2.1. Bifurcating Markov chain: the model

We denote by ${\mathbb N}$ the set of nonnegative integers and ${\mathbb N}^*= {\mathbb N} \setminus \{0\}$ . If $(E, {\mathcal E})$ is a measurable space, then ${\mathcal B}(E)$ (resp. ${\mathcal B}_b(E)$ , resp. ${\mathcal B}_+(E)$ ) denotes the set of (resp. bounded, resp. nonnegative) ${\mathbb R}$ -valued measurable functions defined on E. For $f\in {\mathcal B}(E)$ , we set $\mathop{\parallel\! {f} \! \parallel}\nolimits_\infty =\sup\{|f(x)|$ , $x\in E\}$ . For a finite measure $\lambda$ on $(E,{\mathcal E})$ and $f\in {\mathcal B}(E)$ we shall write $\langle \lambda,f \rangle$ for $\int f(x) \, \textrm{d}\lambda(x)$ whenever this integral is well defined. For $p\geq 1$ and $f\in {\mathcal B}(E)$ , we set $\|f\|_{L^{p}(\lambda)} =\langle \lambda,|f|^p \rangle^{1/p}$ and we define the space $L^{p}(\lambda)= \left\{f \in {\mathcal B}(E);\, \|f\|_{L^{p}(\lambda)} < +\infty\right\}$ of $p\text{-}$ integrable functions with respect to $\lambda$ . For $n\in {\mathbb N}^*$ , the product space $E^n$ is endowed with the product $\sigma$ -field ${\mathcal E}^{\otimes n}$ .

Let $(S, {\mathscr S}\,)$ be a measurable space. Let Q be a probability kernel on $S \times {\mathscr S}\,$ ; that is, $Q(\cdot , A)$ is measurable for all $A\in {\mathscr S}\,$ , and $Q(x,\cdot)$ is a probability measure on $(S,{\mathscr S}\,)$ for all $x \in S $ . For any $f\in {\mathcal B}_b(S)$ , we set, for $x\in S$ ,

(1) \begin{equation}(Qf)(x)=\int_{S} f(y)\; Q(x,\textrm{d} y).\end{equation}

We define (Qf), or simply Qf, for $f\in {\mathcal B}(S)$ as soon as the integral (1) is well defined, and we have $Q f\in {\mathcal B}(S)$ . For $n\in {\mathbb N}$ , we denote by $Q^n$ the nth iterate of Q, defined by $Q^0=I_d$ , the identity map on ${\mathcal B}(S)$ , and $Q^{n+1}f=Q^n(Qf)$ for $f\in {\mathcal B}_b(S)$ .

Let P be a probability kernel on $S \times {\mathscr S}\,^{\otimes 2}$ ; that is, $P(\cdot , A)$ is measurable for all $A\in {\mathscr S}\,^{\otimes 2}$ , and $P(x,\cdot)$ is a probability measure on $\big(S^2,{\mathscr S}\,^{\otimes 2}\big)$ for all $x \in S$ . For any $g\in {\mathcal B}_b\big(S^3\big)$ and $h\in {\mathcal B}_b\big(S^2\big)$ , we set, for $x\in S$ ,

(2) \begin{equation}(Pg)(x)=\int_{S^2} g(x,y,z)\; P(x,\textrm{d} y,\textrm{d} z)\quad \text{and}\quad (Ph)(x)=\int_{S^2} h(y,z)\; P(x,\textrm{d} y,\textrm{d} z).\end{equation}

We define (Pg) (resp. (Ph)), or simply Pg for $g\in {\mathcal B}\big(S^3\big)$ $\big($ resp. Ph for $h\in {\mathcal B}\big(S^2\big)\big)$ , as soon as the corresponding integral (2) is well defined, and we have that Pg and Ph belong to ${\mathcal B}(S)$ .

We now introduce some notation related to the regular binary tree. We set $\mathbb{T}_0=\mathbb{G}_0=\{\emptyset\}$ , $\mathbb{G}_k=\{0,1\}^k$ , $\mathbb{T}_k = \bigcup _{0 \leq r \leq k} \mathbb{G}_r$ for $k\in {\mathbb N}^*$ , and $\mathbb{T} = \bigcup _{r\in {\mathbb N}} \mathbb{G}_r$ . The set $\mathbb{G}_k$ corresponds to the kth generation, $\mathbb{T}_k$ to the tree up to the kth generation, and $\mathbb{T}$ the complete binary tree. For $i\in \mathbb{T}$ , we denote by $|i|$ the generation of i ( $|i|=k$ if and only if $i\in \mathbb{G}_k$ ), and $iA=\{ij;\, j\in A\}$ for $A\subset \mathbb{T}$ , where ij is the concatenation of the two sequences $i,j\in \mathbb{T}$ , with the convention that $\emptyset i=i\emptyset=i$ .

We recall the definition of a bifurcating Markov chain from [Reference Guyon9].

Let $X=(X_i, i\in \mathbb{T})$ be a BMC on a measurable space $(S, {\mathscr S}\,)$ with initial probability distribution $\nu$ and probability kernel ${\mathcal P}$ . We define three probability kernels $P_0$ , $P_1$ , and ${\mathcal Q}$ on $S\times {\mathscr S}\,$ by

\begin{align*}P_0(x,A) & ={\mathcal P}(x, A\times S), \quad P_1(x,A)={\mathcal P}(x, S\times A) \quad \text{for $(x,A)\in S\times {\mathscr S}\,$, and} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad {\mathcal Q}=\mathop{\frac{1}{ {2}}}\nolimits(P_0+P_1).\end{align*}

Notice that $P_0$ (resp. $P_1$ ) is the restriction of the first (resp. second) marginal of ${\mathcal P}$ to S. Following [Reference Guyon9], we introduce an auxiliary Markov chain $Y=(Y_n, n\in {\mathbb N}) $ on $(S,{\mathscr S}\,)$ with $Y_0$ distributed as $X_\emptyset$ and transition kernel ${\mathcal Q}$ . The distribution of $Y_n$ corresponds to the distribution of $X_I$ , where I is chosen independently from X and uniformly at random in generation $\mathbb{G}_n$ . We shall write ${\mathbb E}_x$ when $X_\emptyset=x$ , i.e. the initial distribution $\nu$ is the Dirac mass at $x\in S$ .

We end this section with a useful inequality and the Gaussian BAR model.

