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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1977, Volume 22, Issue 5, Pages 763–770 (Mi mz8098)

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, $M\xi_1=\mu<0$, $M\xi_1^2<\infty$; $S_0=0$, $S_n=\xi_1+\xi_2+…+=xi_n$, $n\ge1$; $\overline S=\sup\{S_n:n\ge0\}$. 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.

Full text: PDF file (450 kB)

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

Bibliographic databases:

UDC: 519.2

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

Citation in format AMSBIB
\Bibitem{Sgi77} \by M.~S.~Sgibnev \paper Distribution of the supremum of sums of independent variables with negative drift \jour Mat. Zametki \yr 1977 \vol 22 \issue 5 \pages 763--770 \mathnet{http://mi.mathnet.ru/mz8098} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=467931} \zmath{https://zbmath.org/?q=an:0373.60061} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 5 \pages 916--920 \crossref{https://doi.org/10.1007/BF01098358}