Convergence rate in precise asymptotics for the law of the iterated logarithm

Abstract

Let X,X ₁ ,X ₂ , . . . be a sequence of independent identically distributed random variables satisfying E[X] = 0, E[X ²] = σ ² > 0, and E[|X| ^q] < ∞ for some q ∈ (2, 3]. Let S _n  = ∑ ⁿ _k = 1 X _k , n ⩾ 1. We prove that $\underset{\mathit{\in}\searrow 0}{ \lim}\left({\displaystyle \sum_{n=3}^{\infty}\frac{1}{n \log n}\mathrm{P}\left(\left|{S}_n\right|\geqslant \mathit{\in}\sqrt{n \log \log n}\right)-{\sigma}^2{\mathit{\in}}^{-2}}\right)=\upzeta -\upgamma,$ where $\upzeta =\underset{n\to \infty }{ \lim}\left({\displaystyle \sum_{j=3}^n\frac{1}{j \log j} \log \log n}\right)\mathrm{and}\ \upgamma ={\displaystyle \sum_{n=3}^{\infty}\frac{1}{n \log n}\mathrm{P}\left({S}_n=0\right).}$