Discretization of the Ergodic Functional Central Limit Theorem

Gilles Pagès () and Clément Rey ()
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Gilles Pagès: LPSM, Sorbonne Université
Clément Rey: Institut Polytechnique de Paris

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 1-44

Abstract: Abstract In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see Bhattacharya in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states the following: Given a stationary and ergodic Markov process $$(X_t)_{t \geqslant 0}$$ ( X t ) t ⩾ 0 with unique invariant measure $$\nu $$ ν and infinitesimal generator A, then, for every smooth enough function f, $$(n^{1/2} \frac{1}{n}\int _0^{nt} Af(X_s){\textrm{d}}s)_{t \geqslant 0}$$ ( n 1 / 2 1 n ∫ 0 nt A f ( X s ) d s ) t ⩾ 0 converges in distribution towards the distribution of the process $$(\sqrt{-2 \langle f, Af \rangle _{\nu }} W_{t})_{t \geqslant 0}$$ ( - 2 ⟨ f , A f ⟩ ν W t ) t ⩾ 0 with $$(W_{t})_{t \geqslant 0}$$ ( W t ) t ⩾ 0 a Wiener process. In particular, we consider the marginal distribution at fixed $$t=1$$ t = 1 , and we show that when $$\int _0^{n} Af(X_s)ds$$ ∫ 0 n A f ( X s ) d s is replaced by a well chosen discretization of the time integral with order q (e.g. Riemann discretization in the case $$q=1$$ q = 1 ), then the CLT still holds but with rate $$n^{q/(2q+1)}$$ n q / ( 2 q + 1 ) instead of $$n^{1/2}$$ n 1 / 2 . Moreover, our results remain valid when $$(X_t)_{t \geqslant 0}$$ ( X t ) t ⩾ 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 (Talay in Stoch Stoch Rep 29:13–36, 1990) of weak order two.

Keywords: Ergodic theory; Markov processes; Invariant measures; Central limit theorem; Stochastic approximation; 60G10; 47A35; 60F05; 60J25; 60J35; 65C20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-023-01237-w

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