2. Model and results

Let $Q= [q_{ij}]_{i,j\in{\mathscr{K}}}$ , with ${\mathscr{K}}:=\{1,\dots,k_\circ\}$ , be an irreducible stochastic rate matrix, and let

(2.1) \begin{equation}\pi\,:=\,\{\pi_1,\dots,\pi_{k_\circ}\!\}\end{equation}

denote its (unique) invariant probability distribution. We fix a constant $\alpha>0$ . For each $n\in N$ , let $J^n$ denote the finite-state irreducible continuous-time Markov chain with state space ${\mathscr{K}}$ and transition rate matrix $n^{\alpha}Q$ . In addition, for each $n\in\mathbb{N}$ and $k\in{\mathscr{K}}$ , let ${\mathfrak{X}}^n \subset\mathbb{R}^d$ be a countable set with no accumulation points in $\mathbb{R}^{d}$ , and let $R^n_k=\bigl[r^n_k(x,y)\bigr]_{x,y\in{\mathfrak{X}}^n}$ be a stochastic rate matrix which gives rise to a non-explosive, irreducible, continuous-time Markov chain.

The transition matrices $\{R^n_k\}$ satisfy the following structural assumptions.

Hypothesis 2.1 is assumed throughout the paper without further mention. We refer the reader to Examples 3.1 to 3.3 for examples of verification of the conditions in Parts (c) and (d).

Consider the stochastic rate matrix $S^n$ on ${\mathfrak{X}}^n\times{\mathscr{K}}$ whose elements are defined by

\begin{equation*}s^n\bigl((x,i),(y,j)\bigr) \,:=\, \begin{cases}r^n_i(x,y) \quad &\text{if }\ i=j, \\[4pt] n^{\alpha} q_{ij} \quad &\text{if}\ x=y, \\[4pt] 0 \quad &\text{otherwise},\end{cases}\end{equation*}

for $x,y\in{\mathfrak{X}}^n$ and $i,j\in{\mathscr{K}}$ . This defines a non-explosive, irreducible Markov chain $(X^n,J^n)$ , where $J^n$ is as described in the preceding paragraph.

In order to simplify some algebraic expressions, we often use the notation

\begin{equation*}\tilde{r}^n_k(x,z) = r^n_k(x,x+z).\end{equation*}

Definition 2.1. Let $\beta:=\max\{1/2,1 - \alpha/2\}$ be fixed. With $x^n_*$ as in Hypothesis 2.1(d), we define the scaled process

\begin{equation*}\widehat{X}^n\,:=\, \frac{X^n - {x}^n_{*}}{n^{\beta}}.\end{equation*}

The state space of $\widehat{X}^n$ is given by

\begin{equation*}\widehat{{\mathfrak{X}}}^n \,:=\, \{\hat{x}^n(x) : x\in{\mathfrak{X}}^n\},\end{equation*}

where $\hat{x} = \hat{x}^n(x) := n^{-\beta}(x - x^n_*)$ for $x\in\mathbb{R}^d$ .

Naturally, $(\widehat{X}^n,J^n)$ is a Markov process, and its extended generator is given by

(2.2) \begin{equation}\widehat{\mathscr{\!\!L}}^{\,\,n} f(\hat{x},k) = {\mathscr{L}}_k^{\kern1.6pt n} f(\hat{x},k)+ {\mathcal{Q}}^n f(\hat{x},k),\quad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

for $f\in C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ , where

(2.3) \begin{equation}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n}f(\hat{x},k) &\,:=\, \sum_{z\in{\mathscr{Z}}^{\,\,n}(x)}\tilde{r}^n_k(n^{\beta}\hat{x} + {x}^n_*,z)\bigl(f(\hat{x} + n^{-\beta}z,k) - f(\hat{x},k)\bigr), \\[4pt] {\mathcal{Q}}^nf(\hat{x},k) &\,:=\, \sum_{\ell\in{\mathscr{K}}}n^{\alpha}q_{k\ell}\bigl(f(\hat{x},\ell) - f(\hat{x},k)\bigr) = \sum_{\ell\in{\mathscr{K}}}n^{\alpha}q_{k\ell}f(\hat{x},\ell).\end{aligned}\end{equation}

One can clearly see that $\,\,\widehat{\mathscr{\!\!L}}^{\,\,n} f$ and ${\mathscr{L}}_k^{\kern1.6pt n}f$ are well defined for $f\in C_b(\mathbb{R}^d)$ , by viewing f as a function on $\mathbb{R}^{d}\times{\mathscr{K}}$ which is constant with respect to its second argument.

Throughout the paper, x and $\hat{x}$ are generic elements of ${\mathfrak{X}}^n$ (or $\mathbb{R}^{d}$ ) and $\widehat{{\mathfrak{X}}}^n$ , respectively.

2.1. Exponential ergodicity

In this subsection, we provide a sufficient condition for the joint process $(\widehat{X}^n,J^n)$ to be exponentially ergodic. We refer the reader to [Reference Meyn and Tweedie20] for the definition of exponential ergodicity and the relevant Foster–Lyapunov criteria. We introduce the following operator, which corresponds to the generator of the ‘averaged’ process.

Definition 2.2. Let

\begin{equation*}\bar{r}^n(x,z) \,:=\, \sum_{k\in{\mathscr{K}}}\pi_k \tilde{r}^n_k(x,z),\end{equation*}

with $\pi_k$ as in (2.1), and

(2.4) \begin{equation}{\mathscr{Z}}^{\,\,n} \,:=\, \cup_{x\in{\mathfrak{X}}^n} \cup_{k\in{\mathscr{K}}} {\mathscr{Z}}^{\,\,n}_k(x).\end{equation}

We define $\overline{\mathscr{L}}^{\,n}: C_b(\mathbb{R}^d\times{\mathscr{K}}\,\kern1.3pt)\mapsto C_b(\mathbb{R}^d\times{\mathscr{K}}\,\kern1.3pt)$ by

(2.5) \begin{equation}\overline{\mathscr{L}}^{\,n} f(\hat{x},k) \,:=\,\sum_{z\in{\mathscr{Z}}^{\,\,n}}\bar{r}^n(n^{\beta}\hat{x} + {x}^n_*,z)\bigl(f(\hat{x} + n^{-\beta}z,k) - f(\hat{x},k)\bigr),\quad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

for $f\in C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ .

In the following theorem, we show that if $\overline{\mathscr{L}}^{\,n}$ satisfies a Foster–Lyapunov inequality with a suitable Lyapunov function, then the original joint process $(\widehat{X}^n,J^n)$ is exponentially ergodic. The proof is given in Section 4.

A function $f:\mathbb{R}^d \mapsto \mathbb{R}_+$ is called norm-like if $f(x) \rightarrow\infty$ as $\vert{x}\vert\rightarrow\infty$ ; see, for example, [Reference Meyn and Tweedie20, Section 1.3].

Theorem 2.1. Suppose that there exist a sequence of nonnegative norm-like functions $\{{\mathcal{V}}^n\in C(\mathbb{R}^d): n\in\mathbb{N}\}$ , $n_0\in\mathbb{N}$ , and some positive constants $\varepsilon_0$ , C, $\overline{C}_1$ , $\overline{C}_2$ , not depending on n, such that

(2.6) \begin{equation}\begin{aligned}(1 + \vert{x}\vert)\,\bigl\vert{{\mathcal{V}}^n(x + y) - {\mathcal{V}}^n(x)}\bigr\vert &\,\leq\, C\vert{y}\vert\bigl(1 + {\mathcal{V}}^n(x)\bigr), \\[3pt] \bigl(1 + \vert{x}\vert^2\bigr)\,\bigl\vert{{\mathcal{V}}^n\bigl(x + y + z\bigr) - {\mathcal{V}}^n(x+ y)\mspace{100mu}} &\\ -{\mathcal{V}}^n(x+ z) + {\mathcal{V}}^n(x)\bigr\vert &\,\leq\, C\vert{y}\vert\vert{z}\vert\bigl(1 + {\mathcal{V}}^n(x)\bigr),\end{aligned}\end{equation}

for any $y,z\in B_0(\varepsilon_0)\setminus\{0\}$ , $x\in\mathbb{R}^d$ , and $n\in\mathbb{N}$ , and

(2.7) \begin{equation}\overline{\mathscr{L}}^{\,n} {\mathcal{V}}^n(\hat{x}) \,\leq\, \overline{C}_1 - \overline{C}_2{\mathcal{V}}^n(\hat{x})\qquad \forall\,\hat{x}\in{\widehat{{\mathfrak{X}}}^n},\ \forall\, n > n_0.\end{equation}

Then there exist functions $\widehat{{\mathcal{V}}}^n\in C(\mathbb{R}^d\times{\mathscr{K}}\,)$ and positive constants $\widehat{C}_1$ , $\widehat{C}_2$ , and $n_1\in\mathbb{N}$ such that, for all $n\geq n_1$ , we have

(2.8) \begin{equation}\frac{1}{2}\bigl({\mathcal{V}}^n(\hat{x}) - 1\bigr) \,\leq\, \widehat{{\mathcal{V}}}^n(\hat{x},k) \,\leq\,\frac{3}{2}{\mathcal{V}}^n(\hat{x}) +\frac{1}{2}\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

and

(2.9) \begin{equation}\widehat{\mathscr{\!\!L}}^{\,\,n} \widehat{{\mathcal{V}}}^n(\hat{x},k) \,\leq\, \widehat{C}_1 - \widehat{C}_2\widehat{{\mathcal{V}}}^n(\hat{x},k) \qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\ \forall\, n>n_1.\end{equation}

As a consequence, $(\widehat{X}^n,J^n)$ is exponentially ergodic for all $n> n_1$ , and its invariant probability distributions are tight.

Remark 2.2. It follows from the proof of Theorem 2.1 that $\widehat{C}_2$ can be selected arbitrarily close to $\overline{C}_2$ , so the rates of convergence of the ‘averaged’ system and the Markov-modulated one become asymptotically close.

Remark 2.3. A sufficient condition for a function ${\mathcal{V}}^n\in C^{2,1}(\mathbb{R}^{d})$ to satisfy (2.6) is

(2.10) \begin{equation}\begin{aligned}\vert{\nabla {\mathcal{V}}^n(x)}\vert &\,\leq\, c\frac{1+{\mathcal{V}}^n(x)}{1+\vert{x}\vert}\quad\text{and}\\[5pt] \vert{\nabla^2 {\mathcal{V}}^n(x)}\vert +\bigl[{\mathcal{V}}^n\bigr]_{2,1;B_{\epsilon}(x)}&\,\leq\, c\frac{1+{\mathcal{V}}^n(x)}{1+\vert{x}\vert^2}\quad\forall\,x\in\mathbb{R}^{d},\end{aligned}\end{equation}

for some fixed positive constants $\epsilon$ and c.

In the next corollary, we relax the incremental growth hypothesis in Hypothesis 2.1(c). The proof is contained in Section 4. In Example 3.2, we show that this result can be applied in the study of exponential ergodicity for Markov-modulated $M/M/n +M$ queues.

We replace Hypothesis 2.1(c) by the following weaker assumption.

Assumption 2.1 Suppose that Parts (a), (b), and (d) of Hypothesis 2.1 are satisfied, and $\tilde r^n_k$ can be decomposed into

\begin{equation*}\tilde{r}^n_k(x,z) = \phi^n_k(x,z) + \psi^n_k(x,z),\quad x\in{\mathfrak{X}}^n, \quad z\in{\mathscr{Z}}^{\,\,n}_k(x),\end{equation*}

where $\phi^n_k(x,z)$ and $\psi^n_k(x,z)$ , $k\in{\mathscr{K}}$ , are locally bounded functions on ${\mathfrak{X}}^n\times{\mathscr{Z}}^{\,\,n}$ . In addition, using without loss of generality the same constant, there exist $\delta_1,\delta_2 \in[0,1]$ such that

(2.11) \begin{equation}\vert{\psi^n_k(x,z) - \psi^n_k(y,z)}\vert\,\leq\, C_0\bigl(n^{\alpha/2} + \vert{x - y}\vert^{\delta_1}\bigr) \qquad \forall\,k\in{\mathscr{K}}, \ \forall\,x,y\in{\mathfrak{X}}^n, \ \forall\,z\in{\mathscr{Z}}^{\,\,n},\end{equation}

and

(2.12) \begin{equation}\vert{\psi^n_k(x,z)}\vert \,\leq\, C_0\bigl(n^{1\vee\alpha/2}+ \vert{x - x^n_*}\vert^{\delta_2}\bigr) \qquad \forall\,k\in{\mathscr{K}},\ \forall\,(x,z)\in{\mathfrak{X}}^n\times{\mathscr{Z}}^{\,\,n},\end{equation}

with $x^n_*\in\mathbb{R}^d$ as in Hypothesis 2.1(d), and for $n\in\mathbb{N}$ .

Corollary 2.1. Grant Assumption 2.1. Let ${\mathcal{G}}^n_k: C_b(\mathbb{R}^d\times{\mathscr{K}}\,)\mapsto C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ be defined by

(2.13) \begin{equation}{\mathcal{G}}^n_k f(\hat{x},k) \,:=\,\sum_{z\in{\mathscr{Z}}^{\,\,n}}\bigl(\phi^n_k(n^{\beta}\hat{x} + {x}^n_*,z)+ \bar{\psi}^n(n^{\beta}\hat{x} + {x}^n_*,z)\bigr)\bigl(f(\hat{x} + n^{-\beta}z,k) - f(\hat{x},k)\bigr)\end{equation}

for $(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}$ and $f\in C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ , and with $x^n_*$ as in Assumption 2.1, where $\bar{\psi}^n(x,z) := \sum_{k\in{\mathscr{K}}}\pi_k\psi^n_k(x,z)$ . Suppose that (2.6) holds with the second inequality replaced by

\begin{equation*}\bigl(1 + \vert{x}\vert^{1+\delta_2}\bigr)\,\bigl\vert{{\mathcal{V}}^n\bigl(x + y + z\bigr) - {\mathcal{V}}^n(x+ y) -{\mathcal{V}}^n(x+ z) + {\mathcal{V}}^n(x)}\bigr\vert \,\leq\, C\vert{y}\vert\vert{z}\vert\bigl(1 + {\mathcal{V}}^n(x)\bigr),\end{equation*}

where $\delta_2$ is as in Assumption 2.1, and there exist $n_2\in\mathbb{N}$ and some positive constants $C_1$ and $C_2$ such that

(2.14) \begin{equation}{\mathcal{G}}^n_k {\mathcal{V}}^n(\hat{x}) \,\leq\, C_1 - C_2{\mathcal{V}}^n(\hat{x})\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}, \ \forall\, n > n_2.\end{equation}

Then the results in (2.8) and (2.9) hold.

In the following corollary, we show that under some stronger assumptions on the transition rate functions and the scaling parameters, (2.6) can be weakened. The proof is given in Section 4.

