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

2014/07/04 by Peter Major, Major, Peter
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1407.1234

arxiv created 2014/07/04 · arxiv updated 2014/07/07

Abstract

We take an L₁-dense class of functions \Cal F on a measurable space (X,\Cal X) and a sequence of i.i.d. X-valued random variables ξ₁,…,ξ_n, and give a good estimate on the tail behaviour of sup_{f∈\Cal F}∑_j=1ⁿf(ξ_j) if the conditions sup_{x∈ X}|f(x)|≤1, Ef(ξ₁)=0 and Ef(ξ₁)²<σ² with some 0≤σ≤1 hold for all f∈\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.

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