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

M. S. Sgibnev

Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 763-770.

Résumé

Let $\{\xi_n\}$ be a sequence of identically distributed independent random variables, $M\xi_1=\mu0$, $M\xi_1^2\infty$; $S_0=0$, $S_n=\xi_1+\xi_2+\dots+=xi_n$, $n\ge1$; $\overline S=\sup\{S_n:n\ge0\}$. 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.