Darling-Erdös type theorem for self-normalized sums

Research Report SRR02-008

Darling-Erdös type theorem for self-normalized sums

Qiying Wang

Abstract: Let X, X₁, X₂,..., be i.i.d. non-degenerate random variables with zero mean, S_n=\sum_j=1ⁿ X_j, V_n²=\sum_j=1ⁿ X_j² and l(x)=EX²I_{(|X|\le
x)}. In the note, a Darling-Erdös type theorem for the maximum of self-normalized sums, max_{1\le k\le
n} S_k/V_k, is derived under the condition that l(x) is a slowly varying function at \infty, satisfying l(x)\le c₁exp{c₂(log x)^\beta} for some c₁>0, c₂>0 and 0\le \beta<1/2. This extends a result give by Csörgö, Szyszkowicz and Wang (2002).

Primary AMS Classification: 60F15
Date: 6 November 2002

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