Remark 2.1. By convention, for $f,g\in {\mathcal B}(S)$ , we define the function $f\otimes g\in {\mathcal B}\big(S^2\big)$ by $(\,f\otimes g)(x,y)=f(x)g(y)$ for $x,y\in S$ and introduce the notation

\begin{equation*}f\otimes _{\textrm{sym}} g= \mathop{\frac{1}{ {2}}}\nolimits(\,f\otimes g + g\otimes f) \quad \text{and}\quad f\otimes ^2= f\otimes f.\end{equation*}

Notice that ${\mathcal P}(g\otimes _{\textrm{sym}} {\bf 1})={\mathcal Q}(g)$ for $g\in {\mathcal B}_+(S)$ . For $f \in {\mathcal B}_+(S)$ , as $ f\otimes f \leq f^2 \otimes _{\textrm{sym}} {\bf 1} $ , we get

(3) \begin{equation}{\mathcal P}\big(\,f\otimes ^2\big)={\mathcal P}(\,f\otimes f) \leq {\mathcal P}\big(\,f^2\otimes _{\textrm{sym}} {\bf 1}\big) = {\mathcal Q}\big(\,f^2\big).\end{equation}

Example 2.1 (Gaussian bifurcating autoregressive process). We will consider the real-valued Gaussian bifurcating autoregressive process (BAR) $X=(X_{u},u\in\mathbb{T})$ where for all $ u \in \mathbb{T}$ ,

\begin{equation*}\begin{cases}X_{u0} = a_{0} X_{u} + b_{0} + \varepsilon_{u0}, \\[3pt] X_{u1} = a_{1} X_{u} + b_{1} + \varepsilon_{u1},\end{cases}\end{equation*}

with $a_{0}, a_{1} \in ({-}1,1)$ , $b_{0}, b_{1} \in \mathbb{R}$ , and $((\varepsilon_{u0},\varepsilon_{u1}),\, u \in \mathbb{T})$ an independent sequence of bivariate Gaussian ${\mathcal N}(0,\Gamma)$ random vectors independent of $X_{\emptyset}$ with covariance matrix as follows, where $\sigma>0$ and $\rho\in {\mathbb R}$ satisfy $|\rho| < \sigma^2$ :

\begin{equation*}\Gamma = \begin{pmatrix} \sigma^{2} \quad \rho \\[2pt] \rho \quad \sigma^{2} \end{pmatrix}.\end{equation*}

Then the process $X=(X_{u},u\in\mathbb{T})$ is a BMC with transition probability ${\mathcal P}$ given by

\begin{equation*}{\mathcal P}(x,dy,dz) = \frac{1}{2\pi \sqrt{\sigma^{4} - \rho^{2}}}\, \exp\!\left( -\frac{\sigma^{2}}{2\big(\sigma^{4} - \rho^{2}\big)}\, g(x,y,z) \right)\! dydz,\end{equation*}

with

\begin{equation*}g(x,y,z)=(y-a_{0}x-b_{0})^{2} - 2\rho\sigma^{-2}(y - a_{0}x -b_{0})(z - a_{1}x - b_{1}) + (z - a_{1}x - b_{1})^{2}.\end{equation*}

The transition kernel ${\mathcal Q}$ of the auxiliary Markov chain is defined by

\begin{equation*}{\mathcal Q}(x,dy) = \frac{1}{2\sqrt{2\pi\sigma^2}}\left(\mathop {\textrm{e}^{{-\big(y - a_{0}x - b_{0}\big)^{2}/2\sigma^{2}}}} + \mathop {\textrm{e}^{{-\big(y - a_{1}x - b_{1}\big)^{2}/2\sigma^{2}}}}\right)\! dy.\end{equation*}

2.2. Assumptions

We assume that $\mu$ is an invariant probability measure for ${\mathcal Q}$ .

We state first some regularity assumptions on the kernels ${\mathcal P}$ and ${\mathcal Q}$ and the invariant measure $\mu$ we will use later on. Notice first that by Cauchy–Schwarz we have, for $f,g\in L^4(\mu)$ ,

\begin{equation*}|{\mathcal P} (\,f\otimes g)| ^2\leq {\mathcal P} \big(\,f^2 \otimes 1\big) \,{\mathcal P}\big(1\otimes g^2\big)\leq 4 {\mathcal Q}\big(\,f^2\big)\, {\mathcal Q}\big(g^2\big),\end{equation*}

so that, as $\mu$ is an invariant measure of ${\mathcal Q}$ ,

(4) \begin{equation}\mathop{\parallel\! {\mathcal P} (\,f\otimes g) \! \parallel}\nolimits_{L^{2}(\mu)} \leq 2 \mathop{\big\| {\mathcal Q}\big(\,f^2\big) \big\|}\nolimits_{L^{2}(\mu)}^{1/2} \,\mathop{\big\|{\mathcal Q} \big(g^2\big) \big\|}\nolimits_{L^{2}(\mu)}^{1/2} \leq 2\mathop{\parallel\! {f} \! \parallel}\nolimits_{L^4(\mu)}\, \mathop{\parallel\! {g} \! \parallel}\nolimits_{L^4(\mu)},\end{equation}

where we use Jensen’s inequality for the second inequality. Similarly, for $f,g\in L^2(\mu)$ ,s

(5) \begin{equation}\langle \mu, {\mathcal P} (\,f\otimes g)\rangle \leq2\mathop{\parallel\! f \! \parallel}\nolimits_{L^{2}(\mu)}\, \mathop{\parallel\! g \! \parallel}\nolimits_{L^{2}(\mu)}.\end{equation}

We shall in fact assume that ${\mathcal P}$ (in fact only its symmetrized version) is in a sense an $L^2(\mu)$ operator; see also Remark 2.2 below.

Remark 2.2. Let $\mu$ be an invariant probability measure of ${\mathcal Q}$ . If there exists a finite constant M such that for all $f,g \in L^{2}(\mu)$ ,

(9) \begin{equation}\mathop{\parallel\! {\mathcal P} (\,f\otimes g) \! \parallel}\nolimits_{L^{2}(\mu)} \leq M \mathop{\parallel\! f \! \parallel}\nolimits_{L^{2}(\mu)}\mathop{\parallel\! g \! \parallel}\nolimits_{L^{2}(\mu)},\end{equation}

then we deduce that (6), (7), and (8) hold. Condition (9) is much more natural and simpler than the latter ones, and it allows us to give shorter proofs. However, Condition (9) appears to be too strong even in the simplest case of the symmetric BAR model developed in Example 2.1 with $a_0=a_1$ and $b_0=b_1$ . Let a denote the common value of $a_0$ and $a_1$ . In fact, according to the value of $a\in ({-}1, 1)$ in the symmetric BAR model, there exists $k_1\in {\mathbb N}$ such that for all $f,g \in L^{2}(\mu)$ ,

(10) \begin{equation}\mathop{\big\| {\mathcal P} \big({\mathcal Q}^{k_1} f \otimes {\mathcal Q}^{k_1} g\big) \big\|}\nolimits_{L^{2}(\mu)}\leq M \mathop{\parallel\! f \! \parallel}\nolimits_{L^{2}(\mu)} \mathop{\parallel\! g \! \parallel}\nolimits_{L^{2}(\mu)} ,\end{equation}

with $k_1$ increasing with $|a|$ . Since Assumption 2.2(i) is only necessary for the asymptotic normality in the case $|a|\in \big[0, 1/\sqrt{2}\big]$ (corresponding to the subcritical and critical regimes), it will be enough to consider $k_1=1$ (but not sufficient to consider $k_1=0$ ). For this reason, we consider (6), that is, (10) with $k_1=1$ . A similar remark holds for (7) and (8). In a sense Condition (10) (as well as similar extensions of (7) and (8)) is in the same spirit as Assumption 2.2(ii): one uses iterates of ${\mathcal Q}$ to get smoothness on the kernel ${\mathcal P}$ and the initial distribution $\nu$ .

