Berry-Esseen bounds for the χ^2 -distance in the Central Limit Theorem: a Markovian approach

Claire Delplancke (1) , Laurent Miclo (2, 3)

1 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 TSE-R - Toulouse School of Economics

This article presents a new proof of the rate of convergence to the normal distribution of sums of independent, identically distributed random variables in chi-square distance, which was also recently studied by Bobkov, Chistyakov and Götze. Our method consists of taking advantage of the underlying time non-homogeneous Markovian structure and studying the spectral properties of the non-reversible transition operator, which allows to find the optimal rate in the convergence above under matching moments assumptions. Our main assumption is that the random variables involved in the sum are independent and have polynomial density; interestingly, our approach allows to relax the identical distribution hypothesis.

Mots clés: Berry-Esseen bounds; Central Limit Theorem; non-homogeneous Markov chain; Hermite-Fourier decomposition; χ 2 -distance

Domaines: Probabilités [math.PR]

Claire Delplancke, Laurent Miclo. Berry-Esseen bounds for the χ^2 -distance in the Central Limit Theorem: a Markovian approach. 2019. ⟨hal-02322392⟩