The Davis-Gut law and Lai law for finitely inhomogeneous walks

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Abstract

Let {Xn, 1} be a sequence of independent and identically distributed random variables with partial sums Sn = ∑k=1 Xk , n 1 . Davis-Gut law states that ∞ ∑ n=1 1 n P { |Sn| > (1+ ε) √ 2n log logn }{< ∞, if ε > 0, = ∞, if ε < 0 if and only if EX1 = 0 and EX2 1 = 1 . Lai law states that ∞ ∑ n=1 nr−1P{|Sn| > (1+ ε) √ 2rn logn} { < ∞, if ε > 0, = ∞, if ε < 0 if and only if EX1 = 0 , EX2 1 = 1 and E(X 2 1 / log |X1|) < ∞ , where r > 0 . The paper will extend those results to the case where {Xn,n 1} are no longer identically distributed, but rather their distributions come from a finite set of distributions.