Corollary 2.2. Grant parts (a) and (b) of Hypothesis 2.1, and suppose that $r^n_k$ satisfies

(2.15) \begin{equation}\bigl\vert{r^n_k (x,x+z) - r^n_k (x^{\prime},x^{\prime}+z)}\bigr\vert\,\leq\, C_0\bigl(1+\vert{x-x^{\prime}}\vert\wedge n\bigr)\end{equation}

and

(2.16) \begin{equation}r^n_k (x^n_*,x^n_*+z)\,\leq\, C_0 n.\end{equation}

If in the assumptions of Theorem 2.1 we replace (2.6) by

(2.17) \begin{equation}\begin{aligned}\bigl\vert{{\mathcal{V}}^n(x + y) - {\mathcal{V}}^n(x)}\bigr\vert &\,\leq\, C\vert{y}\vert\bigl(1 + {\mathcal{V}}^n(x)\bigr), \\\bigl\vert{{\mathcal{V}}^n\bigl(x + y + z\bigr) - {\mathcal{V}}^n(x+ y) -{\mathcal{V}}^n(x+ z) + {\mathcal{V}}^n(x)}\bigr\vert &\,\leq\, C\vert{y}\vert\vert{z}\vert\bigl(1 + {\mathcal{V}}^n(x)\bigr),\end{aligned}\end{equation}

then, provided $\beta$ and $\alpha$ satisfy $2\beta+\alpha>2$ , the conclusions of the theorem still hold.

Note that (2.17) is satisfied for exponential functions.

Remark 2.4. The transition rates of multiclass $M/M/n$ queues, that is, the model in Example 3.2 with no abandonment ( $\gamma_i(k)\equiv0$ ), satisfy (2.15) and (2.16). Uniform exponential ergodicity of this model (with spare capacity, or equivalently, positive safety staffing) is established in [Reference Arapostathis, Hmedi and Pang4] using exponential Lyapunov functions. Thus, we may use exponential Lyapunov functions in (2.7), and take advantage of the results in [Reference Arapostathis, Hmedi and Pang4] to establish exponential ergodicity of Markov-modulated multiclass $M/M/n$ queues with positive safety staffing using the Lyapunov functions in [Reference Arapostathis, Hmedi and Pang4]. We leave it to the reader to verify that for $\alpha\geq1$ , we can in fact establish uniform exponential ergodicity over all work-conserving scheduling policies. For $\alpha<1$ the discontinuities allowed in the policies need to be restricted.

Extending this to the classes of multiclass multi-pool models studied in [Reference Hmedi, Arapostathis and Pang12] is also possible.

2.2. Steady-state approximations

Here, we use a function $\xi^n_z(x,k)$ for $(x,z)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ and $k\in{\mathscr{K}}$ which interpolates the transition rates in the sense that

\begin{equation*}\xi^n_{z}(x,k) = r^n_k(x,x+z) \quad \text{if }\ x,x+z\in{\mathfrak{X}}^n.\end{equation*}

Recall the definition of ${\mathscr{Z}}^{\,\,n}$ in (2.4). It is clear that for $z\notin{\mathscr{Z}}^{\,\,n}$ we may let $\xi^n_z\equiv0$ . Thus

\begin{equation*}{\mathscr{Z}}^{\,\,n} = \bigl\{z\in\mathbb{R}^{d}: \exists\, x, k\ \text{such}\ \text{that}\ \xi^n_{z}(x,k)>0\}.\end{equation*}

This of course also implies that

(2.18) \begin{equation}\xi^n_{z}(x,k) = 0 \quad\text{if}\ \vert{z}\vert> m_0\end{equation}

by Hypothesis 2.1(a).

We let ${\mathscr{I}}:=\{1,\dotsc,d\}$ , and define

(2.19) \begin{equation}\begin{aligned}\Xi^n(x,k) &\,:=\, \sum_{z\in{\mathscr{Z}}^{\,\,n}} z\,\xi^n_{z}(x,k),\\[5pt] \Gamma^n_{ij}(x,k) &\,:=\, \sum_{z\in{\mathscr{Z}}^{\,\,n}}z_iz_j\xi^n_z(x,k),\quad i,j\in{\mathscr{I}},\end{aligned}\end{equation}

for $(x,k)\in \mathbb{R}^d\times{\mathscr{K}}$ .

We impose the following structural assumptions on the function $\xi^n$ .

Assumption 2.2 The following hold.

1. (i) The cardinality of the set $\{z\in\mathbb{R}^{d}: \xi^n_{z}(x,k)>0\}$ does not exceed $\widetilde{N}_0$ .

2. (ii) For each $n\in\mathbb{N}$ , there exists $x^n_*\in\mathbb{R}^{d}$ satisfying

   (2.20) \begin{equation}\sum_{k\in{\mathscr{K}}}\pi_k\Xi^n({x}^n_*,k) = 0.\end{equation}

3. (iii) The function $\xi^n_z$ is uniformly Lipschitz continuous in its first argument; that is, there exists some positive constant $\widetilde{C}$ such that

   (2.21) \begin{equation}\vert{\xi^n_z(x,k) - \xi^n_z(y,k)}\vert\,\leq\, \widetilde{C}\vert{x - y}\vert \qquad \forall\,k\in{\mathscr{K}},\ \forall\,x,y\in\mathbb{R}^d, \ \forall\,z\in{\mathscr{Z}}^{\,\,n},\end{equation}
   for all $n\in\mathbb{N}$ . In addition, using without loss of generality the same constant, we assume that
   (2.22) \begin{equation}\max_{z\in\mathbb{R}^{d}}\,\xi^n_z(x^n_*, k)\,\leq\,\widetilde{C} n\qquad \forall\, k\in{\mathscr{K}},\ \forall\,n\in\mathbb{N}.\end{equation}

4. (iv) The matrix $\Gamma^n(x^n_*,k)$ is positive definite, and

   (2.23) \begin{equation}\frac{1}{n}\, \Gamma^n(x^n_*, k) \,\xrightarrow[n\to\infty]{}\, \bar{\Gamma}(k),\end{equation}
   where $\bar{\Gamma}(k)$ is also a positive definite $d\times d$ matrix, for all $k\in{\mathscr{K}}$ .

We note here that the nondegeneracy hypothesis in Assumption 2.2(iv) is used in Lemma 5.3 to derive gradient estimates of the solution of a Poisson equation.

Remark 2.5. Equation (2.21) is of course much stronger than Hypothesis 2.1(c). This is needed for the results in this section which rely on certain Schauder estimates for solutions of the Poisson equation for the generator of an approximating diffusion equation.

Let $\{A^n_z : z\in {\mathscr{Z}}^{\,\,n}\}$ be a family of independent unit-rate Poisson processes, independent of $J^n$ , and $\tilde{A}^n_{z}(t) := A^n_{z}(t) - t$ . Then the d-dimensional process $X^n(t)$ is governed by the equation

\begin{equation}\begin{aligned}X^n(t) & = X^n(0) +\sum_{z\in{\mathscr{Z}}^{\,\,n}} z\, A^n_{z}\biggl(\int_0^t \xi^n_{z}\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s}\biggr)\nonumber\\ & = X^n(0) + M^n(t) +\int_0^t \Xi^n\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s} ,\end{aligned}\end{equation}

where

\begin{equation*}M^n(t)\,:=\,\sum_{z\in{\mathscr{Z}}^{\,\,n}}z \,\tilde{A}^n_{z}\biggl(\int_0^t\xi^n_{z}\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s}\biggr).\end{equation*}

Note that $M^n(t)$ is a local martingale with respect to the filtration

\begin{equation*}\begin{aligned}\mathcal{F}^n_t &\,:=\, \sigma\biggl\{X^n(0),J^n(s),\tilde{A}^n_{z}\biggl(\int_0^t\xi^n_{z}\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s}\biggr), \\[5pt] &\mspace{120mu} \int_0^t\xi^n_{z}\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s}: z\in{\mathscr{Z}}^{\,\,n}, s\leq t\biggr\}.\end{aligned}\end{equation*}

The locally predictable quadratic variation of $M^n$ satisfies

\begin{equation*}\langle M^n \rangle (t) = \int_0^t \Gamma^n\bigl(X^n(s),J^n(s)\bigr)\,\textrm{d}{s},\quad t\geq0,\end{equation*}

where the function $\Gamma^n = [\Gamma^n_{ij}]: \mathbb{R}^d\times{\mathscr{K}} \mapsto \mathbb{R}^{d\times d}$ is given in (2.19).

By (2.21), it is evident that given $x^n(0) \in\mathbb{R}^d$ , there exists a unique solution $x^n(t)$ satisfying

\begin{equation*}x^n(t) = x^n(0) + \sum_{k\in{\mathscr{K}}}\pi_k\int_0^t\Xi^n(x^n(s),k)\,\textrm{d}{s} .\end{equation*}

We refer to this as the nth ‘averaged’ fluid model.

In this section, the scaled process is defined as in Definition 2.1, with the exception that $x^n_*\in\mathbb{R}^{d}$ is as specified in Assumption 2.2. Note that in the extended generator in (2.2) and (2.3) we may replace $\tilde{r}^n_k(n^{\beta}\hat{x} + {x}^n_*,z)$ by $\xi^n_z(n^{\beta}\hat{x} + {x}^n_*, k)$ . It is evident from (2.24) that $\widehat{X}^n$ satisfies

(2.24) \begin{equation}\widehat{X}^n(t) = \widehat{X}^n(0) + \widehat{M}^n(t)+ \int_0^t\widehat{\Xi}^n\bigl(\widehat{X}^n(s),J^n(s)\bigr)\,\textrm{d}{s},\end{equation}

where

(2.25) \begin{equation}\widehat{M}^n\,:=\, \frac{M^n}{n^{\beta}},\qquad \text{and} \quad\widehat{\Xi}^n(\hat{x},k)\,:=\,\frac{\Xi^n(n^{\beta}\hat{x} + {x}^n_{*},k)}{n^{\beta}},\qquad (\hat{x},k)\in\mathbb{R}^d\times{\mathscr{K}}.\end{equation}

The locally predictable quadratic variation of $\widehat{M}^n$ is given by

\begin{equation*}\langle \widehat{M}^n \rangle (t) = \int_0^{t}\bar{\Gamma}^n\bigl(\widehat{X}^n(s),J^n(s)\bigr)\,\textrm{d}{s},\qquad t\geq0,\end{equation*}

with

(2.26) \begin{equation}\bar{\Gamma}^n(\hat{x},k) \,:=\, \frac{1}{n^{2\beta}}\,\Gamma^n(n^{\beta}\hat{x} + {x}^n_*,k),\qquad (\hat{x},k)\in\mathbb{R}^d\times{\mathscr{K}}.\end{equation}

We next introduce a sequence of processes that approximate $\widehat{X}^n$ . Let $\widehat{Y}^n$ be the strong solution to the Itô d-dimensional stochastic differential equation

(2.27) \begin{equation}\textrm{d} \widehat{Y}^n(t) = \bar{b}^n\bigl(\widehat{Y}^n(t)\bigr)\,\textrm{d}{t}+ \sigma^n \textrm{d} W(t),\end{equation}

with $\widehat{Y}^n(0) = y_0$ , where W(t) is a d-dimensional standard Brownian motion,

\begin{equation*}\bar{b}^n_i(\hat{y}):= \sum_{k\in{\mathscr{K}}}\pi_k\widehat{\Xi}^n_i(\hat{y},k),\qquad \,\hat{y}\in\mathbb{R}^d, \ i\in{\mathscr{I}},\end{equation*}

with $\widehat{\Xi}^n$ defined in (2.25). The diffusion matrix $\sigma^n$ is characterized as follows. Let

(2.28) \begin{equation}\Upsilon \,:=\, (\Pi - Q)^{-1} - \Pi\end{equation}

denote the deviation matrix corresponding to the transition rate matrix Q [Reference Coolen-Schrijner and van Doorn7]. Let $\Theta^n = [\theta^n_{ij}]$ be defined by

(2.29) \begin{equation}\theta^n_{ij}\,:=\, 2\sum_{\ell\in{\mathscr{K}}}\sum_{k\in{\mathscr{K}}}\frac{\Xi^n_i({x}^n_*,k)\Xi^n_j({x}^n_*,\ell)}{n^{\alpha + 2\beta}}\pi_k\Upsilon_{k\ell} ,\qquad i,j\in{\mathscr{I}},\end{equation}

and

\begin{equation*}\bar{a}^n(x) = [\bar{a}^n_{ij}](x)\,:=\, \sum_{k\in{\mathscr{K}}}\pi_k\bar{\Gamma}^n(x,k), \qquad x\in\mathbb{R}^d.\end{equation*}

Then, by Assumption 2.2(iv), and using the spectral decomposition, $\sigma^n$ satisfies

(2.30) \begin{equation}\Sigma^{n} := (\sigma^n)^{\mathsf{T}}\sigma^n = \bar{a}^n(0) + \Theta^n.\end{equation}

The generator of $\widehat{Y}^n$ , denoted by ${\mathscr{A}}^n$ , is given by

(2.31) \begin{equation}{\mathscr{A}}^n f(x) = \sum_{i\in{\mathscr{I}}}\bar{b}^n_i(x)\,\partial_i f(x) +\frac{1}{2}\sum_{i,j\in{\mathscr{I}}}\Sigma^{n}_{ij}\,\partial_{ij}f(x),\qquad f\in C^2(\mathbb{R}^d).\end{equation}

We borrow the following definitions from [Reference Gurvich11]. We say that a function $f\in C^2(\mathbb{R}^d)$ is sub-exponential if $f \geq 1$ and there exists some positive constant c such that

\begin{equation*}\vert{\nabla f(x)}\vert + \bigl\vert{\nabla^2 f(x)}\bigr\vert \,\leq\, c\,\textrm{e}^{c\vert{x}\vert} \qquad\forall\,x\in\mathbb{R}^d\end{equation*}

and

\begin{equation*}\sup_{\{z: \vert{z}\vert\leq1\}}\frac{f(x+z)}{f(x)} \,\leq\, c \qquad\forall\,x\in\mathbb{R}^d.\end{equation*}

We also let ${\mathscr{B}}_x$ denote the open ball around $x\in\mathbb{R}^{d}$ of radius $(1+\vert{x}\vert)^{-1}$ , and define

\begin{equation*}\Vert{f}\Vert_{C^{0,1}({\mathscr{B}}_x)} \,:=\, \sup_{y\in{\mathscr{B}}_x}\vert{f(x)}\vert+ \sup_{y,z\in{\mathscr{B}}_x}\frac{\vert{f(y)-f(z)}\vert}{\vert{y-z}\vert},\qquad f\in C^{0,1}(\mathbb{R}^{d}).\end{equation*}

The following assumption concerning the ergodic properties of $\widehat{Y}^n$ plays a crucial role in the proofs for steady-state approximations.

Assumption 2.3 There exist a sub-exponential norm-like function ${\mathscr{V}}\in C^2(\mathbb{R}^d)$ , a positive constant $\kappa$ , and an open ball ${\mathscr{B}}$ such that

\begin{equation*}{\mathscr{A}}^n {\mathscr{V}}(x) \,\leq\, \mathbb{1}_{{\mathscr{B}}}(x) - \kappa{\mathscr{V}}(x) \qquad\forall\, x\in\mathbb{R}^d,\ \forall\,n\in\mathbb{N}.\end{equation*}

We continue with the main result of this section. Its proof is given in Section 5. Let $\nu^n\in{\mathcal{P}}(\mathbb{R}^d)$ denote the steady-state distribution of $\widehat{Y}^n$ .

