathbf z)|=|x|$ for $ x<0$ . It also shows that, for each fixed $ (x_0,\mathbf z_0)$ with $ x_0<0$ ,

(130) \begin{equation}\Phi_2(x_0,\mathbf z_0)=\begin{cases}- \overline A^{-1} A \overline{\mathbf z},& \text{if } \Phi_1(x_0,\mathbf z_0)=x_0;\\[4pt]- A^{-1}{\overline A \mathbf z},& \text{if }\Phi_1(x_0,\mathbf z_0)=-x_0,\end{cases}\qquad\text{when }x_0<0.\end{equation}

Define the sets $ \widetilde{V}_{\pm}=\{(x,\mathbf z) \in \widetilde{V} :\Phi_1(x,\mathbf z) = \pm x\}$ . In view of (128), (129), and (130), $ \widetilde{V}_-$ and $ \widetilde{V}_+$ partition $ \widetilde{V}$ . Now define the functions

\begin{equation*}\Psi(s,\mathbf z) = \Big(x,- \overline A^{-1}A \overline{\mathbf z}\mathbf 1_{\{x<0\}} - A^{-1}\overline{A \mathbf z}\mathbf 1_{\{x>0\}}\Big),\quad \gamma(x,\mathbf z) = ({-}x,\overline{\mathbf z}).\end{equation*}

Then we can write

(131) \begin{equation}\Phi(x,\mathbf z) = \Psi(x,\mathbf z)\mathbf 1_{\widetilde{V}_+}(s,\mathbf z)+(\Psi\circ \gamma) (x,\mathbf z)\mathbf 1_{\widetilde{V}_-}(s,\mathbf z).\end{equation}

Observe that $ \Psi,\gamma$ are bijective, and also that

(132) \begin{equation}\widetilde \kappa \circ \gamma^{-1}=\widetilde \kappa.\end{equation}

Consider any set $ B \in {\mathcal B}(\mathbb{R}^p)$ . For notational simplicity, write $ [0,1]\times B= S_+ \cup S_-$ , where $ S_\pm = ([0,1]\times B)\cap \widetilde{V}_{\pm}$ . Since $ S_-$ and $ S_+$ are disjoint, we have

(133) \begin{equation}\widetilde \kappa \circ \Phi^{-1}(S_+ \cup S_-) = \widetilde \kappa \circ \Phi^{-1}(S_+) + \widetilde \kappa \circ \Phi^{-1}(S_-).\end{equation}

Therefore, by the relations (131), (132), and (133),

\begin{equation*}\widetilde \kappa([0,1]\times B) = \widetilde \kappa \circ \Phi^{-1}([0,1]\times B)=\widetilde \kappa \circ \Phi^{-1}(S_+ \cup S_-)=\widetilde \kappa \circ \Psi^{-1}(S_+) + \widetilde \kappa\circ \gamma^{-1}\circ \Psi^{-1}(S_-)\end{equation*}

\begin{equation*}= \widetilde \kappa \circ \Psi^{-1}(S_+)+ \widetilde \kappa \circ \Psi^{-1}(S_-)= \widetilde \kappa\circ \Psi^{-1}(B) = \widetilde \kappa\big([0,1]\times({-} A^{-1} \overline{A B})\big).\end{equation*}

This shows (122). Therefore, (66) holds.

Appendix D. On the uniqueness of stochastic integral representations

To characterize the uniqueness of stochastic integral representations, we first recap the concept of isomorphism between measurable spaces, as well as related notions.

In the following proposition, we establish the uniqueness of finite-second-moment stochastic integral representations of processes based on compensated Poisson random measures. The proof is omitted since it is similar to that of Theorem 2.17 in Kabluchko and Stoev [Reference Kabluchko and Stoev32], originally involving univariate integrals.

