Distribution of the supremum of sums of independent variables with negative drift

M. S. Sgibnev

Institute of Mathematics, Siberian Branch of USSR Academy of Sciences

Abstract: Let $\{\xi_n\}$ be a sequence of identically distributed independent random variables, $M\xi_1=\mu<0$, $M\xi_1^2<\infty$; $S_0=0$, $S_n=\xi_1+\xi_2+\dots+=xi_n$, $n\ge1$; $\overline S=\sup\{S_n:n\ge0\}$. The asymptotic behavior of $P(\overline S\ge t)$ as $t\to\infty$ is studied. If $\int_t^\infty P(\xi_1\ge x)\,dx=O(\tau(t))$, then
$$ P(\overline S\ge t)-\frac1{|\mu|}\int_t^\infty P(\xi_1\ge x)\,dx=O(\tau(t)/t), $$
$\tau(t)$ is a positive function, having regular behavior at infinity.

UDC: 519.2

Received: 14.01.1977

 English version: Mathematical Notes, 1977, 22:5, 916–920

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