An extension of the Davis-Gut law and Lai law

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Abstract

Let {X ,Xn,n 1} be a sequence of i.i.d. random variables with EX = 0 and EX2 = 1 and the partial sums Sn = ∑k=1 Xk , n 1 . Assume that f (x) and g(x) are positive functions defined on [0,∞) . In this short note, under some suitable conditions, we establish the following results ∞ ∑ n=1 f (n)P{|Sn| > β √ ng(n)} < ∞ or = ∞ according as ∞ ∑ n=1 f (n) g(n) exp{− 2 2 g2(n)(1+α(n))}< ∞ or = ∞ where α(n) = EX2I{|X | > √ng(n)}/EX2I{|X | √ng(n)} , β > 0 . The results extend and generalize the known Davis-Gut Law and Lai Law.