Discretization of the Ergodic Functional Central Limit Theorem

In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see \cite{Bhattacharya_1982}) which states the following: Given a stationary and ergodic Markov process $(X_t)_{t \geqslant 0}$ with unique invariant measure $\nu$ and infinitesimal generator $A$, then, for every smooth enough function $f$, $(n^{1/2} \frac{1}{n}\int_0^{nt} Af(X_s)ds)_{t \geqslant 0}$ converges in distribution towards the distribution of the process $(\sqrt{-2 \langle f, Af \rangle_{\nu}} W_{t})_{t \geqslant 0}$ with $(W_{t})_{t \geqslant 0}$ a Wiener process. In particular, we consider the marginal distribution at fixed $t=1$, and we show that when $\int_0^{n} Af(X_s)ds$ is replaced by a well chosen discretization of the time integral with order $q$ ($e.g.$ Riemann discretization in the case $q=1$), then the CLT still holds but with rate $n^{q/(2q+1)}$ instead of $n^{1/2}$. Moreover, our results remain valid when $(X_t)_{t \geqslant 0}$ is replaced by a $q$-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order $q$ for the time integral and the $q$-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay's scheme \cite{Talay_1990} of weak order two.