Remark 2.3. Let $\mu$ be an invariant probability measure of ${\mathcal Q}$ and assume that the transition kernel ${\mathcal P}$ has a density, denoted by p, with respect to the measure $\mu^{\otimes 2}$ ; that is, ${\mathcal P}(x, dy, dz) = p(x,y,z) \, \mu(dy)\mu(dz)$ for all $x\in S$ . Then the transition kernel ${\mathcal Q}$ has a density, denoted by q, with respect to $\mu$ ; that is, ${\mathcal Q}(x, dy)= q(x,y) \mu(dy)$ for all $x \in S$ with $q(x,y)=2^{-1} \int_S (p(x,y,z)+p(x,z,y))\, \mu(dz)$ . We set

(11) \begin{equation}{\mathfrak h}(x) = \left(\int_{S} q(x,y)^{2}\, \mu(dy)\right)^{1/2}.\end{equation}

Assume that

(12) \begin{align}\mathop{\parallel\! {\mathcal P}\big({\mathfrak h}\otimes ^2\big) \! \parallel}\nolimits_{L^{2}(\mu)} &<+\infty ,\end{align}

(13) \begin{align}\mathop{\parallel\! {\mathcal P}({\mathcal P}\big({\mathfrak h}\otimes ^2\big)\otimes _{\textrm{sym}} {\mathfrak h}) \! \parallel}\nolimits_{L^{2}(\mu)} &<+\infty ,\end{align}

and that there exists a finite constant C such that for all $f\in L^4(\mu)$ ,

(14) \begin{equation}\mathop{\parallel\! {\mathcal P}\big(\,f\otimes _{\textrm{sym}} {\mathfrak h}\big) \! \parallel}\nolimits_{L^{2}(\mu)} \leq C \,\mathop{\parallel\! {f} \! \parallel}\nolimits_{L^4(\mu)}.\end{equation}

Since $|{\mathcal Q} f| \leq \mathop{\parallel\! f \! \parallel}\nolimits_{L^{2}(\mu)} {\mathfrak h}$ , we deduce that (12), (13), and (14) imply respectively (6), (7), and (8).

We consider the following ergodic property of ${\mathcal Q}$ , which in particular implies that $\mu$ is indeed the unique invariant probability measure for ${\mathcal Q}$ . We refer to [Reference Douc, Moulines, Priouret and Soulier8, Section 22] for a detailed account of $L^2(\mu)$ -ergodicity (see in particular Definition 22.2.2 on the exponentially convergent Markov kernel).

Assumption 2.3. The Markov kernel ${\mathcal Q}$ has a (unique) invariant probability measure $\mu$ , and ${\mathcal Q}$ is $L^2(\mu)$ exponentially convergent; that is, there exist $\alpha \in (0,1)$ and M finite such that for all $f \in L^{2}(\mu)$ ,

(15) \begin{equation}\mathop{\parallel\! {\mathcal Q}^{n}f - \langle \mu, f\rangle \! \parallel}\nolimits_{L^{2}(\mu)}\leq M \alpha^{n} \mathop{\parallel\! f \! \parallel}\nolimits_{L^{2}(\mu)} \quad \text{for all $n \in \mathbb{N}$}.\end{equation}

We consider the stronger ergodic property based on a second spectral gap. (Notice in particular that Assumption 2.2 implies Assumption 2.2.)

Assumption 2.4. The Markov kernel ${\mathcal Q}$ has a (unique) invariant probability measure $\mu$ , and there exist $\alpha \in (0,1)$ ; a finite nonempty set J of indices; distinct complex eigenvalues $\{\alpha_j, \, j\in J\}$ of the operator ${\mathcal Q}$ with $|\alpha_j|=\alpha$ ; nonzero complex projectors $\{{\mathcal R}_j, \, j\in J\}$ defined on ${\mathbb C} L^2(\mu)$ , the ${\mathbb C}$ -vector space spanned by $L^2(\mu)$ , such that ${\mathcal R}_j\circ {\mathcal R}_{j^{\prime}}={\mathcal R}_{j^{\prime}}\circ {\mathcal R}_{j}=0$ for all $j\neq j^{\prime}$ (so that $\sum_{j\in J} {\mathcal R}_j$ is also a projector defined on ${\mathbb C} L^2(\mu)$ ); and a positive sequence $(\beta_n, n\in {\mathbb N})$ converging to $0$ , such that for all $f\in L^2(\mu)$ , with $\theta_j=\alpha_j/\alpha$ ,

(16) \begin{equation}\|{\mathcal Q}^{n}f - \langle \mu, f\rangle - \alpha^{n}\sum_{j\in J} \theta_{j}^{n} {\mathcal R}_{j}(\,f)\|_{L^{2}(\mu)} \leq \beta_{n}\alpha^{n} \|f\|_{L^{2}(\mu)} \quad \text{for all $n \in \mathbb{N}$}.\end{equation}

Assumptions 2.3 and 2.4 stated in an $L^2$ framework correspond to [Reference Bitseki Penda and Delmas4, Assumptions 2.4 and 2.6] stated in a pointwise framework. The structural Assumption 2.2 on the transition kernel ${\mathcal P}$ replaces the structural [Reference Bitseki Penda and Delmas4, Assumptions 2.2] on the set of functions considered.

Remark 2.4. Assume that ${\mathcal Q}$ has a density q with respect to an invariant probability measure $\mu$ such that ${\mathfrak h}\in L^2(\mu)$ , where ${\mathfrak h}$ is defined in (11); that is,

\begin{equation*}\int_{S^2} q(x,y)^2 \, \mu(dx)\mu(dy) < +\infty.\end{equation*}

Then the operator ${\mathcal Q}$ is a nonnegative Hilbert–Schmidt operator (and thus a compact operator) on $L^2(\mu)$ . It is well known that in this case, except for the possible value 0, the spectrum of ${\mathcal Q}$ is equal to the set $\sigma_{p}({\mathcal Q})$ of eigenvalues of ${\mathcal Q}$ ; $\sigma_{p}({\mathcal Q})$ is a countable set with 0 as the only possible accumulation point, and for all $\lambda \in \sigma_{p}({\mathcal Q})\setminus \{0\}$ , the eigenspace associated to $\lambda$ is finite-dimensional (we refer for example to [Reference Beauzany2, Chapter 4] for more details). In particular, if 1 is the only eigenvalue of ${\mathcal Q}$ with modulus 1 and if it has multiplicity 1 (that is, the corresponding eigenspace is reduced to the constant functions), then Assumptions 2.3 and 2.4 also hold. Let us mention that $q(x,y)>0$ $\mu(dx)\otimes \mu(dy)$ -almost surely (a.s.) is a standard condition which implies that 1 is the only eigenvalue of ${\mathcal Q}$ with modulus 1 and that it has multiplicity 1; see for example [Reference Baxter and Rosenthal1].

