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Discretization of the Ergodic Functional Central Limit Theorem

Gilles Pagès, Clément Rey

TL;DR

The paper extends Bhattacharya's ergodic Functional CLT to discretized-time observables by analyzing a discretization of the time integral with order $q$, establishing a rate of convergence $n^{q/(2q+1)}$ for the weighted empirical measures. It develops a Talay-type scheme with decreasing time steps and proves a $q$-order ergodic CLT, including an infinitesimal approximation up to second order, under Lyapunov and step-weight hypotheses. The results yield a first-order CLT consistent with Euler-type discretizations ($n^{1/3}$ rate) and a faster second-order CLT for the second weak-order Talay scheme ($n^{2/5}$ rate), with explicit limiting variances and possible biases depending on the regime. Applications include approximating stationary regimes of Markov Brownian diffusions and recovering known results while highlighting improved rates for higher-order schemes.

Abstract

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 $ν$ 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 $(\sqrt{-2 \langle f, Af \rangle_ν} W_{t})_{t \geqslant 0}$ 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.

Discretization of the Ergodic Functional Central Limit Theorem

TL;DR

The paper extends Bhattacharya's ergodic Functional CLT to discretized-time observables by analyzing a discretization of the time integral with order , establishing a rate of convergence for the weighted empirical measures. It develops a Talay-type scheme with decreasing time steps and proves a -order ergodic CLT, including an infinitesimal approximation up to second order, under Lyapunov and step-weight hypotheses. The results yield a first-order CLT consistent with Euler-type discretizations ( rate) and a faster second-order CLT for the second weak-order Talay scheme ( rate), with explicit limiting variances and possible biases depending on the regime. Applications include approximating stationary regimes of Markov Brownian diffusions and recovering known results while highlighting improved rates for higher-order schemes.

Abstract

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 with unique invariant measure and infinitesimal generator , then, for every smooth enough function , converges in distribution towards the distribution of the process with a Wiener process. In particular, we consider the marginal distribution at fixed , and we show that when is replaced by a well chosen discretization of the time integral with order ( Riemann discretization in the case ), then the CLT still holds but with rate instead of . Moreover, our results remain valid when is replaced by a -weak order approximation (not necessarily stationary). This paper presents both the discretization method of order for the time integral and the -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.

Paper Structure

This paper contains 17 sections, 6 theorems, 54 equations.

Table of Contents

  1. Application - The Talay second weak order scheme
  2. Notations.
  3. Presentation of the main result.
  4. Recursive control
  5. Case $p\geqslant 1$.
  6. Case $p\in(0,1)$.
  7. Infinitesimal approximation
  8. Growth control
  9. Case $2p\leqslant s$.
  10. Case $2p\geqslant s$.
  11. Proof of Theorem \ref{['th:cv_was_Talay']}
  12. Proof of Theorem \ref{['th:cv_was_Talay']} point \ref{['th:cv_was_Talay_point_A']}
  13. Step 1. Mean reverting recursive control
  14. Step 2. Step weight assumption
  15. Step 3. Growth control assumption
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let $p >0,a \in (0,1]$, $s \geqslant 1, \rho \in [1,2]$ and, $\psi_p(y)=y^p$, $\phi(y)=y^a$ and $\epsilon_{\mathcal{I}}(\gamma)=\gamma^{\rho/2}$. Let $\alpha>0$ and $\beta \in \mathbb{R}$.

Theorems & Definitions (13)

  • Remark 3.4
  • Theorem 4.1
  • Remark 4.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lemme:Talay_infinitesimal_approx']}
  • Lemma 4.2
  • ...and 3 more