Theorem 2.2. Grant Assumptions 2.2 and 2.3. Assume that $(\widehat{X}^n,J^n)$ is ergodic, and its steady-state distribution $\pi^n\in{\mathcal{P}}(\mathbb{R}^d\times{\mathscr{K}}\,)$ satisfies

(2.32) \begin{equation}\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^d\times{\mathscr{K}}}{\mathscr{V}}(\hat{x})(1 + \vert{\hat{x}}\vert)^5\pi^n(\textrm{d}{\hat{x}},\textrm{d}{k})\,<\,\infty.\end{equation}

Then, for any $f : \mathbb{R}^d \mapsto \mathbb{R}$ such that $\Vert{f}\Vert_{C^{0,1}({\mathscr{B}}_x)}\leq {\mathscr{V}}(x)$ , and any $\alpha >0$ , we have

(2.33) \begin{equation}\vert{\pi^n(f) - \nu^n(f)}\vert = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge 1/2 }}\biggr).\end{equation}

Theorem 2.2 concerns the gap between the moments of the marginal distribution of the steady-state $\hat{X}^n$ and those of $\nu^n$ . The order of the function in (2.32) is determined by the estimates in Lemma 5.2 and the gradient estimates of the solutions to the Poisson equation in Lemma 5.3. In the following corollary, we provide a sufficient condition for (2.32). We give its proof in Section 5. In Section 3, we show that this sufficient condition holds in many examples.

Corollary 2.3. Grant Assumption 2.2. Let ${\mathscr{V}}$ and $\widetilde{{\mathscr{V}}}$ be two sub-exponential functions in $C^2(\mathbb{R}^d)$ satisfying Assumption 2.3, such that

(2.34) \begin{equation}{\mathscr{V}}(x)(1 + \vert{x}\vert^5) \,\leq\, \widetilde{{\mathscr{V}}}(x)\end{equation}

and

(2.35) \begin{equation}(1 + \vert{x}\vert)\bigl( \vert{\nabla{\widetilde{{\mathscr{V}}}(x)}}\vert+ \bigl\vert{\nabla^2{\widetilde{{\mathscr{V}}}(x)}}\bigr\vert \bigr)+ (1 + \vert{x}\vert^2)\bigl[\widetilde{{\mathscr{V}}}\bigr]_{2,1;B_{m_0/n^{\beta}}(x)}\,\leq\, C \widetilde{{\mathscr{V}}}(x)\end{equation}

for some positive constant C and any $x\in\mathbb{R}^d$ , with $m_0$ as in (2.18). Then (2.32) holds for ${\mathscr{V}}$ . As a consequence, (2.33) holds.

3. Examples

In this section, we demonstrate how the results of Section 2 can be applied through examples.

Example 3.1. (Markov-modulated $M/M/\infty$ queue.) We consider a process given by

\begin{equation*}{X}^n(t) \,:=\, {X}^n(0)+ A^{n}_1\biggl(\int_0^{t}n\lambda\bigl(J^n(s)\bigr)\,\textrm{d}{s}\biggr)- A^{n}_{-1}\biggl(\int_0^{t}\mu\bigl(J^n(s)\bigr) X^n(s)\,\textrm{d}{s}\biggr),\end{equation*}

where $ A^{n}_{-1}$ and $A^{n}_1$ are mutually independent unit-rate Poisson processes, independent of $J^n$ , for $n\in\mathbb{N}$ . We assume that $\lambda(k) > 0$ and $\mu(k) > 0$ , for $k\in{\mathscr{K}}$ . We let

(3.1) \begin{equation}{x}^n_{*} = n\frac{\sum_{k\in{\mathscr{K}}}\pi_k\lambda(k)}{\sum_{k\in{\mathscr{K}}}\pi_k\mu(k)}.\end{equation}

Recall that $\widehat{X}^n = n^{-\beta}(X^n - x^n_*)$ , and then $\widehat{{\mathfrak{X}}}^n = \{\hat{x}^n(x) : x\in\mathbb{Z}_+ \}$ . It is evident that $\lambda(k)$ and $\mu(k)x$ satisfy Hypothesis 2.1(c)–(d). Let $\bar{\lambda} := \sum_{k\in{\mathscr{K}}}\pi_k\lambda(k)$ and $\bar{\mu} := \sum_{k\in{\mathscr{K}}}\pi_k\mu(k)$ . By Definition 2.2, we obtain

(3.2) \begin{equation}\overline{\mathscr{L}}^{\,n} f(\hat{x}) = n\bar{\lambda}\,\bigl(f(\hat{x} + n^{-\beta}) - f(\hat{x})\bigr)+ \bar{\mu}\,(n^{\beta}\hat{x} + x^n_*)\bigl(f(\hat{x} - n^{-\beta}) - f(\hat{x})\bigr)\end{equation}

for all $\hat{x}\in\widehat{{\mathfrak{X}}}^n$ . Let ${\mathcal{V}}(x) = \vert{x}\vert^m$ , for $x\in\mathbb{R}$ , with $m \geq 2$ an even integer. It is clear that

(3.3) \begin{equation}\vert{\hat{x} \pm n^{-\beta}}\vert^m - \vert{\hat{x}}\vert^m = \pm n^{-\beta}m(\hat{x})^{m-1}+ {\mathscr{O}}(n^{-2\beta}){\mathscr{O}}(\vert{\hat{x}}\vert^{m-2}).\end{equation}

Thus we obtain from (3.1) and (3.2) that

\begin{equation*}\begin{aligned}\overline{\mathscr{L}}^{\,n} {\mathcal{V}}(\hat{x}) & = n\bar{\lambda}\bigl(\vert{\hat{x}+n^{-\beta}}\vert^m - \vert{\hat{x}}\vert^m - n^{-\beta}m\vert{\hat{x}}\vert^{m-1}\bigr)+ \bar{\mu} n^{\beta} \hat{x}\bigl(\vert{\hat{x}-n^{-\beta}}\vert^m - \vert{\hat{x}}\vert^m\bigr) \\ &\qquad + \bar{\mu} {x}^n_{*}\bigl(\vert{\hat{x}-n^{-\beta}}\vert^m - \vert{\hat{x}}\vert^m+ mn^{-\beta}\vert{\hat{x}}\vert^{m-1}\bigr) \\ & = \bar{\lambda}{\mathscr{O}}(n^{1-2\beta}){\mathscr{O}}(\vert{\hat{x}}\vert^{m-2})+ \bar{\mu}\bigl(-\vert{\hat{x}}\vert^m + {\mathscr{O}}(n^{-\beta}){\mathscr{O}}(\vert{\hat{x}}\vert^{m-1}) \\ &\mspace{250mu}+ {\mathscr{O}}(n^{1-2\beta}){\mathscr{O}}(\vert{\hat{x}}\vert^{m-2}) \bigr) \\ & \leq C_1 - C_2{\mathcal{V}}(\hat{x}) \qquad \forall\,\hat{x}\in\widehat{{\mathfrak{X}}}^n,\end{aligned}\end{equation*}

for some positive constants $C_1$ and $C_2$ , where in the second equality we use (3.3), and in the last line we apply Young’s inequality. It is straightforward to verify that ${\mathcal{V}}(x)$ satisfies (2.10). Therefore, the assumptions in Theorem 2.1 hold, and $(\widehat{X}^n,J^n)$ is exponentially ergodic for all large enough n.

Next we verify the assumptions in Corollary 2.3. The equation in (2.20) becomes

(3.4) \begin{equation}\sum_{k\in{\mathscr{K}}}\pi_k\Xi^n(x^n_*,k) = \sum_{k\in{\mathscr{K}}}\pi_k n\lambda(k)- \sum_{k\in{\mathscr{K}}}\pi_k\mu(k)x^n_* = 0.\end{equation}

Note that $x^n_*$ in (3.1) is the unique solution to (3.4). Recall the representation of $\widehat{Y}^n$ in (2.27). In this example, it follows by (3.4) that

\begin{equation*}\bar{b}^n(x) = n^{-\beta}\bar{\mu}x^n_* - n^{-\beta}\bar{\mu} (n^{\beta}x + x^n_*) = -\bar{\mu}x \qquad \forall\,x\in\mathbb{R},\end{equation*}

and

\begin{equation*}\bar{a}^n(0) = n^{-2\beta}(n\bar{\lambda} + \bar{\mu}x^n_*) = n^{1-2\beta}2\bar{\lambda}.\end{equation*}

Let ${\mathscr{V}}(x) = \kappa + \vert{x}\vert^m$ , with $\kappa \geq 1$ , for some integer $m\geq 2$ . We choose some $\tilde{\kappa}\geq1$ such that

\begin{equation*}\widetilde{{\mathscr{V}}}(x) \,:=\, \tilde{\kappa}\bigl(1 + \vert{x}\vert^{5+m}\bigr)\,\geq\, {\mathscr{V}}(x)(1 + \vert{x}\vert^5)\qquad\forall\,x\in\mathbb{R}.\end{equation*}

Then Assumptions 2.2 and 2.3 are satisfied. Indeed, by the discussion following Theorem 3.1 of [Reference Gurvich11], if $\widetilde{{\mathscr{V}}}\in C^3(\mathbb{R}^d)$ in Corollary 2.3, we may replace (2.35) by

(3.5) \begin{equation}(1 + \vert{x}\vert)\bigl(\vert{\nabla{\widetilde{{\mathscr{V}}}(x)}}\vert+ \bigl\vert{\nabla^2{\widetilde{{\mathscr{V}}}(x)}}\bigr\vert \bigr)+ (1 + \vert{x}\vert^2)\bigl\vert{\nabla^3{\widetilde{{\mathscr{V}}}(x)}}\bigr\vert\,\leq\, C \widetilde{{\mathscr{V}}}(x)\end{equation}

for some positive constant C and any $x\in\mathbb{R}^d$ , where

\begin{equation*}\nabla^3 := \frac{\partial^3}{\partial x_1^{\eta_1}\cdots\partial x_d^{\eta_d}}\end{equation*}

with a multi-index $(\eta_1,\dots,\eta_d)$ satisfying $\sum_{i=1}^d\eta_i = 3$ . Then it is straightforward to check that $\widetilde{{\mathscr{V}}}$ chosen above satisfies (3.5). Thus, the result in Corollary 2.3 follows.

The following example concerns Markov-modulated multiclass $M/M/n + M$ queues. Exponential ergodicity for these queues under a static priority scheduling policy has been studied in [Reference Arapostathis, Das, Pang and Zheng3, Theorem 4], which treats a special case of the model considered in this paper. Here we show that by using the result in Corollary 2.1, the proof of [Reference Arapostathis, Das, Pang and Zheng3, Theorem 4] is greatly simplified. We also extend the results in [Reference Arapostathis, Das, Pang and Zheng3, Theorem 4 and Lemma 3] to include a larger class of scheduling policies such that the Markov-modulated queues have exponential ergodicity.

Example 3.2. (Markov-modulated multiclass $M/M/n + M$ queues.) We consider a d-dimensional birth–death process $\{X^n(t): t\geq0\}$ , with state space $\mathbb{Z}^d_+$ , given by

\begin{equation*}\begin{aligned}{X}^n_i(t) &\,:=\, {X}^n_i(0)+ A^{n}_{e_i}\biggl(\int_0^{t}n\lambda_i\bigl(J^n(s)\bigr)\,\textrm{d}{s}\biggr) \\[4pt] &\qquad- A^{n}_{-e_i}\biggl(\int_0^{t}\Bigl(\mu_i\bigl(J^n(s)\bigr)z^n_i(X^n(s))+ \gamma_i\bigl(J^n(s)\bigr)\bigl(X^n_i(s) - z^n_i(X^n(s))\bigr)\Bigr)\,\textrm{d}{s}\biggr)\end{aligned}\end{equation*}

for $i\in{\mathscr{I}}:=\{1,\dots,d\}$ , where $\{A^{n}_{e_i},A^{n}_{-e_i}: i\in{\mathscr{I}}\}$ are mutually independent unit-rate Poisson processes, independent of $J^n$ , and $z^n$ is the static priority policy defined by

\begin{equation*}z^n_i(x) \,:=\, x_i\wedge\biggl(n - \sum_{j=1}^{i-1}x_j\biggr)^+\qquad \forall\,i\in{\mathscr{I}}.\end{equation*}

We assume that $\{\lambda_i(k),\mu_i(k),\gamma_i(k): i\in{\mathscr{I}},k\in{\mathscr{K}}\}$ are strictly positive, and the system is critically loaded, that is, $\sum_{i\in{\mathscr{I}}}\rho_i = 1$ with $\rho_i := \bar{\lambda}_i/\bar{\mu}_i$ . Equation (2.20) becomes

\begin{equation*}\sum_{k\in{\mathscr{K}}}\pi_k \Xi^n_i({x}^n_*,k) = n\bar{\lambda}_i - \bar{\mu}_iz^n_i(x^n_*)- \bar{\gamma}_i(x^n_{*,i} - z^n_i(x^n_*)) = 0\qquad \forall\,i\in{\mathscr{I}},\end{equation*}

which has a unique solution $x^n_* = n\rho$ with $\rho = (\rho_1,\dots,\rho_d)$ .