2.3. Notation for averages of different functions over different generations

Let $X=(X_u, u\in \mathbb{T})$ be a BMC on $(S, \mathcal{S})$ with initial probability distribution $\nu$ and probability kernel ${\mathcal P}$ . Recall that ${\mathcal Q}$ is the induced Markov kernel. We shall assume that $\mu$ is an invariant probability measure of ${\mathcal Q}$ . For a finite set $A\subset \mathbb{T}$ and a function $f\in {\mathcal B}(S)$ , we set

\begin{equation*}M_A(\,f)=\sum_{i\in A} f(X_i).\end{equation*}

We shall be interested in the cases $A=\mathbb{G}_n$ (the nth generation) and $A=\mathbb{T}_n$ (the tree up to the nth generation). We recall from [Reference Guyon9, Theorem 11 and Corollary 15] that under geometric ergodicity assumption, for f a continuous bounded real-valued function defined on S we have the following convergences in $L^2(\mu)$ (resp. a.s.):

(17) \begin{equation} \lim_{n\rightarrow\infty } |\mathbb{G}_n|^{-1} M_{\mathbb{G}_n}(\,f)=\langle \mu, f \rangle \quad \text{and}\quad \lim_{n\rightarrow\infty } |\mathbb{T}_n|^{-1} M_{\mathbb{T}_n}(\,f)=\langle \mu, f \rangle. \end{equation}

Using Lemma 5.1 and the Borel–Cantelli theorem, one can prove that we also have (17) with the $L^2(\mu)$ and a.s. convergences under Assumptions 2.2(ii) and 2.3.

We shall now consider the corresponding fluctuations. We will use frequently the following notation:

\begin{equation*}\tilde f= f - \langle \mu, f \rangle\quad \text{for $f\in L^1(\mu)$.}\end{equation*}

In order to study the asymptotics of $ M_{\mathbb{G}_{n-\ell }}\big(\tilde f\big)$ , we shall consider the contribution of the descendants of the individual $i\in \mathbb{T}_{n-\ell}$ for $n\geq \ell\geq 0$ :

(18) \begin{equation}N^\ell_{n,i}(\,f)=|\mathbb{G}_n|^{-1/2} M_{i\mathbb{G}_{n-|i|-\ell}}\big(\tilde f\big),\end{equation}

where $i\mathbb{G}_{n-|i|-\ell}=\{ij, \, j\in\mathbb{G}_{n-|i|-\ell}\}\subset \mathbb{G}_{n-\ell}$ . For all $k\in {\mathbb N}$ such that $n\geq k+\ell$ , we have

\begin{equation*}M_{\mathbb{G}_{n-\ell}}\big(\tilde f\big)=\sqrt{|\mathbb{G}_n|}\,\, \sum_{i\in \mathbb{G}_k}N^\ell_{n,i}(\,f)=\sqrt{|\mathbb{G}_n|}\, \,N_{n, \emptyset}^\ell(\,f).\end{equation*}

Let ${\mathfrak f}=(\,f_\ell, \ell\in {\mathbb N})$ be a sequence of elements of $L^1(\mu)$ . We set, for $n\in {\mathbb N}$ and $i\in \mathbb{T}_n$ ,

(19) \begin{equation}N_{n,i}({\mathfrak f})=\sum_{\ell=0}^{n-|i|} N_{n,i}^\ell(\,f_\ell)=|\mathbb{G}_n|^{-1/2 }\sum_{\ell=0}^{n-|i|}M_{i\mathbb{G}_{n-|i|-\ell}}\big(\tilde f_\ell\big).\end{equation}

We deduce that $ \sum_{i\in \mathbb{G}_k} N_{n,i}({\mathfrak f})=|\mathbb{G}_n|^{-1/2 }\sum_{\ell=0}^{n-k}M_{\mathbb{G}_{n-\ell}}\big(\tilde f_\ell\big)$ , which gives, for $k=0$ ,

(20) \begin{equation}N_{n, \emptyset}({\mathfrak f})= |\mathbb{G}_n|^{-1/2 }\sum_{\ell=0}^{n} M_{\mathbb{G}_{n-\ell}}\big(\tilde f_{\ell}\big).\end{equation}

The notation $N_{n, \emptyset}$ means that we consider the average from the root $\emptyset$ to the nth generation.

Remark 2.5. We shall consider in particular the following two simple cases. Let $f\in L^1(\mu)$ and consider the sequence ${\mathfrak f}=(\,f_\ell, \ell\in {\mathbb N})$ . If $f_0=f$ and $f_\ell=0$ for $\ell\in {\mathbb N}^*$ , then we get

\begin{equation*}N_{n, \emptyset}({\mathfrak f})= |\mathbb{G}_n|^{-1/2} M_{\mathbb{G}_n}\big(\tilde f\big).\end{equation*}

If $f_\ell=f$ for $\ell \in {\mathbb N}$ , then we shall write ${\bf f}=(\,f,f, \ldots)$ , and we get, as $|\mathbb{T}_n|=2^{n+1} - 1 $ and $|\mathbb{G}_n|= 2^n$ ,

\begin{equation*}N_{n, \emptyset}({\bf f})= |\mathbb{G}_n|^{-1/2} M_{\mathbb{T}_n}\big(\tilde f\big)= \sqrt{2 - 2^{-n}}\,\, |\mathbb{T}_n|^{-1/2} M_{\mathbb{T}_n}\big(\tilde f\big).\end{equation*}

Thus, we will deduce the fluctuations of $M_{\mathbb{T}_n}(\,f)$ and $M_{\mathbb{G}_n}(\,f)$ from the asymptotics of $N_{n, \emptyset}({\mathfrak f})$ .

