On the tail behaviour of the distribution function of the maximum for the partial sums of a class of i.i.d. random variables

TL;DR

This paper provides estimates on the tail behavior of the maximum partial sums over an $L_1$-dense class of functions for i.i.d. random variables, showing the supremum is close to the maximum individual element under certain conditions.

Contribution

It introduces a new approach to estimate tail probabilities for the supremum of partial sums over function classes where classical Gaussian methods fail.

Findings

The supremum of partial sums is not much larger than the maximum element.

The estimates depend on the $L_1$-density and variance bounds of the function class.

The method extends tail behavior analysis beyond Gaussian-like variables.

Abstract

We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ and a sequence of i.i.d. $X$-valued random variables $\xi_1,\dots,\xi_n$, and give a good estimate on the tail behaviour of $\sup\limits_{f\in\Cal F}\sum\limits_{j=1}^nf(\xi_j)$ if the conditions $\sup\limits_{x\in X}|f(x)|\le1$, $Ef(\xi_1)=0$ and $Ef(\xi_1)^2<\sigma^2$ with some $0\le\sigma\le1$ hold for all $f\in\Cal F$. Roughly speaking this estimate states that under some natural conditions the above considered supremum is not much larger than the worst element taking part in it. The proof heavily depends on the main result of paper~[3]. Here we have to deal with such a problem where the classical methods worked out to investigate the behaviour of Gaussian or almost Gaussian random variables do not work.}