A concentration inequality for interval maps with an indifferent fixed point

TL;DR

This paper establishes a concentration inequality for interval maps with an indifferent fixed point, providing bounds on variance for Lipschitz observables and applying these results to various statistical properties.

Contribution

It introduces a new concentration inequality for a class of interval maps with an indifferent fixed point, linking decay of correlation to variance bounds.

Findings

Upper bound for variance of Lipschitz observables

Applications to almost-sure CLT and density estimation

Results on empirical measure and periodogram

Abstract

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.