Because of Assumption 2.2(ii) (which roughly states that after $k_0$ generations, the distribution of the induced Markov chain is absolutely continuous with respect to the invariant measure $\mu$ ), it is better to consider only generations $k\geq k_0$ for some $k_0\in {\mathbb N}$ and thus remove the first $k_0-1$ generations in the quantity $N_{n,\emptyset}({\mathfrak f}) $ defined in (20). To study the asymptotics of $N_{n, \emptyset}({\mathfrak f})$ , it is convenient to write, for $n\geq k\geq 1$ ,

(21) \begin{equation}N_{n, \emptyset}({\mathfrak f})= |\mathbb{G}_n|^{-1/2} \sum_{r=0}^{k-1} M_{\mathbb{G}_r}\big(\tilde f_{n-r}\big) + \sum_{i\in \mathbb{G}_k} N_{n,i}({\mathfrak f}).\end{equation}

If ${\bf f}=(\,f,f, \ldots)$ is the infinite sequence in which each term is the same function f, this becomes

\begin{equation*} N_{n, \emptyset}({\bf f})= |\mathbb{G}_n|^{-1/2} M_{\mathbb{T}_n}\big(\tilde f\big)= |\mathbb{G}_n|^{-1/2}M_{\mathbb{T}_{k-1}}\big(\tilde f\big)+ \ \sum_{i\in \mathbb{G}_k} N_{n,i}({\bf f}).\end{equation*}

3.1. The subcritical case: $2\alpha^2<1$

We shall consider, when well defined, for a sequence ${\mathfrak f}=(\,f_\ell,\ell\in {\mathbb N})$ of measurable real-valued functions defined on S, the quantities

(22) \begin{equation}\Sigma^{\textrm{sub}}({\mathfrak f})=\Sigma^{\textrm{sub}}_1({\mathfrak f})+ 2 \Sigma^{\textrm{sub}}_2({\mathfrak f}),\end{equation}

where

(23) \begin{align} \Sigma^{\textrm{sub}}_1({\mathfrak f})&=\sum_{\ell\geq 0}2^{-\ell} \, \big\langle \mu, \tilde f_\ell^ 2\big\rangle+ \sum_{\ell\geq 0, \, k\geq 0}2^{k-\ell} \, \left\langle \mu, {\mathcal P}\!\left(\left({\mathcal Q}^k \tilde f_\ell\right) \otimes ^2\right)\right\rangle,\end{align}

(24) \begin{align} \Sigma^{\textrm{sub}}_2({\mathfrak f})&=\sum_{0\leq \ell< k}2^{-\ell} \left\langle \mu, \tilde f_k {\mathcal Q}^{k-\ell} \tilde f_\ell\right\rangle+ \sum_{\substack{0\leq \ell< k\\ r\geq 0}} 2^{r-\ell} \left\langle \mu,{\mathcal P}\!\left( {\mathcal Q}^r \tilde f_k \otimes _{\textrm{sym}}{\mathcal Q}^{k-\ell+r} \tilde f_\ell \right)\right\rangle.\end{align}

The proof of the next result is detailed in Section 5.

Theorem 3.1. Let X be a BMC with kernel ${\mathcal P}$ and initial distribution $\nu$ such that Assumptions 2.2 and 2.3 are in force with $\alpha\in \big(0, 1/\sqrt{2}\big)$ . We have the following convergence in distribution for any sequence ${\mathfrak f}=(\,f_\ell, \ell\in {\mathbb N})$ that is bounded in $L^4(\mu)$ (that is, such that $\sup_{\ell\in {\mathbb N}} \mathop{\parallel\! {f_\ell} \! \parallel}\nolimits_{L^4(\mu)}<+\infty $ ):

\begin{equation*}N_{n, \emptyset}({\mathfrak f}) \; \xrightarrow[n\rightarrow \infty]{\text{(d)}} \; G,\end{equation*}

where G is a centered Gaussian random variable with variance $\Sigma^{\textrm{sub}}({\mathfrak f})$ given by (22) which is well defined and finite.

Notice that the variance $\Sigma^{\textrm{sub}}({\mathfrak f})$ already appears in the subcritical pointwise-approach case; see [Reference Bitseki Penda and Delmas4, (15) and Theorem 3.1]. Then, arguing similarly as in [Reference Bitseki Penda and Delmas4, Section 3.1], we deduce that if Assumptions 2.2 and 2.3 are in force with $\alpha\in \big(0, 1/\sqrt{2}\big)$ , then for $f\in L^{4}(\mu)$ , we have the following convergence in distribution:

(25) \begin{equation}|\mathbb{G}_{n}|^{-1/2}M_{\mathbb{G}_{n}}\big(\tilde{f}\big) \; \xrightarrow[n\rightarrow \infty ]{\text{(d)}} \; G_{1} \quad \text{and} \quad |\mathbb{T}_{n}|^{-1/2}M_{\mathbb{T}_{n}}\big(\tilde{f}\big) \; \xrightarrow[n\rightarrow \infty ]{\text{(d)}} \; G_{2},\end{equation}

where $G_{1}$ and $G_2$ are centered Gaussian random variables with respective variances $\Sigma^{\textrm{sub}}_{\mathbb{G}}(\,f)=\Sigma^{\textrm{sub}}({\mathfrak f})$ , with ${\mathfrak f} = (\,f,0,0, \ldots)$ , and $\Sigma^{\textrm{sub}}_{\mathbb{T}}(\,f)=\Sigma^{\textrm{sub}}({\bf f})/2$ , with $ {\bf f}=(\,f,f, \ldots)$ , given in [Reference Bitseki Penda and Delmas4, Corollary 3.3], which are well defined and finite.

3.2. The critical case: $2\alpha^2=1$

In the critical case $\alpha=1/\sqrt{2}$ , we shall denote by ${\mathcal R}_j$ the projector on the eigenspace associated to the eigenvalue $\alpha_j$ with $\alpha_j=\theta_j \alpha$ , $|\theta_j|=1$ and for j in the finite set of indices J. Since ${\mathcal Q}$ is a real operator, we get that if $\alpha_j$ is a nonreal eigenvalue, then so is $\overline \alpha_j$ . We shall denote by $\overline {\mathcal R}_j$ the projector associated to $\overline \alpha_j$ . Recall that the sequence $(\beta_n, n\in {\mathbb N})$ in Assumption 2.2 is nonincreasing and bounded from above by 1. For any measurable real-valued function f defined on S, we set, when this is well defined,

(26) \begin{equation} \hat{f} = \tilde{f} - \sum_{j \in J} {\mathcal R}_{j}(\,f)\quad \text{with}\quad \tilde f= f- \langle \mu, f \rangle.\end{equation}

We shall consider, when well defined, for a sequence ${\mathfrak f}=(\,f_\ell,\ell\in {\mathbb N})$ of measurable real-valued functions defined on S, the quantities

(27) \begin{equation}\Sigma^{\textrm{crit}} ({\mathfrak f})=\Sigma^{\textrm{crit}} _1({\mathfrak f})+ 2\Sigma^{\textrm{crit}} _2({\mathfrak f}),\end{equation}

where

(28) \begin{align}\Sigma^{\textrm{crit}}_1({\mathfrak f})&= \sum_{k\geq 0} 2^{-k} \left\langle \mu, {\mathcal P} f_{k, k}^*\right\rangle= \sum_{k\geq 0} 2^{-k} \sum_{j\in J} \left\langle \mu,{\mathcal P}\!\left({\mathcal R}_{j}(\,f_k) \otimes _{\textrm{sym}} \overline{{\mathcal R}}_{j}(\,f_k)\right) \right\rangle,\end{align}

