Moment bounds for dependent sequences in smooth Banach spaces

TL;DR

This paper establishes moment bounds and concentration inequalities for dependent sequences in smooth Banach spaces, extending classical results to more complex, infinite-dimensional settings and specific dynamical systems.

Contribution

It introduces a Marcinkiewicz-Zygmund type inequality for Banach space-valued variables and derives sharp concentration bounds for empirical measures of nonuniformly expanding maps.

Findings

Proved a Marcinkiewicz-Zygmund inequality for smooth Banach space-valued variables.

Derived sharp concentration inequalities for empirical measures of certain dynamical systems.

Extended classical probabilistic inequalities to dependent, infinite-dimensional contexts.

Abstract

We prove a Marcinkiewicz-Zygmund type inequality for random variables taking values in a smooth Banach space. Next, we obtain some sharp concentration inequalities for the empirical measure of $\{T, T^2, \cdots, T^n\}$, on a class of smooth functions, when $T$ belongs to a class of nonuniformly expanding maps of the unit interval.