We first establish exponential ergodicity and verify Assumption 2.1. Let

\begin{equation*}\psi^n_{e_i}(x,k) = n \lambda_i(k),\qquad \psi^n_{-e_i}(x,k) = n \rho_i \mu_i(k),\end{equation*}

and

\begin{equation*}\phi^n_{-e_i}(x,k) = \mu_i(k)(z^n_i(x) - n\rho_i)+ \gamma_i(k)(x_i - z^n_i(x))\end{equation*}

for $i\in{\mathscr{I}}$ and $(x,k)\in\mathbb{R}^d\times{\mathscr{K}}$ . Then $\bar{\psi}^n_{e_i}(x) = n\bar{\lambda}_i$ and $\bar{\psi}^n_{-e_i}(x) = n\rho_i\bar{\mu}_i = n\bar{\lambda}_i$ . It is evident that the functions $\psi^n_{e_i}$ and $\psi^n_{-e_i}$ satisfy (2.11) and (2.12). Note that $z_i^n(x) \leq x_i$ , and thus Hypothesis 2.1(d) is satisfied. Let ${\mathcal{V}}_{\zeta,m}(x) := \sum_{i\in{\mathscr{I}}}\zeta_i\vert{x_i}\vert^m$ for $x\in\mathbb{R}^d$ , an even integer $m\geq 2$ , and a positive vector $\zeta\in\mathbb{R}^d$ to be chosen later. Recall ${\mathcal{G}}^n_k$ in (2.13). It is straightforward to verify that

\begin{equation*}\begin{aligned}{\mathcal{G}}^n_k{\mathcal{V}}_{\zeta,m}(\hat{x})& = n^{-\beta}\sum_{i\in{\mathscr{I}}} - \phi^n_{-e_i}(n^{\beta}\hat{x}+ n\rho,k)\lambda_i\vert{\hat{x}_i}\vert^{m-1} \\[5pt] &\qquad + n^{-2\beta}\sum_{i\in{\mathscr{I}}}\bigl(2n\bar{\lambda}_i + \phi^n_{-e_i}(n^{\beta}\hat{x}+ n\rho,k)\bigr){\mathscr{O}}(\vert{\hat{x}_i}\vert^{m-2}).\end{aligned}\end{equation*}

Since $\inf_{i,k}\{\mu_i(k),\gamma_i(k)\} > 0$ , it follows by [Reference Arapostathis, Biswas and Pang2, Lemma 5.1] that there exist some positive vector $\lambda$ , $n_0\in\mathbb{N}$ , and positive constants $C_1$ and $C_2$ such that

(3.6) \begin{equation}{\mathcal{G}}^n_k{\mathcal{V}}_{\zeta,m}(\hat{x}) \,\leq\, C_1- C_2{\mathcal{V}}_{\zeta,m}(\hat{x}),\qquad (x,k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\quad n\geq n_0.\end{equation}

Therefore, the result in Corollary 2.1 follows. We remark that the claim in Corollary 2.1 holds for any work-conserving scheduling policy satisfying (3.6), since there is no continuity assumption on $\phi^n_{-e_i}$ . This extends the results of [Reference Arapostathis, Das, Pang and Zheng3, Theorem 4 and Lemma 3]. Indeed the proofs of these results can be greatly simplified by following the approach above, since we only need to consider the constant functions $\psi^n_{e_i}$ and $\psi^n_{-e_i}$ in x.

Next we focus on steady-state approximations for this example. It is straightforward to verify that the coefficients in (2.27) take the form

(3.7) \begin{equation}\begin{aligned}\bar{b}^n_i(x) & = -\frac{\bar{\mu}_i}{n^{\beta}}\bigl(z^n_i(n^{\beta}x + x^n_{*}) - z^n_i(x^n_*)\bigr) \\&\mspace{120mu}- \frac{\bar\gamma_i}{n^{\beta}}\bigl(n^{\beta}x_i- (z^n_i(n^{\beta}x + x^n_{*}) - z^n_i(x^n_*)) \bigr),\qquad i\in{\mathscr{I}},\end{aligned}\end{equation}

and

\begin{equation*}\bar{a}^n_{ii}(0) = \frac{1}{n^{2\beta}}\bigl(n\bar{\lambda}_i + \bar{\mu}_i z^n_i({x}^n_*)+ \bar{\gamma}_i\bigl(x^n_{*,i} - z^n_i(x^n_*)\bigr) = n^{1-2\beta}2\bar{\lambda}_i,\qquad \forall\,i\in{\mathscr{I}},\end{equation*}

and that $\bar{a}^n_{ij}(0) = 0$ for $i\neq j$ . We let ${\mathscr{V}}_{\zeta,m}(x) = \kappa + \sum_{i\in{\mathscr{I}}}\zeta_i\vert{x_i}\vert^m$ for some positive vector $\zeta\in\mathbb{R}^d$ , an even integer $m\geq 2$ , and $\kappa \geq 1$ . We choose $\tilde{\kappa}\geq1$ such that

\begin{equation*}\widetilde{{\mathscr{V}}}_{\zeta,m}(x) \,:=\,\tilde{\kappa}\bigg(1 + \sum_{i\in{\mathscr{I}}}\zeta_i\vert{x}\vert^{6+m}\bigg)\,\geq\, {\mathscr{V}}_{\zeta,m}(x)(1 + \vert{x}\vert^5) \quad\forall\,x\in\mathbb{R}^d.\end{equation*}

Repeating the calculation in [Reference Arapostathis, Biswas and Pang2, Lemma 5.1], we find that there exist some positive vector $\zeta\in\mathbb{R}^d$ and some positive constants $c_1$ and $c_2$ such that

\begin{equation*}\langle \bar{b}^n(x), \nabla{\mathscr{V}}_{\zeta,m}(x)\rangle\,\leq\, c_1 - c_2{\mathscr{V}}_{\zeta,m}(x)\qquad \forall\,x\in\mathbb{R}^d.\end{equation*}

It follows directly by Young’s inequality that there exists some positive constant $c_3$ such that

\begin{equation*}\bigl\vert{\nabla^2{\mathscr{V}}_{\zeta,m}(x)}\bigr\vert \,\leq\, c_3 - \frac{c_2}{2}{\mathscr{V}}_{\zeta,m}(x)\qquad \forall\,x\in\mathbb{R}^d.\end{equation*}

The same holds for $\widetilde{{\mathscr{V}}}_{\zeta,m}$ . Thus, we have verified Assumption 2.3. Since $z^n_i$ is Lipschitz continuous, it is evident that Assumption 2.2 holds. An easy calculation shows that (3.5) holds. As a result, Corollary 2.3 follows.

When $d=1$ , (2.20) becomes

\begin{equation*}\sum_{k\in{\mathscr{K}}}\pi_k \Xi^n({x}^n_*,k) = n\bar{\lambda} - \bar{\mu}\bigl({x}^n_* \wedge n\bigr)- \bar{\gamma}\bigl({x}^n_* - n)^+ = 0,\end{equation*}

which can be solved directly without the critically-loaded assumption. It is straightforward to verify that (3.7) becomes

\begin{equation*}\begin{aligned}\bar{b}^n(x) & = -\bar{\mu}\bigl((x+ n^{-\beta}x^n_*)\wedge n^{1-\beta}- n^{-\beta}x^n_*\wedge n^{1-\beta}\bigr) \\&\mspace{120mu} - \bar{\gamma}\bigl( (x + n^{-\beta}x^n_* - n^{1-\beta})^+-n^{-\beta}(x^n_*-n)^+\bigr).\end{aligned}\end{equation*}

Repeating the procedure as above, we establish Corollary 2.3.

Example 3.3. (Markov-modulated $M/PH/n + M$ queues.) We assume that all customers start service in phase 1, and there are d phases. Given $J^n = k$ , the probability of getting phase j after finishing service in phase i is denoted by $p_{ij}(k)$ . Let $X_1^n$ denote the total number of customers, both in service and queued, in phase 1, and let $X_i^n$ , for $i\neq 1$ , denote the number of customers in service in phase i. (We refer the reader to [Reference Dai, He and Tezcan8] for a detailed description of the model without Markov modulation, and to [Reference Zhu and Prabhu26] for an application of Markov-modulated phase-type distributions in queueing.) Then (2.19) becomes

\begin{equation*}\begin{cases}\Xi^n_1(x,k) = n\lambda(k)- \mu_1(k)\bigl(x_1 - (\langle e, x \rangle - n)^+\bigr)- \gamma(k)\bigl(\langle e,x \rangle - n\bigr)^+, \\[3pt] \begin{aligned}\Xi^n_i(x,k) & = -\mu_i(k)x_i+ \sum_{j\neq i,j\neq 1}p_{ji}(k)\mu_j(k)x_j \\[3pt] &\mspace{200mu} + p_{1i}(k)\mu_1(k)\bigl(x_1 - (\langle e, x \rangle - n)^+\bigr)\quad \text{for}\ i\neq 1,\end{aligned}\end{cases}\end{equation*}

and (2.20) becomes

\begin{align*}\begin{cases}n \bar\lambda- \bar{\mu}_1\bigl(x^n_{*,1} - (\langle e, x^n_* \rangle - n)^+\bigr)- \bar{\gamma}\bigl(\langle e,x^n_* \rangle - n\bigr)^+ = 0,\\[5pt] - \bar{\mu}_ix^n_{*,i}+ \sum_{j\neq i,j\neq 1}\bar{p}_{ji}\bar{\mu}_jx^n_{*,j}+ \bar{p}_{1i}\bar{\mu}_1\bigl(x^n_{*,1} - (\langle e, x^n_* \rangle - n)^+\bigr) = 0\quad \text{for}\ i\neq 1,\end{cases}\end{align*}

where $\bar{\gamma} = \sum_{k\in{\mathscr{K}}}\pi_k\gamma(k)$ , and $\bar{p}_{ij} = \sum_{k\in{\mathscr{K}}}\pi_k p_{ij}(k)$ . Here, $e^{\mathsf{T}}=(1,\dotsc,1)$ as defined in Section 1.2. Assume that $\bar{\lambda} = 1$ . Note that $\{\Xi^n_i: i\in{\mathscr{I}}\}$ are piecewise linear functions in their first argument. It is straightforward to verify that Hypothesis 2.1 and Assumption 2.2 are satisfied. We get $x^n_*= n\rho$ , where

\begin{equation*}\rho \,:=\, \frac{\bar{M}^{-1}e_1}{e^{\mathsf{T}} \bar{M}^{-1}e_1}, \quad \text{and} \quad\bar{M} \,:=\, (I - \bar{P}^{\mathsf{T}})\text{diag}(\bar{\mu}),\end{equation*}

with I the identity matrix and $\bar{P}:= [\bar{p}_{ij}]$ . The coefficients in (2.27) satisfy

\begin{align*}\bar{b}^n(x) & = - \bar{M}x+ \bigl(\bar{M} - \bar{\gamma} I\bigr)e_1\langle e,x \rangle^+,\\[5pt] \bar{a}^n_{ii}(0) & = \begin{cases}n^{1-2\beta}\bigl(1 + \bar{\mu}_1\rho_1\bigr) \quad & \text{if}\ i=1, \\[5pt] n^{1-2\beta}\Bigl(\sum_{j\neq i,j\neq 1}\bar{p}_{ji}\bar{\mu}_j\rho_j + \bar{\mu}_i\rho_i+ \bar{\mu}_1\rho_1\bar{p}_{1i}\Bigr) \quad & \text{if}\ i\neq1,\end{cases}\end{align*}

and

\begin{equation*}\bar{a}^n_{ij}(0) = n^{1-2\beta}\bigl(\bar{p}_{ij}\bar{\mu}_i\rho_i + \bar{p}_{ji}\bar{\mu}_j\rho_j\bigr), \qquad i\neq j.\end{equation*}

By [Reference Arapostathis, Pang and Sandrić5, Theorem 3.5] (see also [Reference Dieker and Gao9, Theorem 3]), there exists a function $\widetilde{{\mathscr{V}}}$ satisfying the assumption in Corollary 2.3. In analogy to [Reference Arapostathis, Pang and Sandrić5, Theorem 3.5], we can show that there exists a function ${\mathcal{V}}(x) = \langle x, Rx \rangle^{m/2}$ , for $m\geq 2$ and some positive definite matrix R, satisfying the conditions in Theorem 2.1.

4. Proofs of Theorem 2.1 and Corollaries 2.1 and 2.2

The range of the transition matrix Q is the subspace $\Delta := \{y\in\mathbb{R}^{k_\circ}\,: \sum_{k\in{\mathscr{K}}} \pi_k y_k=0\}$ . As shown in [Reference Hunter13, Theorem 3.5], if v and u are any vectors in $\mathbb{R}^{k_\circ}$ satisfying $\pi^{\mathsf{T}} v\ne0$ and $u^{\mathsf{T}} e\ne0$ , then the matrix $Q+v u^{\mathsf{T}}$ is nonsingular, and

(4.1) \begin{equation}{\mathcal{T}}:= \bigl(Q+v u^{\mathsf{T}}\bigr)^{-1}\end{equation}

is a generalized inverse of Q; that is, it satisfies $Q{\mathcal{T}} Q=Q$ . This of course means that

(4.2) \begin{equation}Q{\mathcal{T}} y = y\quad\text{for}\ \text{all}\ y\in\Delta.\end{equation}

We also need the following definition.

Definition 4.1. Recall (2.3) and Definition 2.2. Let $\breve{\mathscr{L}}_k^{\kern1.6pt n}:=\overline{\mathscr{L}}^{\,n}-{\mathscr{L}}_k^{\kern1.6pt n}$ . This operator takes the form

\begin{equation*}\breve{\mathscr{L}}_k^{\kern1.6pt n}f(\hat{x},k) \,:=\, \sum_{z\in{\mathscr{Z}}^{\,\,n}}\breve{r}^n_k(n^{\beta}\hat{x} + x^n_*,z)\bigl(f(\hat{x}+n^{-\beta}z,k)-f(\hat{x},k)\bigr),\qquad (\hat{x},k) \in \widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation*}

for $f\in C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ , where

\begin{equation*}\breve{r}^n_k(x,z) \,:=\,\bar{r}^n(x,z) - \tilde{r}^n_k(x,z),\qquad (x,k)\in {{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation*}

Proof of Theroem 2.1. Let ${\mathcal{T}}=[{\mathcal{T}}_{k\ell}]_{k,\ell\in{\mathscr{K}}}$ be as defined in (4.1).

(4.3) \begin{equation}\widetilde{{\mathcal{V}}}^n(\hat{x},k) \,:=\,\frac{1}{n^{\alpha}}\sum_{\ell\in{\mathscr{K}}}{\mathcal{T}}_{k\ell}\,\breve{\mathscr{L}}^{\,n}_{\ell}{\mathcal{V}}^n(\hat{x}),\qquad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation}

Then

(4.4) \begin{equation}{\mathcal{Q}}^n\widetilde{{\mathcal{V}}}^n(\hat{x},k) = \breve{\mathscr{L}}_k^{\kern1.6pt n}{\mathcal{V}}^n(\hat{x}) \qquad\forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

by (4.2).

We define

(4.5) \begin{equation}\widehat{{\mathcal{V}}}^n(\hat{x},k)\,:=\, {\mathcal{V}}^n(\hat{x}) + \widetilde{{\mathcal{V}}}^n(\hat{x},k),\qquad(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation}

By Hypothesis 2.1(c)–(d), we have

(4.6) \begin{equation}\tilde{r}^n_k(n^{\beta}\hat{x} + x^n_*,z)\,\leq\, C_0 (n^{1\vee\alpha/2}+n^{\beta}\vert{\hat{x}}\vert)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\ \forall\,z\in{\mathscr{Z}}^{\,\,n}, \ \forall\,n\in\mathbb{N}.\end{equation}

We choose $N_1$ large enough so that $m_0 \leq \varepsilon_0 N_1^\beta$ , with $m_0$ as defined in Hypothesis 2.1(a). By Hypothesis 2.1(a)–(b), (2.6), and (4.6), we have

(4.7) \begin{equation}\bigl\vert{\breve{\mathscr{L}}^{\,n}_{k} {\mathcal{V}}^n(\hat{x})}\bigr\vert\,\leq\, N_0 C_0 (n^{1\vee\alpha/2} +n^{\beta}\vert{\hat{x}}\vert)\, C m_0\frac{1+{\mathcal{V}}^n(\hat{x})}{n^\beta (1+\vert{\hat{x}}\vert)}\end{equation}

for all $n\geq N_1$ . Therefore, since $\alpha + \beta - 1 \geq \alpha/2$ for $\alpha > 0$ , it follows by (4.5)–(4.7) that there exists $n_1\in\mathbb{N}$ , $n_1\geq N_1$ , such that (2.8) holds.

