Exact asymptotic behaviour of the renewal measure in the “critical” case

M. S. Sgibnev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $\{S_{k}\}$ be a random walk drifting to $-\infty$. The exact asymptotic behaviour of $\sum\limits_{k=1}^{\infty}\mathsf P(S_{k}\geq x)$ is considered under the following moment conditions: for some $\gamma>0$, $\mathsf Ee^{\gamma S_{1}}=1$, $\mathsf E|S_{1}|e^{\gamma S_{1}}<\infty$ and, in general, $\mathsf ES_{1}^{2}e^{\gamma S_{1}}=\infty$.

UDC: 519.218.4

Received: 01.05.2000

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