(29) \begin{align}\Sigma^{\textrm{crit}}_2({\mathfrak f})&= \sum_{0\leq \ell <k} 2^{-(k+\ell)/2} \left\langle \mu, {\mathcal P}f_{k, \ell}^* \right\rangle,\end{align}

with, for $k, \ell\in {\mathbb N}$ ,

\begin{equation*}f_{k, \ell}^*= \sum_{j\in J}\,\, \theta_j^{\ell -k}\, {\mathcal R}_j(\,f_k)\otimes _{\textrm{sym}} \overline {\mathcal R}_j (\,f_\ell).\end{equation*}

Notice that $f_{k, \ell}^*=f_{\ell, k}^*$ and that $f_{k, \ell}^*$ is real-valued as

\begin{equation*}\overline {\theta_j^{\ell -k}\, {\mathcal R}_j (\,f_k)\otimes \overline {\mathcal R}_j (\,f_\ell)}= \theta_{j^{\prime}}^{\ell -k}\, {\mathcal R}_{j^{\prime}} (\,f_k)\otimes \overline {\mathcal R}_{j^{\prime}} (\,f_\ell)\end{equation*}

for j ^′ such that $\alpha_{j^{\prime}}=\overline\alpha _j$ and thus ${\mathcal R}_{j^{\prime}}=\overline {\mathcal R}_j$ .

The technical proof of the next result is omitted, as it is an adaptation of the proof of Theorem 3.1 in the subcritical case, in the same spirit as [Reference Bitseki Penda and Delmas4, Theorem 3.4] (critical case) is an adaptation of the proof of [Reference Bitseki Penda and Delmas4, Theorem 3.1] (subcritical case). The interested reader can find the details in [Reference Bitseki Penda and Delmas3].

Theorem 3.2. Let X be a BMC with kernel ${\mathcal P}$ and initial distribution $\nu$ such that Assumptions 2.2 (with $k_0\in {\mathbb N}$ ), 2.3, and 2.4 are in force with $\alpha = 1/\sqrt{2}$ . We have the following convergence in distribution for any sequence ${\mathfrak f}=(\,f_\ell,\ell\in {\mathbb N})$ that is bounded in $L^4(\mu)$ $\big($ that is, such that $\sup_{\ell\in {\mathbb N}}\mathop{\parallel\! {f_\ell} \! \parallel}\nolimits_{L^4(\mu)}<+\infty\big) $ :

\begin{equation*}n^{-1/2}N_{n, \emptyset}({\mathfrak f}) \; \xrightarrow[n\rightarrow \infty]{\text{(d)}} \; G,\end{equation*}

where G is a centered Gaussian random variable with variance $\Sigma^{\textrm{crit}}({\mathfrak f})$ given by (27), which is well defined and finite.

Notice that the variance $\Sigma^{\textrm{crit}}({\mathfrak f})$ already appears in the critical pointwise-approach case; see [Reference Bitseki Penda and Delmas4, (20) and Theorem 3.4]. Then, arguing similarly as in [Reference Bitseki Penda and Delmas4, Section 3.2], we deduce that if Assumptions 2.2 (with $k_0\in {\mathbb N}$ ), 2.3, and 2.4 are in force with $\alpha = 1/\sqrt{2}$ , then for $f\in L^{4}(\mu)$ , we have the following convergence in distribution:

(30) \begin{equation}(n|\mathbb{G}_{n}|)^{-1/2}M_{\mathbb{G}_{n}}\big(\tilde{f}\big) \; \xrightarrow[n\rightarrow\infty ]{\text{(d)}} \; G_{1},\quad \text{and} \quad (n|\mathbb{T}_{n}|)^{-1/2}M_{\mathbb{T}_{n}}\big(\tilde{f}\big) \; \xrightarrow[n\rightarrow \infty ]{\text{(d)}} \; G_{2},\end{equation}

where $G_{1}$ and $G_2$ are centered Gaussian random variables with respective variances $\Sigma^{\textrm{crit}}_{\mathbb{G}}(\,f)=\Sigma^{\textrm{crit}}({\mathfrak f})$ , with ${\mathfrak f} = (\,f,0,0, \ldots)$ , and $\Sigma^{\textrm{crit}}_{\mathbb{T}}(\,f)=\Sigma^{\textrm{crit}}({\bf f})/2$ , with $ {\bf f}=(\,f,f, \ldots)$ , given in [Reference Bitseki Penda and Delmas4, Corollary 3.6], which are well defined and finite.

3.3. The supercritical case $2\alpha^2>1$

We consider the supercritical case $\alpha\in \big(1/\sqrt{2}, 1\big)$ . This case is very similar to the supercritical case in the pointwise approach; see [Reference Bitseki Penda and Delmas4, Section 3.3]. So we only mention the most interesting results without proof. The interested reader can find the details in [Reference Bitseki Penda and Delmas3].

We shall assume that Assumptions 2.2(ii) and 2.4 hold. In particular we do not assume Assumption 2.4(i). Recall (16) with the eigenvalues $\big\{\alpha_j=\theta_j \alpha, j\in J\big\} $ of ${\mathcal Q}$ , with modulus equal to $\alpha$ (i.e. $|\theta_j|=1$ ) and the projector ${\mathcal R}_j$ on the eigenspace associated to eigenvalue $\alpha_j$ . Recall that the sequence $(\beta_n, n\in {\mathbb N})$ in Assumption 2.4 can (and will) be chosen to be nonincreasing and bounded from above by 1. We shall consider the filtration ${\mathcal H}=({\mathcal H}_n, n\in )$ defined by ${\mathcal H}_{n} = \sigma(X_{i},i\in\mathbb{T}_{n})$ . The next lemma exhibits martingales related to the projector ${\mathcal R}_j$ .

Lemma 3.1. Let X be a BMC with kernel ${\mathcal P}$ and initial distribution $\nu$ such that Assumptions 2.2(ii) and 2.4 are in force with $\alpha\in \big(1/\sqrt{2}, 1\big)$ in (16). Then, for all $j\in J$ and $f\in L^{2}(\mu)$ , the sequence $M_{j}(\,f)=\big(M_{n,j}(\,f), n\in {\mathbb N}\big)$ , with

\begin{equation*}M_{n,j}(\,f) = \big(2 \alpha_j\big) ^{-n} \, M_{\mathbb{G}_n} \big({\mathcal R}_j(\,f)\big),\end{equation*}

is an ${\mathcal H}$ -martingale which converges a.s. and in $L^{2}(\nu)$ to a random variable, say $M_{\infty,j}(\,f)$ .

The next result corresponds to [Reference Bitseki Penda and Delmas4, Corollary 3.13] in the pointwise approach.

