FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 3, PAGES 911-920

Exact asymptotic behaviour of the renewal measure in the "critical" case

M. S. Sgibnev

Abstract

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Let $\{S$_k} be a random walk drifting to $$-¥. The exact asymptotic behaviour of $$S_k=1^¥P (S_k ³ x) is considered under the following moment conditions: for some $$g > 0, $E\; e$g S₁ = 1, $E\; |S$₁| e^{g S₁} < ¥ and, in general, $E\; S$₁² e^{g S₁} = ¥.

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