Functional central limit theorem for Euler--Maruyama scheme with decreasing step sizes

TL;DR

This work analyzes the Euler--Maruyama scheme for dissipative SDEs under a nonincreasing step-size schedule ${\eta_k}$ and shows that the EM time series converges to a subordinated Brownian motion $\{B_{a(t)}\}$ rather than plain Brownian motion, with time-change $a(t)$ determined by the step sizes. The authors develop a Poisson-equation-based decomposition to prove a central limit theorem with a novel scaling $\dfrac{n}{\sqrt{T_n}}$ (where $T_n=\sum_{k=1}^n 1/\eta_k$) and a functional central limit theorem yielding a Brownian motion time-changed by $a(t)$; a key technical achievement is bounding the Wasserstein-2 distance between the EM iterates and the corresponding continuous-time process. A crucial byproduct is a quantitative $W_2$ convergence bound $W_2(\theta_k,X_{t_k})\le C(1+|z|^{1/2})\eta_k^{1/4}$ obtained via reflection coupling, which is essential to handle the inhomogeneous dependence structure. The results clarify how decreasing step sizes induce polynomial ergodicity, alter scaling, and may lead to nonexistence of CLT/FCLT at the critical $\eta_k=1/k$ regime, with simulations supporting this conjecture. Overall, the paper provides a rigorous framework for CLT/FCLT under nonuniform time discretization and offers coupling techniques of independent interest for inhomogeneous Markov chains.

Abstract

We consider the Euler--Maruyama (EM) scheme of a family of dissipative SDEs, whose step sizes $η_{1}\geη_{2}\ge \cdots$ are decreasing, and prove that the EM scheme weakly converges to a subordinated Brownian motion $\{B_{a(t)}\}_{0\le t\le 1}$ rather than $\{B_{t}\}_{0\le t\le 1}$, where $a(t)$ is an increasing function depending on $\{η_{k}\}_{k \ge 1}$, for instance, $a(t)=t^{1+α}$ if $η_k =k^{-α}$. Compared to the EM scheme with constant step size, there are substantial differences as the following: (i) the EM time series is inhomogeneous and weakly converges to the ergodic measure in a polynomial speed; (ii) we have a special number $T_n =\frac{1}{η_1 }+\cdots+\frac{1}{η_n }$ which roughly measures the dependence of the EM time series; (iii) the normalized number in the CLT is $T_n ^{-1/2}n$ rather than $\sqrt{n}$, in particular, $T_n ^{-1/2}n \propto n^{(1-β)/2}$ when $η_{k}=1/k^β$ with $β\in(0,1)$; (iv) in the critical choice $η_{k}=1/k$, we have $T_{n}^{-1/2}n=O(1)$ and thus conjecture that the CLT and FCLT do not hold. This conjecture has been verified by simulations. A key distinction arises between the constant and decreasing step size implementations of the EM scheme. Under a constant step size, the time series is homogeneous. This allows one to use a stationary initialization, which automatically eliminates several complex terms in the subsequent proof of the CLT. Conversely, the time series generated by the EM scheme with decreasing step sizes forms an inhomogeneous Markov chain. To manage the analogous difficult terms in this case, that is, when the test function $h$ is Lipschitz, we must instead establish a bound for the Wasserstein-2 distance $W_{2}(θ_k ,X_{t_k })$. This technique for handling the inhomogeneous case could be of independent interest beyond the current proof.

Figures (3)

• Figure 1: The map of proof
• Figure 2: Distribution of $\frac{1}{\sqrt{T_{n}}}\sum_{k=0}^{n-1}[h(\theta_{k})-\pi(h)]$, $h(x)=\frac{1}{1+x^{2}}$, $n=10^{5}$.
• Figure 3: Distribution of $\frac{1}{\sqrt{T_{n}}}\sum_{k=0}^{n-1}[h(\theta_{k})-\pi(h)]$, $\eta_{k}=\frac{1}{k}$, $n=10^{9}$.

Theorems & Definitions (26)

• Remark 2.1
• Theorem 2.4
• Remark 2.2
• Theorem 2.6
• Theorem 2.7
• Remark 2.3
• Theorem 2.8
• Lemma 3.1: Lemma 3.3 in DFL2024
• Lemma 3.2: Moment estimations
• Lemma 3.3