Recall the definitions in (2.2), (2.3), and (2.5). We have

\begin{equation*}\overline{{\mathscr{L}}}^n{\mathcal{V}}^n(\hat{x}) = {\mathscr{L}}_k^n{\mathcal{V}}^n(\hat{x}) + \breve{\mathscr{L}}_k^{\kern1.6pt n}{\mathcal{V}}^n(\hat{x}) = {\mathscr{L}}_k^n{\mathcal{V}}^n(\hat{x}) + {\mathcal{Q}}^n \widetilde{{\mathcal{V}}}^n(\hat{x},k)\end{equation*}

by (4.4). Therefore, since ${\mathcal{Q}}^n {\mathcal{V}}^n(\hat{x}) = 0$ , we obtain

(4.8) \begin{equation}\begin{aligned}\widehat{{\mathscr{\!\!L}}}^{\,\,n} \widehat{{\mathcal{V}}}^n(\hat{x},k) & = {\mathscr{L}}_k^{\kern1.6pt n}{\mathcal{V}}^n(\hat{x}) + {\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k)+ {\mathcal{Q}}^n \widetilde{{\mathcal{V}}}^n(\hat{x},k) \\[4pt] & = \,\,\overline{{\mathscr{\!\!L}}}^{\,\,n}{\mathcal{V}}^n(\hat{x})+ {\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^{n}\times{\mathscr{K}}.\end{aligned}\end{equation}

We define the function

\begin{equation*}G^n_k(\hat{x},z) \,:=\,\breve{r}^n_k\bigl(n^{\beta}\hat{x} + x^n_{*},z\bigr)\bigl({\mathcal{V}}^n(\hat{x}+n^{-\beta}z) - {\mathcal{V}}^n(\hat{x})\bigr).\end{equation*}

It is straightforward to verify, using (4.3), that

(4.9) \begin{equation}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k)\,=&\, \sum_{h\in{\mathscr{Z}}^{\,\,n}}\tilde{r}^n_k(n^{\beta}\hat{x} + {x}^n_*,h)\bigl(\widetilde{{\mathcal{V}}}^n(\hat{x} + n^{-\beta}h,k)- \widetilde{{\mathcal{V}}}^n(\hat{x},k)\bigr)\\[4pt] \,=&\, \frac{1}{n^{\alpha}} \sum_{h,z\in{\mathscr{Z}}^{\,\,n}}\tilde{r}^n_{k}(n^{\beta}\hat{x} + {x}^n_*,h) \sum_{\ell\in{\mathscr{K}}}{\mathcal{T}}_{k\ell}\bigl(G^n_{\ell}(\hat{x} + n^{-\beta}h,z) - G^n_{\ell}(\hat{x},z)\bigr).\end{aligned}\end{equation}

On the other hand, it follows by Hypothesis 2.1(c) and a triangle inequality that

(4.10) \begin{equation}\begin{aligned}&\vert{G^n_k(\hat{x} + n^{-\beta}h,z) - G^n_k(\hat{x},z)}\vert \\ &\mspace{100mu}\,\leq\,2 C_0(n^{\alpha/2}+\vert{h}\vert)\,\bigl\vert{{\mathcal{V}}^n(\hat{x}+n^{-\beta}z) - {\mathcal{V}}^n(\hat{x})}\bigr\vert \\ &\mspace{150mu}+\bigl\vert{\breve{r}^n_k(n^{\beta}\hat{x} + x^n_{*} + h,z)}\bigr\vert\bigl\vert{{\mathcal{V}}^n(\hat{x} + n^{-\beta}z + n^{-\beta}h)} \\ &\mspace{200mu}- {\mathcal{V}}^n(\hat{x} + n^{-\beta}h) - {\mathcal{V}}^n(\hat{x}+ n^{-\beta}z)+ {\mathcal{V}}^n(\hat{x})\bigr\vert\end{aligned}\end{equation}

for all $h,z\in{\mathscr{Z}}^{\,\,n}$ . As in (4.6), we have

(4.11) \begin{equation}\vert{\breve{r}^n_k(n^{\beta}\hat{x} + x^n_*+h,z)}\vert\,\leq\, C_0 (n^{1\vee\alpha/2}+n^{\beta}\vert{\hat{x}}\vert + \vert{h}\vert)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\ \forall\,h,z\in{\mathscr{Z}}^{\,\,n},\end{equation}

for all $n\in\mathbb{N}$ . By (2.6) and Hypothesis 2.1(a), we have

(4.12) \begin{equation}\begin{aligned}\bigl\vert{{\mathcal{V}}^n(\hat{x}+n^{-\beta}z) - {\mathcal{V}}^n(\hat{x})}\bigr\vert&\,\leq\, C m_0\frac{1+{\mathcal{V}}^n(\hat{x})}{n^\beta (1+\vert{\hat{x}}\vert)},\\[3pt]\bigl\vert{{\mathcal{V}}^n(\hat{x} + n^{-\beta}z + n^{-\beta}h)- {\mathcal{V}}^n(\hat{x} + n^{-\beta}h)\mspace{50mu}}&\\- {\mathcal{V}}^n(\hat{x}+ n^{-\beta}z) + {\mathcal{V}}^n(\hat{x})\bigr\vert&\,\leq\, C m_0^2\frac{1+{\mathcal{V}}^n(\hat{x})}{n^{2\beta} (1+\vert{\hat{x}}\vert^2)}\end{aligned}\end{equation}

for all $h,z\in B_{m_0}$ , $\hat{x}\in\widehat{\mathfrak{X}}^n$ , and $n\in\mathbb{N}$ . Hence, using (4.9) together with the estimates in (4.6) and (4.10)–(4.12) and Hypothesis 2.1(a)–(b), we obtain

(4.13) \begin{equation}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n} & \widetilde{{\mathcal{V}}}^n(\hat{x},k)\\ &\,\leq\,N_0C_0C m_0\sum_{k,k^{\prime}\in{\mathscr{K}}}\vert{{\mathcal{T}}_{k\ell}}\vert\biggl(2(n^{\alpha/2} + m_0)(n^{1\vee\alpha/2} +n^{\beta}\vert{\hat{x}}\vert)\frac{1+{\mathcal{V}}^n(\hat{x})}{n^{\alpha+\beta} (1+\vert{\hat{x}}\vert)}\\&\mspace{40mu}+N_0C_0m_0 (n^{1\vee\alpha/2}+n^{\beta}\vert{\hat{x}}\vert)(n^{1\vee\alpha/2} +n^{\beta}\vert{\hat{x}}\vert + m_0)\frac{1+{\mathcal{V}}^n(\hat{x})}{n^{\alpha+2\beta} (1+\vert{\hat{x}}\vert^2)}\biggr).\end{aligned}\end{equation}

Using the property $\beta=\max\{1/2,1 - \alpha/2\}$ , we deduce from (4.13) that for any $\epsilon>0$ there exists some constant $C_\circ(\epsilon)$ such that

(4.14) \begin{equation}{\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k)\,\leq\, C_\circ(\epsilon) + \epsilon {\mathcal{V}}^n(\hat{x})\qquad\forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\ \forall n\in\mathbb{N}.\end{equation}

Therefore, choosing $\epsilon=\frac{1}{2}\overline{C}_2$ , and using (2.7), (2.8), (4.8), and (4.14), we obtain

\begin{equation*}\widehat{{\mathscr{\!\!L}}}^{\,\,n} \widehat{{\mathcal{V}}}^n(\hat{x},k) \,\leq\, \overline{C}_1+ C_\circ\bigl(\overline{C}_2/2\bigr)+\frac{1}{6}\overline{C}_2 - \frac{1}{3}\overline{C}_2\widehat{{\mathcal{V}}}^n(\hat{x},k) \qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\ \forall\, n>n_1.\end{equation*}

This completes the proof.

Proof of Corollary 2.1. Recall ${\mathcal{G}}^n_k$ in (2.13), and let $\breve{\mathcal{G}}^n_k := {\mathcal{G}}^n_k - {\mathscr{L}}_k^{\kern1.6pt n}$ . Then $\breve{\mathcal{G}}^n_k$ takes the form

\begin{equation*}\breve{\mathcal{G}}^n_k f(\hat{x},k) = \sum_{z\in{\mathscr{Z}}^{\,\,n}} \breve\psi^n_k(n^\beta\hat{x} + x^n_*,z)\bigl(f(\hat{x}+n^{-\beta}z,k)-f(\hat{x},k)\bigr),\qquad (\hat{x},k) \in \widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation*}

for $f\in C_b(\mathbb{R}^d\times{\mathscr{K}}\,)$ , where

\begin{equation*}\breve{\psi}^n_k(x,z)\,:=\,\bar{\psi}^n(x,z) - \psi^n_k(x,z), \qquad k\in{\mathscr{K}}, \quad (x,z)\in{\mathfrak{X}}^n\times{\mathscr{Z}}^{\,\,n}.\end{equation*}

Compare this to Definition 4.1. We let

\begin{equation*}\widetilde{{\mathcal{V}}}^n(\hat{x},k) \,:=\,\frac{1}{n^{\alpha}}\sum_{\ell\in{\mathscr{K}}}{\mathcal{T}}_{k\ell}\,\breve{\mathcal{G}}^n_{\ell}{\mathcal{V}}^n(\hat{x}),\qquad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation*}

As in (4.4), we have

\begin{equation*}{\mathcal{Q}}^n\widetilde{{\mathcal{V}}}^n(\hat{x},k) = \breve{\mathcal{G}}^n_k{\mathcal{V}}^n(\hat{x}), \qquad(\hat{x},k)\in\widehat{{\mathfrak{X}}}^{n}\times{\mathscr{K}}.\end{equation*}

In analogy to (4.8), we get

\begin{equation*}\widehat{\mathscr{\!\!L}}^{\,\,n} \widehat{{\mathcal{V}}}^n(\hat{x},k) = {\mathcal{G}}^n_k{\mathcal{V}}^n(\hat{x})+ {\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k),\qquad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^{n}\times{\mathscr{K}}.\end{equation*}

In obtaining an estimate for ${\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k)$ , the proof is the same as that of Theorem 2.1, except that we replace $\breve r^n$ with $\breve\psi^n$ , and use (2.11) and (2.12). Applying (2.11) and (2.12) again, we may show (2.8). Then the claim in (2.9) follows by (2.14).

Proof of Corollary 2.2. We present only some crucial estimates that are different from those in the proof of Theorem 2.1. Indeed, it follows by (2.15) and (2.16) that

(4.15) \begin{equation}\tilde{r}^n_k(n^{\beta}\hat{x} + x^n_*,z) \,\leq\, C_0(1 + n),\qquad \breve{r}^n_k(n^{\beta}\hat{x} + x^n_*,z) \,\leq\, C_0(1 + n),\end{equation}

and

(4.16) \begin{align}\bigl\vert{\breve{r}^n_k(n^{\beta}(\hat{x} + n^{-\beta}h) + x^n_*,z) - \breve{r}^n_k(n^{\beta}\hat{x} + x^n_*,z)} \bigr\vert\,\leq\, C_0(1 + \vert{h}\vert\wedge n)\end{align}

for some positive constant $C_0$ . By Hypothesis 2.1(a)–(b), (4.15), and (2.17), (4.7) becomes

(4.17) \begin{equation}\bigl\vert{\breve{\mathscr{L}}^{\,n}_{k} {\mathcal{V}}^n(\hat{x})}\bigr\vert\,\leq\, N_0 C_0 (1 + n)\, C m_0\frac{1+{\mathcal{V}}^n(\hat{x})}{n^\beta}\end{equation}

for all large n. Using (2.17) and (4.15)–(4.17), together with Hypothesis 2.1(a)–(b), (4.13) becomes

(4.18) \begin{equation}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n} \widetilde{{\mathcal{V}}}^n(\hat{x},k) &\,\leq\,N_0C_0C m_0\sum_{k,k^{\prime}\in{\mathscr{K}}}\vert{{\mathcal{T}}_{k\ell}}\vert\biggl(2(1 + m_0)(1+n)\frac{1+{\mathcal{V}}^n(\hat{x})}{n^{\alpha+\beta}}\\&\mspace{175mu}+N_0C_0m_0 (1+n)(1+n)\frac{1+{\mathcal{V}}^n(\hat{x})}{n^{\alpha+2\beta}}\biggr).\end{aligned}\end{equation}

Since $\alpha + 2\beta > 2$ implies $\alpha + \beta >1$ , it follows by (4.18) that (4.14) holds for all large n. The rest of the proof is the same as that of Theorem 2.1.

5. Proofs of Theorem 2.2 and Corollary 2.3

We need to introduce some additional notation to facilitate the proofs. Recall the definitions of $\widehat{\Xi}^n$ , $\bar{\Gamma}^n$ , $\bar{b}^n$ , and $\bar{a}^n$ in (2.25)–(2.27) and (2.30), respectively. For $f\in C^2(\mathbb{R}^{d})$ and $n\in\mathbb{N}$ , let

(5.1) \begin{equation}\begin{aligned}\breve{g}^n_1[f](x,k)&\,:=\, \sum_{i\in{\mathscr{I}}}\bigl(\bar{b}^n_i(x) -\bigl(\widehat{\Xi}^n_i(x,k) - \widehat{\Xi}^n_i(0,k) \bigr)\bigr)\partial_i f(x) \\&\mspace{180mu}+ \frac{1}{2}\sum_{i,j\in{\mathscr{I}}}\bigl(\bar{a}^n_{ij}(x)- \bar{\Gamma}^n_{ij}(x,k)\bigr)\partial_{ij}f(x)\end{aligned}\end{equation}

and

(5.2) \begin{equation}\begin{aligned}\breve{g}^n_2[f](x,k)&\,:=\, \frac{1}{n^{\alpha + 2\beta}}\sum_{z}\sum_{h\in{\mathscr{K}}}\biggl(\sum_{l\in{\mathscr{K}}}\pi_l\xi^n_z(n^{\beta}x + {x}^n_{*},l)\Upsilon_{lh} \\&\mspace{100mu}- \xi^n_z(n^{\beta}x + {x}^n_{*},k)\Upsilon_{kh}\biggr)\sum_{j\in{\mathscr{I}}}\Xi^n_j({x}^n_{*},h)\sum_{i\in{\mathscr{I}}}z_i\partial_{ij} f(x),\end{aligned}\end{equation}

with $\Upsilon$ as defined in (2.28). It follows by the identity

\begin{equation*}\sum_{k\in{\mathscr{K}}}\Bigl(\sum_{l\in{\mathscr{K}}}\pi_l\xi^n_z(n^{\beta}x + {x}^n_{*},l)\Upsilon_{lh}- \xi^n_z(n^{\beta}x + {x}^n_{*},k)\Upsilon_{kh}\Bigr)\,\equiv\,0\end{equation*}

that $\sum_{k\in{\mathscr{K}}}\pi_k \breve{g}^n_2[f](x,k) = 0$ . It is clear that $\sum_{k\in{\mathscr{K}}}\pi_k \breve{g}^n_1[f](x,k) = 0$ . Recall the matrix ${\mathcal{T}}$ in (4.1) and (4.2). We define

(5.3) \begin{equation}g^n_{i}[f](x,k) \,:=\, \frac{1}{n^{\alpha}}\sum_{\ell\in{\mathscr{K}}}{\mathcal{T}}_{k\ell}\, \breve{g}^n_i[f](x,\ell),\qquad i=1,2,\end{equation}

and thus

(5.4) \begin{equation}{\mathcal{Q}}^ng^n_i[f](x,k) = \breve{g}^n_i[f](x,k),\qquad i=1,2.\end{equation}

For $f\in C^2(\mathbb{R}^{d})$ and $n\in\mathbb{N}$ , let

(5.5) \begin{equation}g^n_{3}[f](x,k)\,:=\, \frac{1}{n^{\alpha+\beta}}\sum_{h\in{\mathscr{K}}}\sum_{j\in{\mathscr{I}}}\Xi^n_j({x}^n_{*},h)\Upsilon_{kh}\,\partial_jf(x).\end{equation}

Note that the function $g^n_3[f]$ corresponds to the covariance of the background Markov process $J^n$ . We let $g^n[f]$ denote the sum of the above functions, that is,

(5.6) \begin{equation}g^n[f](x,k) \,:=\, g^n_{1}[f](x,k)+ g^n_{2}[f](x,k) + g^n_{3}[f](x,k),\qquad (x,k)\in\mathbb{R}^d\times{\mathscr{K}}.\end{equation}

To keep the algebraic expressions in the proofs manageable, we adopt the notation introduced in the following definition.