Proposition D.1 For $ \emptyset \neq T \subseteq \mathbb{R}$ , let $ X = \{X(t)\}_{t \in T}$ be an ID stochastic process with stochastic representation of the form (55). For $ i=1,2$ , let

(135) \begin{equation}\Big\{f^{(i)}_t\Big\}_{t \in T} \subseteq L^2\big(\overline{\Omega}_i,{\mathcal B}_i,\kappa_i\big)\end{equation}

be two minimal representations of X, where

(136) \begin{equation}\overline{\Omega}_i = \mathbb{R} \times \mathbb{R}^{q} \in {\mathcal B}\big(\mathbb{R} \times \mathbb{R}^{q}\big), \quad {\mathcal B}_i = {\mathcal B}\big(\overline{\Omega}_i\big).\end{equation}

1. (i) If $ \big(\overline{\Omega}_1,{\mathcal B}_1, \kappa_1\big)$ is a ( $ \sigma$ -finite) Borel space, then there is a measurable map $ \Phi\,:\,\overline{\Omega}_2 \rightarrow \overline{\Omega}_1$ such that $ \kappa_1 = \kappa_2 \circ \Phi^{-1}$ and, for all $ t \in T$ ,

   \begin{equation*} f^{(2)}_t({\boldsymbol \varpi}) = f^{(1)}_t [\Phi({\boldsymbol \varpi})] \quad \text{for $\kappa_2$-almost all ${\boldsymbol \varpi} \in \overline{\Omega}_2$}. \end{equation*}

2. (ii). If both $ \big(\overline{\Omega}_i,{\mathcal B}_i, \kappa_i\big)$ , $ i = 1,2$ , are ( $ \sigma$ -finite) Borel spaces, then the mapping $ \Phi$ in (i) is a measure space isomorphism and it is unique modulo null sets.

Appendix E. Auxiliary results

As pointed out in Remark 3.2, there are instances of ofLm—hence, stochastic processes with proper distributions—whose stochastic integral representations have integrands with deficient rank a.e. In fact, for $ x \neq 0$ , define the integrand

(137) \begin{align}L^2(\mathbb{R},M(p,\mathbb{R})) \ni \widetilde{h}_t(x) & = \Re \widetilde{g}_t(x) - \Im \widetilde{g}_t(x)\nonumber\\ & =\bigg\{\frac{\sin(tx)}{x}-\frac{(1-\cos(tx))}{x}\bigg\} |x|^{-D} \Re(A)\nonumber\\ & \quad - \bigg\{\frac{\sin(tx)}{x}+ \frac{(1-\cos(tx))}{x}\bigg\}\Big\{x^{-D}_{+}-x^{-D}_{-}\Big\}\Im(A), \quad t \in \mathbb{R}.\end{align}

Also, in (36), suppose $ \mu_{\mathbb{R}^{2p}}(d{\mathbf z}) = \delta_{{\mathbf 1}}(d{\mathbf z})$ , where $ {\mathbf 1} = (1,\ldots,1)^* \in \mathbb{R}^{2p}$ . Then, for $ {\mathbf 1} \in \mathbb{R}^{p}$ and by the relation (79), we can express the rhofLm $ \widetilde{X}_H$ as

(138) \begin{equation}\mathbb{E} \widetilde{X}_H(t)\widetilde{X}_H(t)^* = \int_{\mathbb{R}} \widetilde{h}_t(x){\mathbf 1}{\mathbf 1}^* \widetilde{h}_t(x)^* d x, \quad t \neq 0.\end{equation}

Note that the integrand on the right-hand side of (138) has rank 1 a.e. However, the properness condition (30) may still be satisfied, as we show next.

Proof. Fix $ t \neq 0$ . Let

(139) \begin{equation}M\,:\!=\, \Im(A){\mathbf 1}{\mathbf 1}^* \Im(A)^*=\left(\begin{array}{c@{\quad}c}1 & 1\\1 & 1\end{array}\right).\end{equation}

