Discretization of the Ergodic Functional Central Limit Theorem

In this paper, we study the discretization of ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states following: Given a stationary and Markov process $$(X_t)_{t \geqslant 0}$$ with unique invariant measure $$\nu $$ infinitesimal generator A, then, for every smooth enough function f, $$(n^{1/2} \frac{1}{n}\int _0^{nt} Af(X_s){\textrm{d}}s)_{t converges distribution towards $$(\sqrt{-2 \langle Af \rangle _{\nu }} W_{t})_{t $$(W_{t})_{t Wiener process. particular, consider marginal at fixed $$t=1$$ , show that when $$\int _0^{n} Af(X_s)ds$$ is replaced well chosen time integral order q (e.g. Riemann case $$q=1$$ ), then CLT still holds but rate $$n^{q/(2q+1)}$$ instead $$n^{1/2}$$ . Moreover, our results remain valid q-weak approximation (not necessarily stationary). This paper presents both method q-order derive from them. We finally propose applications concerning first Brownian diffusion regimes Euler scheme (where recover existing literature) second using Talay’s (Talay Stoch Rep 29:13–36, 1990) weak two.