Definition 5.1. We define the operators $[{\mathcal{D}}^n_z]^{0}$ and $[{\mathcal{D}}^n_z]^{1}_j$ , $j\in{\mathscr{I}}$ , by

\begin{align*}[{\mathcal{D}}^n_z]^{0}f(x) & := f(x+n^{-\beta}z) - f(x)- n^{-\beta}\sum_{i\in{\mathscr{I}}}z_i\partial_if(x)- n^{-2\beta}\sum_{i,j\in{\mathscr{I}}}z_{i}z_{j}{\partial_{ij}}\,f(x),\\ \,[\,{\mathcal{D}}^{n}_{z}\,]^{1}_{j}f(x) & := \partial_jf(x+n^{-\beta}z) - \partial_jf(x)- n^{-\beta}\sum_{i\in{\mathscr{I}}}z_{i}\partial_{ij}\, f(x),\end{align*}

for $f\in C^2(\mathbb{R}^d)$ and $z\in{\mathscr{Z}}^{\,\,n}$ . In addition, we define

\begin{equation*}\begin{aligned}{\mathcal{R}}^n_1[f](\hat{x},k) &\,:=\, \sum_{z}^{}\xi^n_{z}(n^{\beta}\hat{x}+ {x}^n_{*},k)\,[{\mathcal{D}}^n_z]^{0}f(\hat{x}),\\{\mathcal{R}}^n_2[f](\hat{x}) &\,:=\, \frac{1}{2}\sum_{i,j\in{\mathscr{I}}}\sum_{k\in{\mathscr{K}}}\pi_k\bigl(\bar{\Gamma}^n_{ij}(\hat{x},k) - \bar{\Gamma}^n_{ij}(0,k)\bigr)\partial_{ij}f(\hat{x}),\\{\mathcal{R}}^n_3[f](\hat{x},k) &\,:=\, \frac{1}{n^{\alpha+2\beta}}\sum_{i,j\in{\mathscr{I}}}\sum_{h,l\in{\mathscr{K}}}\bigl(\Xi^n_i({x}^n_* + n^{\beta}\hat{x},l) - \Xi^n_i({x}^n_*,l)\bigr)\Xi^n_j({x}^n_{*},h)\pi_l\Upsilon_{lh}\partial_{ij}f(\hat{x}),\\{\mathcal{R}}^n_4[f](\hat{x},k) &\,:=\, \frac{1}{n^{\alpha+\beta}}\sum_{z}\sum_{h\in{\mathscr{K}}}\xi^n_z(n^{\beta}\hat{x} + {x}^n_{*},k)\Upsilon_{kh}\sum_{j\in{\mathscr{I}}}\Xi^n_j({x}^n_{*},h)[{\mathcal{D}}^n_z]^{1}_jf(\hat{x}),\\{\mathcal{R}}^n_5[f](\hat{x},k) &\,:=\, {\mathscr{L}}_k^{\kern1.6pt n}\,g^n_{1}[f](\hat{x},k),\\[3pt]{\mathcal{R}}^n_6[f](\hat{x},k) &\,:=\, {\mathscr{L}}^{\,n}_{k}\,g^n_{2}[f](\hat{x},k).\end{aligned}\end{equation*}

The following lemma establishes a useful identity involving the generator of $(\widehat{X}^n,J^n)$ in (2.2) and that of $\widehat{Y}^n$ in (2.31) and the operators ${\mathcal{R}}^n_i$ in Definition 5.1.

Lemma 5.1. Under Assumption 2.2(ii), for $f\in C^2(\mathbb{R}^d)$ , we have

(5.7) \begin{equation}\widehat{\mathscr{\!\!L}}^{\,\,n} f(\hat{x}) + \widehat{\mathscr{\!\!L}}^{\,\,n} g^n[f](\hat{x},k) = {\mathscr{A}}^n f(\hat{x}) + \sum_{i=1}^6 {\mathcal{R}}^n_i[f](\hat{x},k),\qquad (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation}

Proof. By (2.2) we have

(5.8) \begin{equation}\widehat{\mathscr{\!\!L}}^{\,\,n} g^n[f](\hat{x},k) = \sum_{i=1}^3\Bigl({\mathscr{L}}_k^{\kern1.6pt n}g^n_{i}[f](\hat{x},k)+ {\mathcal{Q}}^ng^n_{i}[f](\hat{x},k)\Bigr),\end{equation}

and $\,\,\widehat{\mathscr{\!\!L}}^{\,\,n}f(\hat{x}) = {\mathscr{L}}_k^{\kern1.6pt n} f(\hat{x})$ for any $f\in C^2(\mathbb{R}^d)$ .

We first show that

(5.9) \begin{equation}\begin{aligned}&{\mathscr{L}}_k^{\kern1.6pt n}f(\hat{x}) + {\mathcal{Q}}^ng^n_{1}[f](\hat{x},k)+ {\mathcal{Q}}^ng^n_{3}[f](\hat{x},k) \\&\mspace{80mu} = \sum_{i\in{\mathscr{I}}}\bar{b}^n_i(\hat{x}) \partial_if(\hat{x})+ \frac{1}{2}\sum_{i,j\in{\mathscr{I}}}\bar{a}^n_{ij}\partial_{ij}f(\hat{x})+ {\mathcal{R}}^n_1[f](\hat{x},k) + {\mathcal{R}}^n_2[f](\hat{x}).\end{aligned}\end{equation}

Using (2.3) and (5.5), we obtain

(5.10) \begin{equation}{\mathcal{Q}}^ng^n_{3}[f](\hat{x},k) = \sum_{h\in{\mathscr{K}}}\sum_{\ell\in{\mathscr{K}}} q_{k\ell}\Upsilon_{\ell h}\sum_{j\in{\mathscr{I}}}\frac{\Xi^n_j({x}^n_*,h)}{n^\beta}\partial_jf(\hat{x}).\end{equation}

Since $Q\Upsilon = \Pi - I$ , where I denotes the identity matrix, it follows by (2.20) that

(5.11) \begin{equation}\sum_{h\in{\mathscr{K}}}\sum_{\ell\in{\mathscr{K}}}q_{k\ell}\Upsilon_{\ell h}\Xi^n_j({x}^n_{*},h) = \sum_{h\in{\mathscr{K}}}\pi_h\Xi^n_j({x}^n_*,h) - \Xi^n_j({x}^n_*,k) = - \Xi^n_j({x}^n_*,k),\end{equation}

where in the second equality we use Assumption 2.2(ii). Thus, by (5.10) and (5.11), we have

(5.12) \begin{equation}{\mathcal{Q}}^ng^n_{3}[f](\hat{x},k) = \sum_{j\in{\mathscr{I}}} -\frac{\Xi^n_j({x}^n_*,k)}{n^{\beta}}\,\partial_jf(\hat{x}) = \sum_{j\in{\mathscr{I}}} -\widehat{\Xi}^n_j(0,k)\,\partial_jf(\hat{x}).\end{equation}

By (2.3) and a standard identity, we obtain

(5.13) \begin{equation}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n}f(\hat{x}) & = \sum_{z\in{\mathscr{Z}}^{\,\,n}}\xi^n_z(n^{\beta}\hat{x}+ x^n_*,k)\biggl(\sum_{i\in{\mathscr{I}}}n^{-\beta}z_i\partial_i f(\hat{x}) \\ &\mspace{200mu}+ \sum_{i,j\in{\mathscr{I}}}n^{-2\beta}z_iz_j\partial_{ij}f(\hat{x})+ [{\mathcal{D}}^n_z]^0 f(\hat{x})\biggr)\\[5pt] & = \sum_{i\in{\mathscr{I}}}\widehat{\Xi}^n_i(\hat{x},k)\partial_if(\hat{x}) +\sum_{i,j\in{\mathscr{I}}} \bar{\Gamma}^n_{ij}(\hat{x},k) \partial_{ij}f(\hat{x})+ {\mathcal{R}}^n_1[f](\hat{x},k).\end{aligned}\end{equation}

Thus (5.9) follows from (5.1), (5.4), (5.12), and (5.13).

Next, we show that

(5.14) \begin{equation}{\mathscr{L}}_k^{\kern1.6pt n}\,g^n_{3}[f](\hat{x},k) + {\mathcal{Q}}^ng^n_{2}[f](\hat{x},k) = \frac{1}{2}\sum_{i,j\in{\mathscr{I}}}\theta^n_{ij}\partial_{ij}f(\hat{x})+ {\mathcal{R}}^n_3[f](\hat{x},k) + {\mathcal{R}}^n_4[f](\hat{x},k).\end{equation}

We have

\begin{equation*}\begin{aligned}{\mathscr{L}}_k^{\kern1.6pt n} & \,g^n_3[f](\hat{x},k) \\& = \frac{1}{n^{\alpha + \beta}}\sum_{z}\xi^n_{z}(n^{\beta}\hat{x} + {x}^n_{*},k)\sum_{h\in{\mathscr{K}}}\sum_{j\in{\mathscr{I}}}\Xi^n_j({x}^n_{*},h)\Upsilon_{kh}\,\bigl(\partial_jf(\hat{x} + n^{-\beta}z) - \partial_jf(\hat{x})\bigr)\end{aligned}\end{equation*}

by (2.3). It is clear that

\begin{equation*}\partial_jf(\hat{x}+n^{-\beta}z) - \partial_jf(\hat{x}) = n^{-\beta}\sum_{i\in{\mathscr{I}}}z_i\partial_{ij} f(\hat{x}) +[{\mathcal{D}}^n_z]^{1}_jf(\hat{x})\end{equation*}

and

\begin{equation*}\sum_{z}z_i\xi^n_{z}(n^{\beta}\hat{x} + {x}^n_{*},k) = \Xi^n_i({x}^n_*,k)+ \bigl(\Xi^n_i({x}^n_* + n^{\beta}\hat{x},k) - \Xi^n_i({x}^n_*,k)\bigr).\end{equation*}

Therefore, (5.14) follows from combining these identities with (5.2) and (5.4).

Hence, we obtain (5.7) by adding (5.8), (5.9), and (5.14), and using the definitions of ${\mathcal{R}}^n_i[f]$ for $i=5,6$ . This completes the proof.

The following lemma provides needed estimates for ${\mathcal{R}}^n_5$ and ${\mathcal{R}}^n_6$ .

Lemma 5.2. Under Assumption 2.2 (i)–(iii), there exists some positive constant C such that

(5.15) \begin{equation}\begin{aligned}&\bigl\vert{{\mathcal{R}}^n_5[f](\hat{x},k)}\bigr\vert\\[4pt] &\,\leq\,C \biggl[\biggl(\frac{1}{n^{\alpha}}\vert{\hat{x}}\vert+ \frac{1}{n^{\alpha+\beta -1}} \biggr) \vert{\nabla f(\hat{x})}\vert+ \biggl(\frac{1}{n^{\alpha+\beta}}\vert{\hat{x}}\vert + \frac{1}{n^{\alpha+2\beta-1}} \biggr)\bigl\vert{\nabla^2 f(\hat{x})}\bigr\vert\\[4pt] &\quad + \biggl(\frac{1}{n^{\alpha - \beta}}\vert{\hat{x}}\vert^2+ \frac{1}{n^{\alpha-1}}\vert{\hat{x}}\vert\biggr)\,\max_{z\in{\mathscr{Z}}^{\,\,n}}\vert{\nabla f(\hat{x} + n^{-\beta}z) - \nabla f(\hat{x})}\vert\\[4pt] &\quad + \biggl(\frac{1}{n^{\alpha}}\vert{\hat{x}}\vert^2+ \frac{1}{n^{\alpha+\beta-1}}\vert{\hat{x}}\vert+ \frac{1}{n^{\alpha + 2\beta - 2}}\biggr)\,\max_{z\in{\mathscr{Z}}^{\,\,n}}\bigl\vert{ \nabla^2 f(\hat{x}+n^{-\beta}z) - \nabla^2 f(\hat{x})}\bigr\vert\biggr]\end{aligned}\end{equation}

and

(5.16) \begin{equation}\begin{aligned}\bigl\vert{{\mathcal{R}}^n_6[f](\hat{x},k)}\bigr\vert &\,\leq\, C\biggl[\biggl(\frac{1}{n^{2\alpha+\beta -1}}\vert{\hat{x}}\vert+ \frac{1}{n^{2\alpha+2\beta-2}}\biggr)\,\bigl\vert{\nabla^2 f(\hat{x})}\bigr\vert \\[4pt] &\mspace{40mu} +\biggl(\frac{1}{n^{2\alpha - 1}}\vert{\hat{x}}\vert^2+ \frac{1}{n^{2\alpha + \beta -2}}\vert{\hat{x}}\vert \\[4pt] &\mspace{100mu}+ \frac{1}{n^{2\alpha + 2\beta - 3}}\biggr)\max_{z\in{\mathscr{Z}}^{\,\,n}}\,\bigl\vert{\nabla^2 f(\hat{x}+n^{-\beta}z) - \nabla^2 f(\hat{x})}\bigr\vert\biggr],\end{aligned}\end{equation}

for any $(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}$ .