Corollary 3.1. Let X be a BMC with kernel ${\mathcal P}$ and initial distribution $\nu$ such that Assumptions 2.2(ii) and 2.4 are in force with $\alpha \in \big(1/\sqrt{2},1\big) $ in (16). Assume $\alpha$ is the only eigenvalue of ${\mathcal Q}$ with modulus equal to $\alpha$ (and thus J is reduced to a singleton, say $\{j_0\}$ ). Then, for $f\in L^{2}(\mu)$ , we have

\begin{equation*} (2\alpha)^{-n}M_{\mathbb{G}_{n}}\big(\tilde{f}\big) \underset{n \rightarrow \infty}{\overset{\mathbb{P}}{\xrightarrow{}}} M_{\infty}(\,f) \quad \text{and} \quad (2\alpha)^{-n}M_{\mathbb{T}_{n}}\big(\tilde{f}\big) \underset{n \rightarrow \infty}{\overset{\mathbb{P}}{\xrightarrow{}}} \frac{2\alpha}{2\alpha - 1}M_{\infty, j_0}(\,f) ,\end{equation*}

where $M_{\infty, j_0}(\,f)$ is the random variable defined in Lemma 3.1.

4.1. Symmetric BAR

We consider a particular case from [Reference Cowan and Staudte7] of the real-valued bifurcating autoregressive process (BAR) from Example 2.1. We keep the same notation. Let $a\in ({-}1, 1)$ and assume that $a=a_{0} = a_{1}$ , $b_{0} = b_{1} = 0$ , and $\rho = 0$ . In this particular case the BAR has symmetric kernel as

\begin{equation*}{\mathcal P}(x, dy, dz)={\mathcal Q}(x, dy) {\mathcal Q}(x, dz).\end{equation*}

We have ${\mathcal Q} f(x)={\mathbb E}[f(ax +\sigma G)]$ and more generally

\begin{equation*}{\mathcal Q}^n f(x)={\mathbb E}\!\left[f\!\left(a^n x + \sqrt{1- a^{2n}} \sigma_a G\right)\right],\end{equation*}

where G is a standard ${\mathcal N}(0, 1)$ Gaussian random variable and $\sigma_a=\sigma \big(1- a^2\big)^{-1/2}$ . The kernel ${\mathcal Q}$ admits a unique invariant probability measure $\mu$ , which is ${\mathcal N}\big(0, \sigma_a^2\big)$ and whose density, still denoted by $\mu$ , with respect to the Lebesgue measure is given by

\begin{equation*}\mu(x) = \frac{\sqrt{1 - a^{2}}}{\sqrt{2\pi\sigma^2}} \exp\!\left({-}\frac{\big(1 - a^{2}\big)x^{2}}{2 \sigma^{2}}\right).\end{equation*}

The densities p (resp. q) of the kernel ${\mathcal P}$ (resp. ${\mathcal Q}$ ) with respect to $\mu^{\otimes 2}$ (resp. $\mu$ ) are given by

\begin{equation*}p(x,y,z) = q(x,y)q(x,z)\end{equation*}

and

\begin{equation*}q(x,y) = \frac{1}{\sqrt{1 - a^{2}}} \exp\!\left({-}\frac{(y-ax)^{2}}{2\sigma^{2}} + \frac{\big(1-a^{2}\big)y^{2}}{2\sigma^{2}}\right) = \frac{1}{\sqrt{1 - a^{2}}} \mathop {\textrm{e}^{ {- \big(a^2 y^2 + a^2 x^2 -2axy \big)/ 2\sigma^ 2}}}.\end{equation*}

Notice that q is symmetric. The operator ${\mathcal Q}$ $\big($ in $L^2(\mu)\big)$ is a symmetric integral Hilbert–Schmidt operator whose eigenvalues are given by $\sigma_{p}({\mathcal Q}) = (a^{n}, n \in \mathbb{N})$ , their algebraic multiplicity is one, and the corresponding eigenfunctions $(\bar{g}_{n}(x), n \in \mathbb{N})$ are defined for $n\in {\mathbb N}$ by

\begin{equation*}\bar{g}_{n}(x) = g_{n}\!\left(\sigma_a^{-1} \, x\right),\end{equation*}

where $g_{n}$ is the Hermite polynomial of degree n ( $g_0=1$ and $g_1(x)=x$ ). Let ${\mathcal R}$ be the orthogonal projection on the vector space generated by $\bar{g}_{1}$ ; that is, ${\mathcal R} f= \langle \mu, f\bar g_1 \rangle\, \bar g_1$ , or equivalently, for $x\in {\mathbb R}$ ,

(31) \begin{equation}{\mathcal R} f(x)=\sigma_a^{-1}\, x \, {\mathbb E}\!\left[G f(\sigma_aG)\right].\end{equation}

Recall ${\mathfrak h}$ defined in (11). It is not difficult to check that

\begin{equation*}{\mathfrak h}(x)=\big(1-a^4\big)^{-1/4}\exp\!\left(\frac{a^2\big(1-a^2\big)}{1+a^2} \, \frac{x^2}{2\sigma^2}\right) \quad \text{for $x\in {\mathbb R}$},\end{equation*}

and ${\mathfrak h} \in L^2(\mu)$ $\big($ that is, $\int_{{\mathbb R}^2} q(x,y)^2 \, \mu(x) \mu(y)\, dx dy<+\infty\big) $ . Using elementary computations, it is possible to check that ${\mathcal Q} {\mathfrak h}\in L^4(\mu)$ if and only if $|a|< 3^{-1/4}$ $\big($ whereas ${\mathfrak h}\in L^4(\mu)$ if and only if $|a|<3^{-1/2}\big)$ . As ${\mathcal P}$ is symmetric, we get ${\mathcal P}\big({\mathfrak h}\otimes ^2\big)\leq ({\mathcal Q}{\mathfrak h})^2$ and thus (12) holds for $|a|< 3^{-1/4}$ . We also get, using the Cauchy–Schwarz inequality, that

\begin{equation*}\mathop{\parallel\! {\mathcal P}\big(\,f\otimes _{\textrm{sym}} {\mathfrak h}\big) \! \parallel}\nolimits_{L^{2}(\mu)}=\mathop{\parallel\! ({\mathcal Q} f) ({\mathcal Q} {\mathfrak h}) \! \parallel}\nolimits_{L^{2}(\mu)} \leq \mathop{\parallel\! {f} \! \parallel}\nolimits_{L^4(\mu)}\,\mathop{\parallel\! {{\mathcal Q} ({\mathfrak h})} \! \parallel}\nolimits_{L^4(\mu)},\end{equation*}

and thus (14) holds for $|a|< 3^{-1/4}$ . Some elementary computations give that (13) also holds for $|a|\leq 0.724$ (but (13) fails for $|a|\geq 0.725$ ). $\big($ Notice that $2^{-1/2}< 0.724< 3^{-1/4}\big)$ . As a consequence of Remark 2.3, if $|a|\leq 0.724$ , then (6)–(8) are satisfied and thus Assumption 2.2(i) holds.

