Convergence rate in precise asymptotics for Davis law of large numbers

TL;DR

This paper extends existing results on convergence rates in classical probability theorems to the Davis law of large numbers, providing precise asymptotics for the sum of probabilities involving partial sums of i.i.d. variables.

Contribution

It generalizes Klesov's convergence rate results from Heyde's theorem to the Davis law of large numbers, offering new precise asymptotic analysis.

Findings

Derived precise asymptotics for Davis law of large numbers

Extended Klesov's convergence rate results to new setting

Provided detailed asymptotic behavior of probability sums

Abstract

Let $\{X_n,n\geq 1\}$ be a sequence of i.i.d. random variables with partial sums $\{S_n,n\geq 1\}$. Based on the classical Baum-Katz theorem, a paper by Heyde in 1975 initiated the precise asymptotics for the sum $\sum_{n\geq 1}\mbox{P}(|S_n|\geq\epsilon n)$ as $\epsilon$ goes to zero. Later, Klesov determined the convergence rate in Heyde's theorem. The aim of this paper is to extend Klesov's result to the precise asymptotics for Davis law of large numbers, a theorem in Gut and Sp\u{a}taru [2000a].