Proof. Recall the functions $g^n_1[f]$ and $g^n_{2}[f]$ in (5.3). It follows by (2.21) and (2.22) that

(5.17) \begin{align}\vert{\xi^n_z(x, k)}\vert \,\leq\,\widetilde{C}(\vert{x - x^n_*}\vert + n)\end{align}

and

(5.18) \begin{align}\vert{\xi^{n}_z(x+x^{n}_*,k) - \xi^n_z(x^{n}_*,k)}\vert \,\leq\, \widetilde{C}\vert{x}\vert\end{align}

for $(x,k)\in\mathbb{R}^d\times{\mathscr{K}}$ , $z\in{\mathscr{Z}}^{\,\,n}$ , and $n\in\mathbb{N}$ . By Assumption 2.2(i), and applying (2.20) and (5.18) it is straightforward to verify that

(5.19) \begin{equation}\Biggl| \sum_{k\in{\mathscr{K}}} \pi_k \widehat\Xi^n(\hat{x},k) \Biggr|\,\leq\, \widetilde{C}\widetilde{N}_0m_0\vert{\hat{x}}\vert\qquad \forall\,\hat{x}\in\mathbb{R}^{d}.\end{equation}

Thus, by (5.18) and (5.19),we have

(5.20) \begin{equation}\bigl\vert{\bar{b}^n(\hat{x}) - \bigl(\widehat{\Xi}^n(\hat{x},k)- \widehat{\Xi}^n(0,k)\bigr)}\bigr\vert\,\leq\, 2\widetilde{C}\widetilde{N}_0m_0\vert{\hat{x}}\vert\qquad \forall\,(\hat{x},k)\in\mathbb{R}^d\times{\mathscr{K}}.\end{equation}

Applying (5.17), we obtain

(5.21) \begin{equation}\bigl\vert{\bar{a}^n(\hat{x}) - \bar{\Gamma}^n(\hat{x},k)}\bigr\vert\,\leq\, 2\widetilde{C}\widetilde{N}_0m_0^2\bigl(n^{-\beta}\vert{\hat{x}}\vert + n^{1- 2\beta}\bigr)\qquad \forall (\hat{x},k)\in\mathbb{R}^d\times{\mathscr{K}},\end{equation}

and

(5.22) \begin{equation}\Biggl| \sum_{l\in{\mathscr{K}}}\pi_l\xi^n_z(n^{\beta}\hat{x} + {x}^n_{*},l)\Upsilon_{lh}- \xi^n_z(n^{\beta}\hat{x} + {x}^n_{*},k)\Upsilon_{kh} \Biggr|\,\leq\, C_1\bigl(n^{\beta}\vert{\hat{x}}\vert + n\bigr) \qquad\forall\, \hat{x}\in\mathbb{R}^d\end{equation}

and all $k,h\in{\mathscr{K}}$ and $z\in{\mathscr{Z}}^{\,\,n}$ , for some positive constant $C_1$ . We have

(5.23) \begin{equation}\vert{\Xi^n(x^n_*,k)}\vert \,\leq\, \widetilde{C}\widetilde{N}_0m_0n\qquad \forall\,k\in{\mathscr{K}},\ n\in\mathbb{N},\end{equation}

by (2.22), and

(5.24) \begin{equation}\vert{\xi^n_z(n^{\beta}\hat{x} + x^n_*,k)}\vert \,\leq\,\widetilde{C}(n^{\beta}\vert{\hat{x}}\vert +n) \qquad\forall\,(\hat{x},k) \in\mathbb{R}^d\times{\mathscr{K}},\ z\in{\mathscr{Z}}^{\,\,n},\ n\in\mathbb{N},\end{equation}

by (5.17). From (2.21), we obtain

(5.25) \begin{equation}\bigl\vert{\widehat{\Xi}^n(\hat{x} + n^{-\beta}z,k) - \widehat{\Xi}^n(\hat{x},k)}\bigr\vert\,\leq\, n^{-\beta}\widetilde{C}\widetilde{N}_0m_0^2\end{equation}

and

(5.26) \begin{align}\bigl\vert{\bar{\Gamma}^n(\hat{x} + n^{-\beta}z,k) - \bar{\Gamma}^n(\hat{x},k)}\bigr\vert\,\leq\, n^{-2\beta}\widetilde{C}\widetilde{N}_0m_0^3\end{align}

for $(\hat{x},k)\in\mathbb{R}^d\times{\mathscr{K}}$ , $z\in{\mathscr{Z}}^{\,\,n}$ , and $n\in\mathbb{N}$ . Repeating similar calculations as in (4.10) and (4.13), and applying (5.20), (5.21), and (5.24)–(5.26), we have

\begin{equation*}\begin{aligned}&\bigl\vert{{\mathcal{R}}^n_5[f](\hat{x}),k}\bigr\vert\\ &\mspace{20mu}\leq \widetilde{C}\widetilde{N}_0m_0\sum_{k,\ell\in{\mathscr{K}}}\vert{{\mathcal{T}}_{k\ell}}\vert\biggl(2\widetilde{C}\widetilde{N}_0m_0^2\frac{(n^{\beta}\vert{\hat{x}}\vert +n)}{n^{\alpha+\beta}}\vert{\nabla f(\hat{x})}\vert\\ &\mspace{40mu} + 2\widetilde{C}\widetilde{N}_0m_0\vert{\hat{x}}\vert\frac{(n^{\beta}\vert{\hat{x}}\vert +n)}{n^{\alpha}}\max_{z\in{\mathscr{Z}}^{\,\,n}}\vert{\nabla f(\hat{x} + n^{-\beta}z) - \nabla f(\hat{x})}\vert\\ &\mspace{40mu} + \widetilde{C}\widetilde{N}_0m_0^3\frac{(n^{\beta}\vert{\hat{x}}\vert +n)}{n^{\alpha+2\beta}} \vert{\nabla^2 f(\hat{x})}\vert\\[4pt] &\mspace{40mu} + 2\widetilde{C}\widetilde{N}_0m_0^2\frac{\bigl(n^{-\beta}\vert{\hat{x}}\vert + n^{1- 2\beta}\bigr)(n^{\beta}\vert{\hat{x}}\vert +n)}{n^{\alpha}}\max_{z\in{\mathscr{Z}}^{\,\,n}}\vert{\nabla f(\hat{x} + n^{-\beta}z) - \nabla f(\hat{x})}\vert\biggr),\end{aligned}\end{equation*}

which establishes (5.15). The estimate for ${\mathcal{R}}^n_6$ in (5.16) obtained in a similar manner by applying (2.21) and (5.22)–(5.24). This completes the proof.

We borrow the following estimates for solutions to the Poisson equation for the operator ${\mathscr{A}}^n$ from [Reference Gurvich11, Theorem 4.1] and the discussion following this theorem. Recall that $\nu^n$ is the steady-state distribution of $\hat{Y}^n$ in (2.27).

Lemma 5.3. Grant Assumption 2.2, and fix a function ${\mathscr{V}}$ in Assumption 2.3. Let $f\in C^{0,1}(\mathbb{R}^d)$ be such that $\Vert{f}\Vert_{C^{0,1}({\mathscr{B}}_x)} \leq {\mathscr{V}}(x)$ and $\nu^n(f) = 0$ . Then the function $u^n_f\in C^2(\mathbb{R}^d)$ defined by

\begin{equation*}u^n_f(x) \,:=\, \int_0^{\infty} \mathbb{E}_x\bigl[f\bigl(\widehat{Y}^n(s)\bigr)\bigr]\,\textrm{d}{s}\end{equation*}

is the unique (up to an additive constant) solution to the Poisson equation

(5.27) \begin{equation}{\mathscr{A}}^n u = -f,\end{equation}

and satisfies

(5.28) \begin{equation}\vert{\nabla u^n_f(x)}\vert \,\in\, {\mathscr{O}}\bigl((1 + \vert{x}\vert){\mathscr{V}}(x)\bigr),\qquad \bigl\vert{\nabla^2 u^n_f(x)}\bigr\vert \,\in\,{\mathscr{O}}\bigl((1 + \vert{x}\vert^2){\mathscr{V}}(x)\bigr),\end{equation}

and

(5.29) \begin{equation}\bigl[u^n_f\bigr]_{2,1;B_{\frac{m_0}{\sqrt{n}}}(x)} \,\in\,{\mathscr{O}}\bigl((1 + \vert{x}\vert^3){\mathscr{V}}(x) \bigr).\end{equation}

In the following lemma, we consider the solution of the Poisson equation in (5.27) and establish an estimate for the sum of terms ${\mathcal{R}}^n_i[u^n_f]$ , $i=1,\dotsc,6$ , given in Definition 5.1.

Lemma 5.4. Grant Assumption 2.2, and fix a function ${\mathscr{V}}$ in Assumption 2.3. Let f and $u^n_f$ be as in Lemma 5.3. Then

(5.30) \begin{equation}\sum_{j=1}^{6}{\mathcal{R}}^n_j[u^n_f](\hat{x},k) = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge1/2}}\biggr){\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^5){\mathscr{V}}(\hat{x})\bigr)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation}

Proof. Note that

\begin{equation*}[{\mathcal{D}}^n_z]^{0}u^n_f(\hat{x}) = n^{-2\beta}\sum_{i,j\in{\mathscr{I}}}z_iz_j\partial_{ij}\bigl(u^n_f(\hat{x} + \varepsilon^n_{x,z}) - u^n_f(\hat{x})\bigr)\end{equation*}

for $\varepsilon^n_{\hat{x},z}\in\prod_{i\in{\mathscr{I}}}[\hat{x}_i, \hat{x}_i + n^{-\beta}z_i]$ . Applying (4.6) and (5.29), we obtain

(5.31) \begin{equation}{\mathcal{R}}^n_1[u^n_f](\hat{x},k) = \frac{1}{n^{\beta}}{\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^4){\mathscr{V}}(\hat{x})\bigr)\qquad \forall\, (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}.\end{equation}

By (5.18), we have

\begin{equation*}\vert{\bar{\Gamma}^n_{ij}(\hat{x},k) - \bar{\Gamma}^n_{ij}(0,k)}\vert \,\leq\,\widetilde{C}\widetilde{N}_0m_0^2n^{-\beta}\vert{\hat{x}}\vert\qquad \forall\, (\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation*}

and thus it follows by (5.28) that

(5.32) \begin{equation}{\mathcal{R}}^n_2[u^n_f](\hat{x}) = \frac{1}{n^{\beta}}{\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^3){\mathscr{V}}(\hat{x})\bigr).\end{equation}

Applying Definition 5.1, (5.18), (5.23), and (5.28), we obtain

(5.33) \begin{equation}{\mathcal{R}}^n_3[u^n_f](\hat{x},k) = \frac{1}{n^{\alpha + \beta - 1}}{\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^3){\mathscr{V}}(\hat{x})\bigr)\qquad \forall\,k\in{\mathscr{K}}.\end{equation}

Repeating the above procedure, and using Definition 5.1, (5.17), (5.23), and (5.29), we obtain

(5.34) \begin{equation}{\mathcal{R}}^n_4[u^n_f](\hat{x},k) = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha+3\beta-2}}\biggr){\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^4){\mathscr{V}}(\hat{x})\bigr) \qquad \forall\,k\in{\mathscr{K}} .\end{equation}

It follows by Lemma 5.2, (5.28), and (5.29) that

(5.35) \begin{align}{\mathcal{R}}^n_5[u^n_f](\hat{x},k) & = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha + \beta- 1}}\biggr){\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^5){\mathscr{V}}(\hat{x})\bigr) \end{align}

and

(5.36) \begin{align}{\mathcal{R}}^n_6[u^n_f](\hat{x},k) & = {\mathscr{O}}\biggl(\frac{1}{n^{2\alpha + 3\beta- 3}}\biggr){\mathscr{O}}\bigl((1 + \vert{\hat{x}}\vert^5){\mathscr{V}}(\hat{x})\bigr) \end{align}

for all $k\in{\mathscr{K}}$ . On the other hand, when $\alpha > 1$ , $\beta = \frac{1}{2}$ , $\alpha + \beta - 1 \geq \beta$ and $2\alpha + 3\beta -3 \geq \beta$ , and when $\alpha \leq 1$ , $\alpha + \beta - 1 = 2\alpha + 3\beta - 3 = \alpha/2$ and $\alpha + 3\beta - 2 = \beta$ . Then, by using (5.31)–(5.36), we have shown (5.30). This completes the proof.

Proof of Theorem 2.2. Without loss of generality, we assume that $\nu^n(f) = 0$ (see [Reference Gurvich11, Remark 3.2]). Recall the function $g^n$ in (5.6). Applying Lemma 5.1, it follows that

(5.37) \begin{equation}\begin{aligned}&\mathbb{E}_{\pi^n}\Bigl[u^n_f\bigl(\widehat{X}^n(T)\bigr)+ g^n[u^n_f]\bigl(\widehat{X}^n(T),J^n(T)\bigr)\Bigr] \\[4pt] &\mspace{60mu} = \mathbb{E}_{\pi^n}\Bigl[u^n_f(\widehat{X}^n(0))+ g^n[u^n_f](\widehat{X}^n(0),J^n(0))\Bigr] \\[4pt] &\mspace{100mu}+ \mathbb{E}_{\pi^n}\biggl[\int_0^{T}{\mathscr{A}}^n u^n_f\bigl(\widehat{X}^n(s)\bigr)\,\textrm{d}{s}\biggr]\\[4pt] &\mspace{150mu}+\sum_{j=1}^{6}\mathbb{E}_{\pi^n}\biggl[\int_0^{T}{\mathcal{R}}^n_j[u_f^n]\bigl(\widehat{X}^n(s),J^n(s)\bigr)\,\textrm{d}{s}\biggr].\end{aligned}\end{equation}

By Lemma 5.4, we have

(5.38) \begin{equation}\begin{aligned}\Biggl|\sum_{j=1}^{6}\mathbb{E}_{\pi^n}\biggl[\int_0^{T} & {\mathcal{R}}^n_j[u_f^n]\bigl(\widehat{X}^n(s),J^n(s)\bigr)\,\textrm{d}{s}\biggr] \Biggr|\\[4pt] &\,\leq\, {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge1/2}}\biggr)\mathbb{E}_{\pi^n}\biggl[\int_0^{T}\Bigl(1 + {\mathscr{V}}\bigl(\widehat{X}^n(s)\bigr)\bigl(1 + \vert{\widehat{X}^n(s)}\vert^5\bigr)\Bigr)\,\textrm{d}{s}\biggr]\\[4pt] & = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge1/2}}\biggr)T \int_{\mathbb{R}^d\times{\mathscr{K}}}(1 +{\mathscr{V}}(\hat{x}))(1 + \vert{\hat{x}}\vert)^5\pi^n(\textrm{d}{\hat{x}},\textrm{d}{k}).\end{aligned}\end{equation}

Applying (5.6), (5.24), and (5.28), we obtain

(5.39) \begin{equation}\vert{g^n(\hat{x},k)}\vert \,\leq\, C_1\bigl(1 + (1 + \vert{\hat{x}}\vert^3){\mathscr{V}}(\hat{x})\bigr)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

for some positive constant $C_1$ and all large enough n. Since $\bigl\vert{u^n_f}\bigr\vert\in{\mathscr{O}}({\mathscr{V}})$ by the claim in (22) of [Reference Gurvich11], it follows by (5.39) that

