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

We consider the Euler--Maruyama (EM) scheme of a family of dissipative SDEs, whose step sizes $\eta_{1}\ge\eta_{2}\ge \cdots$ 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 $\{\eta_{k}\}_{k \ge 1}$, for instance, $a(t)=t^{1+\alpha}$ if $\eta_k =k^{-\alpha}$. 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 $T_n =\frac{1}{\eta_1 }+\cdots+\frac{1}{\eta_n }$ 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, $T_n ^{-1/2}n \propto n^{(1-\beta)/2}$ when $\eta_{k}=1/k^{\beta}$ with $\beta\in(0,1)$; (iv) in the critical choice $\eta_{k}=1/k$, 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 $W_{2}(\theta_k ,X_{t_k })$. This technique for handling the inhomogeneous case could be of independent interest beyond the current proof.