Sharp estimate on the supremum of a class of partial sums of small i.i.d. random variables

TL;DR

This paper provides a sharp estimate for the tail distribution of the supremum of partial sums over an $L_1$-dense class of functions applied to i.i.d. random variables, especially when expected values are small.

Contribution

It offers a novel tail estimate for the supremum of partial sums over a function class with small expectations, advancing understanding of empirical process behavior.

Findings

Good tail bounds for supremum of partial sums

Applicable to classes with small expected values

Foundation for more general bounds in subsequent work

Abstract

We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ together with a sequence of independent, identically distributed $X$-space valued random variables $\xi_1,\dots,\xi_n$ and give a good estimate on the tail distribution of $\sup_{f\in\Cal F}\sum_{j=1}^n f(\xi_j)$ if the expected values $E|f(\xi_1)|$ are very small for all $f\in\Cal F$. In a subsequent paper~[2] we shall give a sharp bound for the supremum of normalized sums of i.i.d. random variables in a more general case. But that estimate is a consequence of the results in this work.