(5.40) \begin{equation}\begin{aligned}&\Bigl\vert{\mathbb{E}_{\pi^n}\Bigl[u^n_f\bigl(\widehat{X}^n(T)\bigr) + g^n[u^n_f] \bigl(\widehat{X}^n(T),J^n(T)\bigr)\Bigr]}\Bigr\vert \\[4pt] &\,\leq\,C_2\biggl( 1 + \int_{\mathbb{R}^d\times{\mathscr{K}}}{\mathscr{V}}(\hat{x})(1 + \vert{\hat{x}}\vert)^3\pi^n(\textrm{d}{\hat{x}},\textrm{d}{k})\biggr)\end{aligned}\end{equation}

for some positive constant $C_2$ . By (5.27),

(5.41) \begin{equation}\mathbb{E}_{\pi^n}\biggl[\int_0^{T}{\mathscr{A}}^n u^n_f\bigl(\widehat{X}^n(s)\bigr)\,\textrm{d}{s}\biggr] = - \mathbb{E}_{\pi^n}\biggl[\int_0^{T}f\bigl(\widehat{X}^n(s)\bigr)\,\textrm{d}{s}\biggr] = - T\pi^n(f).\end{equation}

Since $\pi^n$ is the stationary distribution, the bound in (5.40) also holds for the first term on the right-hand side of (5.37). Thus, applying (5.37), (5.38), (5.40), and (5.41), we obtain

(5.42) \begin{equation}\begin{aligned}T\bigl\vert{\pi^n(f)}\bigr\vert &\,\leq\, 2C_2\biggl( 1 + \int_{\mathbb{R}^d\times{\mathscr{K}}}{\mathscr{V}}(\hat{x})(1 + \vert{\hat{x}}\vert)^3\pi^n(\textrm{d}{\hat{x}},\textrm{d}{k})\biggr) \\[4pt] &\mspace{60mu} + {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge1/2}}\biggr)T \int_{\mathbb{R}^d\times{\mathscr{K}}}(1 +{\mathscr{V}}(\hat{x}))(1 + \vert{\hat{x}}\vert)^5\pi^n(\textrm{d}{\hat{x}},\textrm{d}{k}).\end{aligned}\end{equation}

Therefore, dividing both sides of (5.42) by T, taking $T\rightarrow\infty$ , and applying (2.32), we obtain

\begin{equation*}\vert{\pi^n(f)}\vert = {\mathscr{O}}\biggl(\frac{1}{n^{\alpha/2\wedge1/2}}\biggr).\end{equation*}

This completes the proof.

Proof of Corollary 2.3. We claim that for some positive constants $C_1$ , $\kappa_1$ , and $\kappa_2$ , a ball ${\mathscr{B}}$ , and a sequence $\epsilon_n\rightarrow 0$ as $n\rightarrow\infty$ , we have

(5.43) \begin{equation}\begin{aligned}\widehat{\mathscr{\!\!L}}^{\,\,n} \widetilde{{\mathscr{V}}}(\hat{x}) + \widehat{\mathscr{\!\!L}}^{\,\,n} g^n[\widetilde{{\mathscr{V}}}](\hat{x},k)& = {\mathscr{A}}^n \widetilde{{\mathscr{V}}}(\hat{x}) + \sum_{i=1}^6 {\mathcal{R}}^n_i[\widetilde{{\mathscr{V}}}](\hat{x},k)\\&\,\leq\, \kappa_1\mathbb{1}_{{{\mathscr{B}}}}(\hat{x}) - \kappa_2 \widetilde{{\mathscr{V}}}(\hat{x}) + C_1 +\epsilon_n\widetilde{{\mathscr{V}}}(\hat{x})\end{aligned}\end{equation}

for all $(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}}$ . Indeed the equality in (5.43) follows from Lemma 5.1. Following the calculation in the proof of Lemma 5.4, and using (2.35), the inequality in (5.43) follows from Assumption 2.3 and Lemma 5.2. By Assumption 2.2 and (2.35), we have

(5.44) \begin{equation}C_2\widetilde{{\mathscr{V}}}(\hat{x}) - C_3\,\leq\, \widetilde{{\mathscr{V}}}(\hat{x}) + g^n[\widetilde{{\mathscr{V}}}](\hat{x},k)\,\leq\, C_3(\widetilde{{\mathscr{V}}}(\hat{x}) + 1)\qquad \forall\,(\hat{x},k)\in\widehat{{\mathfrak{X}}}^n\times{\mathscr{K}},\end{equation}

for some positive constants $C_2$ and $C_3$ . Combining (5.43) and (5.44), we see that $V(\hat{x},k):= \widetilde{{\mathscr{V}}}(\hat{x}) + g^n[\widetilde{{\mathscr{V}}}](\hat{x},k)$ satisfies $\widehat{\mathscr{\!\!L}}^{\,\,n} V(\hat{x},k) \leq \kappa_3 \mathbb{1}_{{\mathscr{B}}^{\prime}}(x) -\kappa_4 V(\hat{x},k)$ for some positive constants $\kappa_3$ and $\kappa_4$ , and a ball ${\mathscr{B}}^{\prime}$ . This together with (5.44) and the hypothesis in (2.34) implies (2.32), and completes the proof.

Appendix A. The diffusion limit

Proposition A.1, which follows, shows that under suitable assumptions, the processes $\widehat{X}^n$ in (2.24) and $\widehat{Y}^n$ in (2.27) have the same diffusion limit. This proposition is interesting in its own right.

Let $(\mathbb{D}^d,{\mathcal{J}}_1)$ denote the space of $\mathbb{R}^{d}$ -valued càdlàg functions endowed with the ${\mathcal{J}}_1$ topology (see, e.g., [Reference Billingsley6]).

Proposition A.1. Grant Assumption 2.2. In addition, suppose that $\widehat{X}^n(0) \Rightarrow y_0$ ,

(A.1) \begin{equation}\frac{\xi^n_z(x^n_* + n^{\beta}\hat{x},k) - \xi^n_z(x^n_*,k)}{n^{\beta}}\,\xrightarrow[n\to\infty]{}\, \widehat{\xi}_z(\hat{x},k) \qquad \forall\,(k,z)\in{\mathscr{K}}\times{\mathscr{Z}}^{\,\,n}\end{equation}

uniformly on compact sets in $\mathbb{R}^d$ , $\widehat{M}^n$ is a square-integrable martingale, and

(A.2) \begin{equation}\frac{\Xi^n(x^n_*,k)}{n}\,\xrightarrow[n\to\infty]{}\, \overline{\Xi}(k)\,\in\,\mathbb{R}^d\qquad \forall\,k\in{\mathscr{K}}.\end{equation}

Then $\widehat{X}^n$ and $\widehat{Y}^n$ have the same diffusion limit $\widehat{X}$ in $(\mathbb{D}^d,{\mathcal{J}}_1)$ , and $\widehat{X}$ is the strong solution of the stochastic differential equation

\begin{equation*}\textrm{d} \widehat{X}(t) = \bar{b}\bigl(\widehat{X}(t)\bigr)\,\textrm{d}{t}+ \sigma_\alpha \textrm{d} W(t),\end{equation*}

with $\widehat{X}(0) = y_0$ , where

\begin{equation*}\bar{b}(\hat{x})\,:=\, \sum_{k\in{\mathscr{K}}}\pi_k\sum_{z}z\,\widehat{\xi}_z(\hat{x},k),\end{equation*}

\begin{equation*}(\sigma_\alpha)^{\mathsf{T}}\sigma_\alpha \,:=\, \begin{cases}\sum_{k\in{\mathscr{K}}}\pi_k\bar{\Gamma}(k) &\quad {for }\ \alpha>1, \\[3pt] \sum_{k\in{\mathscr{K}}}\pi_k\bar{\Gamma}(k) + \Theta &\quad {for }\ \alpha = 1, \\[3pt] \Theta &\quad {for }\ \alpha <1,\end{cases}\end{equation*}

and $\Theta = [\theta_{ij}]$ is defined by

\begin{equation*}\theta_{ij} \,:=\,2\sum_{k,\ell\in{\mathscr{K}}}\overline{\Xi}_i(k)\overline{\Xi}_j(\ell)\pi_k\Upsilon_{k\ell},\qquad i,j\in{\mathscr{I}}.\end{equation*}

Proof. Recall that $\sum_{k\in{\mathscr{K}}} \pi_k\Xi^n({x}^n_*,k) = 0$ and $\widehat{\Xi}^n(0,k) = n^{-\beta}\Xi^n({x}^n_*,k)$ . Recall the representation of $\widehat{X}^n$ in (2.24). By [Reference Pang, Talreja and Whitt21, Lemma 5.8], $\widehat{M}^n$ is stochastically bounded; see also the proof of [Reference Arapostathis, Das, Pang and Zheng3, Theorem 2.1(i)]. Since $\widehat{\Xi}^n$ is Lipschitz continuous by (2.21), it follows by the same argument as in the proof of [Reference Pang, Talreja and Whitt21, Lemma 5.5] that $\widehat{X}^n$ is stochastically bounded. Thus, by [Reference Pang, Talreja and Whitt21, Lemma 5.9], $n^{-1}X^n$ converges to the zero process in $(\mathbb{D}^d,{\mathcal{J}}_1)$ . We write $\widehat{X}^n$ as

(A.3) \begin{equation}\begin{aligned}\widehat{X}^n(t) & = \widehat{X}^n(0) + \sum_{k\in{\mathscr{K}}}\int_0^{t}\bigl(\widehat{\Xi}^n(\widehat{X}^n(s),k) - \widehat{\Xi}^n(0,k)\bigr)\mathbb{1}_k\bigl(J^n(s)\bigr)\,\textrm{d}{s} + \widehat{M}^n(t)\\&\mspace{150mu}+ \sum_{k\in{\mathscr{K}}}\frac{\Xi^n({x}^n_*,k)}{n}n^{1 - \beta}\int_0^{t}\bigl(\mathbb{1}_k\bigl(J^n(s)\bigr) - \pi_k\bigr)\,\textrm{d}{s}.\end{aligned}\end{equation}

Let $\hat{S}^n(t)$ and $\hat{R}^n(t)$ be d-dimensional processes denoting the second and fourth terms on the right-hand side of (A.3). It follows by [Reference Anderson, Blom, Mandjes, Thorsdottir and De Turck1, Proposition 3.2] and (2.23) that

(A.4) \begin{equation}\hat{R}^n \,\Rightarrow\,\begin{cases} W_R &\quad \text{for}\ \alpha\,\leq\,1, \\[4pt] 0 &\quad \text{for}\ \alpha\,>\,1,\end{cases}\quad \text{in} \quad (\mathbb{D}^d,{\mathcal{J}}_1),\end{equation}

as $n\rightarrow\infty$ , where $W_R$ is a d-dimensional Wiener process with the covariance matrix $\Theta$ . On the other hand, we have

(A.5) \begin{equation}\begin{aligned}&\hat{S}^n(t) \\& = \sum_{k\in{\mathscr{K}}}\int_0^{t}n^{-\alpha/2}\bigl(\widehat{\Xi}^n(\widehat{X}^n(s),k) - \widehat{\Xi}^n(0,k)\bigr)\,\textrm{d}\,\biggl(n^{\alpha/2}\int_0^{s}\bigl(\mathbb{1}_k(J^n(u))- \pi_k\bigr)\,\textrm{d}{u}\biggr) \\&\mspace{200mu}+ \sum_{k\in{\mathscr{K}}}\pi_k\int_0^{t}\bigl(\widehat{\Xi}^n(\widehat{X}^n(s),k)- \widehat{\Xi}^n(0,k)\bigr)\,\textrm{d}{s}.\end{aligned}\end{equation}

It follows by the convergence of $n^{-1}X^n$ to the zero process that $n^{-\alpha/2}\widehat{X}^n$ also converges to the zero process uniformly on compact sets in probability. Note that, for some constant C, we have $\vert{\widehat{\Xi}^n(\widehat{X}^n(s),k)- \widehat{\Xi}^n(0,k)}\vert \leq C\vert{\widehat{X}^n(s)}\vert$ for all $s\geq 0$ by (2.21). It then follows by [Reference Anderson, Blom, Mandjes, Thorsdottir and De Turck1, Proposition 3.2] and [Reference Jansen, Mandjes, De Turck and Wittevrongel14, Theorem 5.2] that the first term on the right-hand side of (A.5) converges to the zero process uniformly on compact sets in probability, as $n\rightarrow\infty$ . See also the proofs of Lemma 4.4 in [Reference Jansen, Mandjes, De Turck and Wittevrongel14] and Lemma 4.1 in [Reference Arapostathis, Das, Pang and Zheng3]. It is clear by (A.1) that

(A.6) \begin{equation}h^n(\hat{x}) \,:=\, \sum_{k\in{\mathscr{K}}}\pi_k\bigl(\widehat{\Xi}^n(\hat{x},k)- \widehat{\Xi}^n(0,k)\bigr) \,\longrightarrow\,\sum_{k\in{\mathscr{K}}}\pi_k\sum_{z}z\,\widehat{\xi}_z(\hat{x},k)\end{equation}

uniformly on compact sets in $\mathbb{R}^d$ . Note that the function $h^n$ is Lipschitz continuous by (2.21). By [Reference Pang, Talreja and Whitt21, Theorem 4.1] (see also [Reference Jansen, Mandjes, De Turck and Wittevrongel14, Lemma 4.1]), the integral mapping $x^n = \Psi^n(z^n): \mathbb{D}^d\rightarrow\mathbb{D}^d$ defined by

\begin{equation*}x^n(t) = z^n(t) + \int_0^{t} h^n(x^n(s)) \,\textrm{d}{s}\qquad \forall\,n\in\mathbb{N},\end{equation*}

is continuous in $(\mathbb{D}^d,{\mathcal{J}}_1)$ . Thus, applying the continuous mapping theorem and using (A.3)–(A.6), we obtain

\begin{equation*}\widehat{X}^n \,\Rightarrow\, \widehat{X} \quad \text{in} \quad (\mathbb{D}^d,{\mathcal{J}}_1).\end{equation*}

Recall the definitions of $\bar{\Gamma}^n$ and $\Theta^n$ in (2.25) and (2.29), respectively. As $n\rightarrow\infty$ , we have that $\bar{\Gamma}^n(0,k)\rightarrow\bar{\Gamma}(k)$ when $\alpha\geq 1$ , and $\bar{\Gamma}^n(0,k) \rightarrow 0$ when $\alpha < 1$ , by (2.23). Since $\beta = \max\{1-\alpha/2,1/2\}$ , it then follows by (A.2) that $\Theta^n \rightarrow \Theta$ when $\alpha \leq 1$ , and $\Theta^n \rightarrow 0$ when $\alpha > 1$ . It is then straightforward to verify that $\widehat{Y}^n\Rightarrow\widehat{X}$ in $(\mathbb{D}^d,{\mathcal{J}}_1)$ , as $n\rightarrow\infty$ . Therefore, $\widehat{X}^n$ and $\widehat{Y}^n$ have the same diffusion limit.