For $ \widetilde{h}_t$ as in (137), the expression (138) implies that

\begin{equation*}\mathbb{E} \widetilde{X}_H(t)\widetilde{X}_H(t)^* = \int_{\mathbb{R}} \bigg\{ \bigg(\frac{\sin(t x)}{x}\bigg)^2 + \bigg( \frac{1- \cos(t x)}{x}\bigg)^2 \bigg\} |x|^{-D} M |x|^{-D^*} d x\end{equation*}

\begin{equation*}= \int_{\mathbb{R}} 2\frac{(1- \cos(t x))}{x^2} |x|^{-D} M |x|^{-D^*} d x,\qquad \end{equation*}

where we make use of the fact that

\begin{equation*}\frac{\sin(t x)(1- \cos(t x))}{x^2}\end{equation*}

is an odd function. For $ - 1 < \delta < 1$ , we can write

\begin{equation*}\mathbb{R} \ni \int_{\mathbb{R}} 2\frac{(1- \cos(t x))}{x^2} |x|^{-\delta} d x= |t|^{1+\delta}\int_{\mathbb{R}} 2\frac{(1- \cos(y))}{y^2} |y|^{-\delta} d y =\,:\, |t|^{1+\delta}\beta(\delta),\end{equation*}

where we make the change of variable $ y = tx$ . Then,

\begin{equation*}\det\big(\mathbb{E} \widetilde{X}_H(t)\widetilde{X}_H(t)^*\big) = |t|^{2+2(d_1+d_2)}\big(\beta(2d_1)\beta(2d_2) - \beta^2(d_1 + d_2)\big).\end{equation*}

In particular, for all $ t \neq 0$ , $ \det\big(\mathbb{E} \widetilde{X}_H(t)\widetilde{X}_H(t)^*\big) = 0$ if and only if

\begin{equation*}\beta(2d_1)\beta(2d_2) - \beta^2(d_1 + d_2) = 0.\end{equation*}

However, the latter condition cannot hold for every $ - \frac{1}{2} < d_1, d_2 < \frac{1}{2}$ (cf. Remark 4.1 in Didier and Pipiras [Reference Didier and Pipiras23]). This establishes the claim.

The following proposition is used in the proof of Proposition 3.1. It establishes orthogonal-increment random measures that can be used to yield integral representations for maofLm and rhofLm (cf. Rozanov [Reference Rozanov59], Section 1.3).

Proof. Statement (i) can be shown by means of a direct adaptation of the statement of Theorem 3.5 in Marquardt and Stelzer [Reference Marquardt and Stelzer46], which in turn is based on a multivariate generalization of Rozanov [Reference Rozanov59], Theorem 2.1. Therefore we only show (140) and (141). It suffices to establish the statement over intervals (a, b], $ - \infty < a \leq b < \infty$ . Note that the characteristic function of $ \Phi_{\widetilde{{\mathcal M}}}(a,b]$ at $ {\mathbf u } \in \mathbb{R}^p$ is given by

\begin{equation*}\exp\Bigg\{ \int_{\mathbb{R}}\int_{\mathbb{R}^p} \Bigg( e^{{\mathbf i} {\mathbf u}^* \frac{1}{2\pi}\big(\frac{e^{{\mathbf i} s b}-e^{{\mathbf i} s a}}{{\mathbf i} s}\big){\mathbf z}}- 1 - {\mathbf i} {\mathbf u}^* \frac{1}{2\pi}\Bigg(\frac{e^{{\mathbf i} s b}-e^{{\mathbf i} s a}}{{\mathbf i} s}\Bigg){\mathbf z}\Bigg) \mu(d{\mathbf z})ds\Bigg\}.\end{equation*}

By taking the first derivative with respect to $ {\mathbf u}$ and setting $ {\mathbf u} = {\mathbf 0}$ , we conclude that $ \mathbb{E} \Phi_{\widetilde{{\mathcal M}}}(a,b] = {\mathbf 0}$ , which proves (140). Moreover, by taking the second derivative with respect to $ {\mathbf u}$ and setting $ {\mathbf u} = {\mathbf 0}$ , we obtain

\begin{equation*}\mathbb{E} \Phi_{\widetilde{{\mathcal M}}}(a,b]\Phi_{\widetilde{{\mathcal M}}}(a,b]^* = \frac{1}{4\pi^2}\Bigg(\int_{\mathbb{R}} \Bigg|\frac{e^{{\mathbf i} s b}-e^{{\mathbf i} s a}}{{\mathbf i} s}\Bigg|^2 ds \Bigg)\Bigg(\int_{\mathbb{R}^p} {\mathbf z}{\mathbf z}^* \mu(d{\mathbf z})\Bigg).\end{equation*}

