A general logarithmic asymptotic behavior for partial sums of i.i.d. random variables

Yu Miao and Deli Li

Statistics & Probability Letters, 2024, vol. 208, issue C

Abstract: Let 00. Let {X,Xn;n≥1} be a sequence of independent and identically distributed B-valued random variables and set Sn=∑i=1nXi,n≥1. In this note, a general logarithmic asymptotic behavior for {Sn;n≥1} is established. We show that if Sn/n1/p→P0, then, for all s>0, lim supn→∞logP‖Sn‖>sn1/p(logn)θ=−p−θζ¯(p,θ),lim infn→∞logP‖Sn‖>sn1/p(logn)θ=−p−θζ̲(p,θ),where ζ¯(p,θ)=−lim supt→∞logeptP(log‖X‖>t)tθandζ̲(p,θ)=−lim inft→∞logeptP(log‖X‖>t)tθ.The main tools used to prove this result are the symmetrization technique, an auxiliary lemma for the maximum of i.i.d. random variables, a moment inequality, and an exponential inequality.

Keywords: Large deviations; Laws of large numbers; Logarithmic asymptotic behaviors; Sums of i.i.d. random variables (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spl.2024.110043

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