Rates of convergence in the central limit theorem for Banach valued dependent variables

TL;DR

This paper establishes convergence rates in the central limit theorem for Banach space-valued dependent variables, applying projective criteria and mixing conditions to derive explicit bounds, including for empirical distribution functions.

Contribution

It introduces new rates of convergence in the CLT for Banach space-valued dependent variables with finite moments, extending to empirical distribution functions under mixing conditions.

Findings

Rate of order O(n^{-(p-2)/2}) for empirical distribution functions in L^p spaces.

New conditions for optimal Wasserstein convergence rates in the real case.

Applicable to adapted stationary sequences with finite p-moments in Banach 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 $L^p(\mu)$, where $p\geq 2$ and $\mu$ is a real $\sigma$-finite measure: under some $\tau$-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 $p\in ]2,3]$.