Quenched normal approximation for random sequences of transformations

TL;DR

This paper establishes a quenched central limit theorem with convergence rates for random compositions of transformations, even in non-stationary and non-invariant settings, by analyzing fiberwise properties and dissecting the underlying mechanisms.

Contribution

It introduces bounds and techniques for quenched CLTs in non-stationary, non-invariant contexts, advancing understanding of the underlying mechanisms.

Findings

Proves a quenched CLT with convergence rates for non-stationary transformations.

Works with multivariate cases assuming fiberwise centering.

Provides insights into the mechanisms behind quenched CLTs.

Abstract

We study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate, also in the multivariate case, assuming fiberwise centering. For the most part we work with non-stationary randomness and non-invariant, non-product measures. Independently, we believe our work sheds light on the mechanisms that make quenched central limit theorems work, by dissecting the problem into three separate parts.