Thus, by setting $ a = 0$ and $ b = y$ under the condition (39), we further conclude that (141) holds, where we make use of the fact that

\begin{equation*}\frac{1}{4 \pi^2}\int_{\mathbb{R}} \bigg| \frac{e^{{\mathbf i} x y}-1}{{\mathbf i} x}\bigg|^2 dx = y\end{equation*}

(see the expression (9.7) in Taqqu [Reference Taqqu66]). Likewise, the orthogonality of the increments can be verified by Parseval’s theorem.

Statement (ii) can be established by a similar argument. In particular, the random measure (144) is $ \mathbb{R}^p$ -valued because the integrand

\begin{equation*}\frac{e^{-{\mathbf i} xa}-e^{-{\mathbf i} xb}}{{\mathbf i} x}\end{equation*}

is a Hermitian function.

The following lemma is used in Example 4.4.

The following lemma is used in Section 4.

Proof. This is a consequence of the fact that, for any m and any $ t_1,\ldots,t_m$ ,

\begin{equation*}\int_{\mathbb{R}} \int_{\mathbb{C}^p} \Bigg(e^{{\mathbf i} 2 \sum^{m}_{j=1}{\mathbf u}^*_j \Re (\widetilde{g}_{t_j}(x){\mathbf z})}-1 -{\mathbf i} 2 \sum^{m}_{j=1}{\mathbf u}^*_j \Re (\widetilde{g}_{t_j}(x){\mathbf z})\Bigg)\mu(d{\mathbf z}) dx\end{equation*}

\begin{equation*}=\int_{\mathbb{R}} \int_{\mathbb{C}^p} \Bigg(e^{{\mathbf i} 2 \sum^{m}_{j=1}{\mathbf u}^*_j \Re (\widetilde{g}_{t_j}(x^{\prime}){\mathbf z}^{\prime})}-1 -{\mathbf i} 2 \sum^{m}_{j=1}{\mathbf u}^*_j \Re (\widetilde{g}_{t_j}(x^{\prime}){\mathbf z}^{\prime})\Bigg)\mu(\overline{d{\mathbf z}^{\prime}}) dx,\end{equation*}

where we make the change of variables $ \big(x^{\prime},\mathbf z^{\prime}\big)=({-}x,\overline{\mathbf z})$ .

Appendix F. Tempered operator-stable Lévy measures

More specifically, the framework of tempered operator-stable Lévy measures is used in Proposition 3.3. To recap it, we first recall the notion of operator-stable distributions (see, for instance, Meerschaert and Scheffler [Reference Meerschaert and Scheffler50]). Let $ B\in M(p,\mathbb{R})$ , $ \text{eig}B\subseteq \{z\in \mathbb{C}: \Re z \in(1/2,\infty)\}$ . Consider a norm $ \| \cdot \|_B$ on $ \mathbb{R}^q$ with unit sphere $ S_0\,:\!=\,\{x:\|x\|_B=1\}$ satisfying the following:

1. (i) for each $ x\in \mathbb{R}^q\setminus\{0\}$ , $ r\mapsto \big\| r^B x\big\|$ is monotonically increasing for $ r>0$ ;

2. (ii) $ (r,\boldsymbol\theta)\mapsto r^B\boldsymbol\theta$ from $ \mathbb{R}_+\times S_B$ to $ \mathbb{R}^q\setminus\{0\}$ is a homeomorphism.

