Rates of convergence in the central limit theorem for Banach valued dependent variables
Aurélie Bigot
TL;DR
This work develops rates of convergence in the central limit theorem for adapted stationary sequences with Banach-space-valued, dependent observations under projective criteria. The main result, a rate bound of the form $\Delta_n(f)=|\mathbb{E}[f(n^{-1/2}S_n)]-\mathbb{E}[f(G)]| \lesssim n^{-\delta/2}$, holds for test functions in $\Lambda_{2+\delta}(\mathbb{B},M)$ when the CLT holds and $X_0$ has a finite $(2+\delta)$-moment, under 2-smoothness of $\mathbb{B}$ and summable projective coefficients $\gamma(k)$ and $\gamma_{2,\delta}(k)$. The framework specializes to $L^p(\mu)$ spaces (with $p\ge 2$), yielding explicit rates for empirical functionals, such as the empirical distribution function in $L^p(\mu)$ and empirical processes linked to intermittent maps. In the real-valued setting, the paper provides concrete rates in terms of Zolotarev distances and, via Wasserstein comparisons, explicit rates under mixing conditions. The results bridge i.i.d. rate theory to dependent Banach-valued sequences and offer applicable criteria for empirical-process-type functionals in functional spaces.
Abstract
We provide rates of convergence in the central limit theorem in terms of projective criteria for adapted stationary sequences of centered random variables taking values in Banach spaces, with finite moment of order $p \in ]2,3]$ as soon as the central limit theorem holds for the partial sum normalized by $n^{-1/2}$. This result applies to the empirical distribution function in $L^p(μ)$, where $p\geq 2$ and $μ$ is a real $σ$-finite measure: under some $τ$-mixing conditions we obtain a rate of order $O(n^{-(p-2)/2})$. In the real case, our result leads to new conditions to reach the optimal rates of convergence in terms of Wasserstein distances of order $p\in ]2,3]$.