An extension of the Davis-Gut law and Lai law

摘要

Let {X,Xn,n 1} be a sequence of i.i.d. random variables with EX =0 and EX2 =1and the partial sums Sn = Σnk=1 Xk , n 1. Assume that f (x) and g(x) are positive functionsdefined on [0,∞) . In this short note, under some suitable conditions, we establish the followingresults∞Σn=1f (n)P{|Sn| β√ng(n)} ∞ or = ∞according as∞Σn=1f (n)g(n)exp{?β 22g2(n)(1+α(n))} ∞ or = ∞where α(n) = EX2I{|X| √ng(n)}/EX2I{|X| √ng(n)} , β 0. The results extend andgeneralize the known Davis-Gut Law and Lai Law.