Sharp tail distribution estimates for the supremum of a class of sums of i.i.d. random variables

Major, Péter (2016) Sharp tail distribution estimates for the supremum of a class of sums of i.i.d. random variables. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 126 (1). pp. 118-137. ISSN 0304-4149

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Abstract

We take a class of functions F with polynomially increasing covering numbers on a measurable space (X,X) together with a sequence of i.i.d. X-valued random variables ξ1,...,ξn, and give a good estimate on the tail behaviour of [eq found] if the relations [eq found] hold with some [eq found] for all [eq found]. Roughly speaking this estimate states that under some natural conditions the above supremum is not much larger than the largest element taking part in it. The proof heavily depends on the main result of paper Major (2015). We also present an example that shows that our results are sharp, and compare them with results of earlier papers. © 2015 Elsevier B.V. All rights reserved.

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