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

Major, Péter (2016) Sharp estimate on the supremum of a class of sums of small i.i.d. random variables. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 126 (1). pp. 100-117. ISSN 0304-4149

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Abstract

We take a class of functions F with polynomial covering numbers on a measurable space (X,X) together with a sequence of independent, identically distributed X-space valued random variables ξ1,...,ξn, and give a good estimate on the tail distribution of supfε Fj=1nf(ξj) if the expected values E|f(ξ1)| are very small for all fF. In a subsequent paper (Major, in press) we give a sharp bound for the supremum of normalized sums of i.i.d. random variables in a more general case. But the proof of that estimate is based on the results in this work. © 2015 Elsevier B.V. All rights reserved.

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