(See Lemma 6.1.5 in Meerschaert and Scheffler [Reference Meerschaert and Scheffler50].) A full operator-stable distribution (recall that full means a distribution is not supported on any proper subspace of $ \mathbb{R}^p$ ) has a Lévy measure $ \mu$ that can be written as

(146) \begin{equation}\mu_B(A) = \int_{S_0}\int_{\mathbb{R}_+} 1_A\big(r^B\boldsymbol\theta\big)\frac{dr}{r^2}\lambda(d\boldsymbol\theta),\end{equation}

where $ \lambda$ is a finite Borel measure on $ S_0$ (see Theorem 7.2.5 in Meerschaert and Scheffler [Reference Meerschaert and Scheffler50]). Provided $ \Re \lambda_q(B)<1$ , it can be shown that

\begin{equation*}\int_{\|\mathbf z\|\geq 1} \|\mathbf z\| \mu_B(d\mathbf z)<\infty\end{equation*}

(see Corollary 8.2.6 in Meerschaert and Scheffler [Reference Meerschaert and Scheffler50] and Theorem 25.3 in Sato [Reference Sato62]). In particular, under the assumption $ \Re\lambda_q(B)<1$ , the integral

(147) \begin{equation}\psi_B({\mathbf u}) = \int_{\mathbb{R}^q} \Big(e^{{\mathbf i} \langle{\mathbf u}, {\mathbf z}\rangle}-1-{\mathbf i} \langle{\mathbf u}, {\mathbf z}\rangle \Big)\mu_B(d {\mathbf z}),\quad {\mathbf u} \in \mathbb{R}^q\end{equation}

(cf. (82)), exists, and the Lévy symbol $ \psi_B$ satisfies $ \psi(c^B\mathbf u) = c\psi(\mathbf u);$ i.e., the function $ e^{\psi(\mathbf u)}$ is the characteristic function of a strictly operator-stable distribution $ \nu_B$ (see Kremer and Scheffler [Reference Kremer and Scheffler36], p. 4085). Based on $ \nu_B$ , we define the random measures $ L_B$ used in Proposition 3.3 as $ \mathbb{R}^p$ -valued (in (47)) or $ \mathbb{C}^p$ -valued (in (50)) ID independently scattered random measures on $ (\mathbb{R},\mathcal B(\mathbb{R}))$ generated by $ \nu_B$ and the Lebesgue measure on $ \mathbb{R}$ in the sense of Example 3.7(a) and Remark 3.8 in Kremer and Scheffler [Reference Kremer and Scheffler35].

For the purposes of retaining second moments and constructing asymptotically o.s.s. instances of ofLm, we consider tempered counterparts of (146), where the associated Lévy measure is given by

(148) \begin{equation}\mu_{B,q}(A) = \int_{S_0}\int_{\mathbb{R}_+} 1_A\big(r^B\boldsymbol\theta\big)q(r,\boldsymbol\theta)\frac{dr}{r^2}\lambda(d\boldsymbol\theta), \qquad r\in(0,\infty),\quad \boldsymbol\theta\in S_0.\end{equation}

In (148), $ q\,:\,(0,\infty)\times S_0\to [0,1]$ is any Borel measurable function such that, for $ \lambda(d\boldsymbol \theta)$ -a.e. $ \boldsymbol \theta \in S_0$ ,

(149) \begin{equation}q(\cdot,\boldsymbol\theta) \text{ decays to 0}\end{equation}

sufficiently fast as $ r\to \infty$ to guarantee that $ \mu_{B,q}$ has second moments and

(150) \begin{equation}q(0^+,\boldsymbol \theta)=1\end{equation}

for each $ \boldsymbol \theta\in S_0$ (for instance, $ q(r,\boldsymbol\theta)=\mathbf 1_{\{|r|\leq 1\}}$ , $ \boldsymbol \theta \in S_0$ ). When q is a completely monotone function (that is,

\begin{equation*}({-}1)^k \frac{d^k}{dt^k}q(t,\boldsymbol\theta)>0\end{equation*}

for all $ t>0$ and each $ k\geq 0$ ), Lévy measures defined by (148) are called tempered operator-stable Lévy measures; they are studied in Ali [Reference Ali2] (see also the seminal work of Rosiński [Reference Rosiński57] for the tempered stable case). For conditions for the existence of moments in the tempered stable case, see Rosiński [Reference Rosiński57], Proposition 2.7. For the tempered operator-stable case, see Ali [Reference Ali2], Korollar 3.2.5.