Central limit theorem under the Dedecker-Rio condition in some Banach spaces

TL;DR

This paper extends the central limit theorem to certain Banach spaces under the Dedecker-Rio condition, with applications to empirical processes in Sobolev spaces, broadening the theorem's applicability.

Contribution

It generalizes the CLT to adapted stationary ergodic sequences in smooth Banach spaces, including $L^p$ spaces, and provides conditions for empirical processes in Sobolev spaces.

Findings

CLT extended to $L^p$ spaces with $p \,\geq\, 2$

Sufficient conditions for empirical processes in Sobolev spaces

Discussion on optimality of these conditions

Abstract

We extend the central limit theorem under the Dedecker-Rio condition to adapted stationary and ergodic sequences of random variables taking values in a class of smooth Banach spaces. This result applies to the case of random variables taking values in $L^p(\mu)$, with $2 \leq p < \infty$ and $\mu$ a $\sigma$-finite real measure. As an application we give a sufficient condition for empirical processes indexed by Sobolev balls to satisfy the central limit theorem, and discuss about the optimality of these conditions.