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

2014/07/04 by Péter Major, Peter Major, Major, Peter
Decision Sciences · Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #math.PR

paper · pdf · doi:10.48550/arxiv.1407.1224

arxiv created 2014/07/04 · openalex publication_date 2014/07/04 · arxiv updated 2014/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We take an L₁-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 ξ₁,…,ξ_n and give a good estimate on the tail distribution of sup_{f∈\Cal F}∑_j=1ⁿ f(ξ_j) if the expected values E|f(ξ₁)| are very small for all f∈\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.

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