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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.
Received: 14.01.1977
Citation:
M. S. Sgibnev, “Distribution of the supremum of sums of independent variables with negative drift”, Mat. Zametki, 22:5 (1977), 763–770; Math. Notes, 22:5 (1977), 916–920
Linking options:
https://www.mathnet.ru/eng/mzm8098 https://www.mathnet.ru/eng/mzm/v22/i5/p763
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Abstract page: | 208 | Full-text PDF : | 98 | First page: | 1 |
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