Notice that $\nu {\mathcal Q}^k$ is the probability distribution of $a^k X_\emptyset + \sigma_a \sqrt{1- a^{2k}}\, G$ , with G an ${\mathcal N}(0, 1)$ random variable independent of $X_\emptyset$ . So Assumption 2.2(ii) holds in particular if $\nu$ has compact support (with $k_0=1$ ) or if $\nu$ has a density with respect to the Lebesgue measure, which we still denote by $\nu$ , such that $\mathop{\parallel\! {\nu/\mu} \! \parallel}\nolimits_\infty $ is finite (with $k_0\in {\mathbb N}$ ). Notice that if $\nu$ is the probability distribution of ${\mathcal N}\big(0, \rho_0^2\big)$ , then $\rho_0> \sigma_a$ (resp. $\rho_0\leq \sigma_a$ ) implies that Assumption 2.2(ii) fails (resp. is satisfied).

Using the fact that $\big(\bar g_n/\sqrt{n!}, \, n\in {\mathbb N}\big)$ is an orthonormal basis of $L^2(\mu)$ and Parseval’s identity, it is easy to check that Assumption 2.4 holds with $J=\{j_0\}$ , $\alpha_{j_0} = \alpha = a$ , $\beta_n = a^{n}$ , and ${\mathcal R}_{j_0}={\mathcal R}$ .

4.2. Numerical studies: illustration of phase transitions for the fluctuations

We consider the symmetric BAR model from Section 4.1 with $a=\alpha\in (0, 1)$ . Recall that $\alpha$ is an eigenvalue with multiplicity one, and we denote by ${\mathcal R}$ the orthogonal projection on the one-dimensional eigenspace associated to $\alpha$ . The expression for ${\mathcal R}$ is given in (31).

In order to illustrate the effects of the geometric rate of convergence $\alpha$ on the fluctuations, we plot for ${\mathbb A}_n\in \{\mathbb{G}_n, \mathbb{T}_n\}$ the slope, say $b_{\alpha,n}$ , of the regression line ${\log}\big({\textrm{Var}}\big(|{\mathbb A}_{n}|^{-1}M_{{\mathbb A}_{n}}(\,f)\big)\big) $ versus $ {\log}(|{\mathbb A}_{n}|)$ as a function of the geometric rate of convergence $\alpha$ . In the classical cases (e.g. Markov chains), the points are expected to be distributed around the horizontal line $y = -1$ . For n large, we have ${\log}(|{\mathbb A}_{n}|)\simeq n\, {\log}(2)$ , and for the symmetric BAR model, the convergences in (25) for $\alpha<1/\sqrt{2}$ , (30) for $\alpha=1/\sqrt{2}$ , and Corollary 3.1 for $\alpha>1/\sqrt{2}$ yield that $b_{\alpha,n} \simeq h_{1}(\alpha) $ with $h_1(\alpha)=\,{\log}\big(\alpha^2\vee 2^{-1}\big)/{\log}(2)$ as soon as the limiting Gaussian random variable in (25) and (30) or $M_\infty (\,f)$ in Corollary 3.1 is nonzero.

For our illustrations, we consider the empirical moments of order $p\in \{1, \ldots, 4\}$ ; that is, we use the functions $f(x) = x^p$ . As we can see in Figures 1 and 2, these curves present two trends, with a phase transition around the rate $\alpha = 1/\sqrt{2}$ for $p \in \{1,3\}$ and around the rate $\alpha^{2} = 1/\sqrt{2}$ for $p \in \{2,4\}$ . For convergence rates $\alpha \in \big(0,1/\sqrt{2}\big)$ , the trend is similar to that of the classical cases. For convergence rates $\alpha \in \big(1/\sqrt{2},1\big)$ , the trend differs from the classical cases. One can observe that the slope $b_{\alpha,n}$ increases with the value of the geometric convergence rate $\alpha$ . We also observe that for $\alpha > 1/ \sqrt{2}$ , the empirical curve agrees with the graph of $h_1(\alpha)=\,{\log}\big(\alpha^2\vee 2^{-1}\big)/{\log}(2)$ for $f(x) = x^{p}$ when p is odd; see Figure 1. However, the empirical curve does not agree with the graph of $h_1$ for $f(x) = x^{p}$ when p is even (see Figure 2); instead, it agrees with the graph of the function $h_2(\alpha)=\,{\log}\big(\alpha^4\vee 2^{-1}\big)/{\log}(2)$ . This is due to the fact that for p even, the function $f(x)=x^p$ belongs to the kernel of the projector ${\mathcal R}$ (which is clear from the formula (31)), and thus $M_\infty (\,f)=0$ . In fact, in those two cases, one should take into account the projection on the eigenspace associated to the third eigenvalue, which in this particular case is equal to $\alpha^2$ . Intuitively, this indeed gives a rate of order $h_2$ . Therefore, the normalization given for $f(x)=x^p$ when p is even is not correct.

Figure 1. Slope $b_{\alpha, n}$ (empirical mean and confidence interval in black) of the regression line ${\log}\big({\textrm{Var}}\big(|{\mathbb A}_{n}|^{-1}M_{{\mathbb A}_{n}}(\,f)\big)\big) $ versus $ {\log}(|{\mathbb A}_{n}|)$ as a function of the geometric ergodic rate $\alpha$ , for $n=15$ , ${\mathbb A}_{n} \in \{\mathbb{G}_{n}, \mathbb{T}_{n}\}$ , and $f(x) = x^p$ with $p\in \{1, 3\}$ . In this case, we have ${\mathcal R}(\,f) \neq 0$ , where ${\mathcal R}$ is the projector defined from the formula (31). One can see that the empirical curve (in black) is close to the graph (in red) of the function $h_1(\alpha) =\, {\log}\big(\alpha^{2} \vee 2^{-1}\big)/{\log}(2)$ for $\alpha\in (0, 1)$ .

Figure 2. Slope $b_{\alpha, n}$ (empirical mean and confidence interval in black) of the regression line ${\log}\big({\textrm{Var}}\big(|{\mathbb A}_{n}|^{-1}M_{{\mathbb A}_{n}}(\,f)\big)\big) $ versus $ {\log}(|{\mathbb A}_{n}|)$ as a function of the geometric ergodic rate $\alpha$ , for $n=15$ , ${\mathbb A}_{n} \in \{\mathbb{G}_{n}, \mathbb{T}_{n}\}$ , and $f(x) = x^p$ with $p\in \{2, 4\}$ . In this case, we have ${\mathcal R}(\,f) = 0$ , where ${\mathcal R}$ is the projector defined from the formula (31). One can see that the empirical curve (in black) does not agree with the graph (dashed line in red) of the function $h_1(\alpha) =\, {\log}\big(\alpha^{2} \vee 2^{-1}\big)/{\log}(2)$ for $2\alpha^2>1$ , but it is close to the graph (in blue) of the function $h_2(\alpha) =\, {\log}\big(\alpha^{4} \vee 2^{-1}\big)/{\log}(2)$ for $\alpha\in (0, 1)$ .