Convergence rate in precise asymptotics for Davis law of large numbers

Lingtao Kong and Hongshuai Dai

Statistics & Probability Letters, 2016, vol. 119, issue C, 295-300

Abstract: Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with E[X]=0 and E[X2]=σ2∈(0,∞), and set Sn=∑k=1nXk,n≥1. For any δ≥0, let γδ=limn→∞(∑j=1n(logj)δj−(logn)δ+1δ+1)andηδ=∑n=1∞(logn)δnP(Sn=0). Under the moment condition E[X2(log(1+∣X∣))1+δ]<∞, we prove that limϵ↘0[∑n=1∞(logn)δnP(∣Sn∣≥ϵnlogn)−E[∣N∣2δ+2]δ+1σ2δ+2ϵ−(2δ+2)]=γδ−ηδ, which refines Theorem 3 of Gut and Spătaru (2000a).

Keywords: Convergence rate; Precise asymptotics; Davis law of large numbers (search for similar items in EconPapers)
Date: 2016
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spl.2016.08.018

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