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Teoriya Veroyatnostei i ee Primeneniya, 1979, Volume 24, Issue 1, Pages 18–33
(Mi tvp948)
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This article is cited in 10 scientific papers (total in 10 papers)
On the probabilities of large deviations for the maximum of sums of independent random variables
A. K. Aleškevičiene Vilnius
Abstract:
Let $\xi_1,\xi_2,\dots$ be a sequence of independent identically distributed random variables with, non-degenerate distribution function $F(x)$,
$$
a=\mathbf E\xi_1,\quad\sigma^2=\mathbf D\xi_1,\quad S_{n}=\sum_{l=1}^n\xi_l,\quad
\overline S_n=\max_{1\le k\le n}S_k,\quad\overline F(x)=\mathbf P\{\bar S_n<x\}
$$ .
We prove that if $a=0$ and
$$
\int_{-\infty}^{\infty} e^{hy}\,dF(y)< \infty,\qquad |h|\le A,\ A>0,
$$
then for $n\to\infty$, $1<x=o(\sqrt{n})$
$$
\frac{1-\overline F_n(x\overline{\sigma}\sqrt{n})}{1-G(x)}=
\exp\biggl\{\frac{x^{3}}{\sqrt{n}}\lambda\biggl(\frac{x}{\sqrt{n}}\biggr)\biggr\}
\biggl[1+O\biggl(\frac{x}{\sqrt{n}}+e^{-x^2/8}\biggr)\biggr],
$$
where $\displaystyle G(x)=(2/\pi)^{1/2}\int_{0}^x e^{-u^2/2}\,du$ ($x\ge 0$), $G(x)=0$ ($x<0$) and $\lambda(u)$ is a Cramer's power series. Analogous statement is proved for the case $a>0$. We obtain also the theorems on the probabilities of large deviations for $\overline S_n$ in the Linnik's zones.
Received: 26.01.1976
Citation:
A. K. Aleškevičiene, “On the probabilities of large deviations for the maximum of sums of independent random variables”, Teor. Veroyatnost. i Primenen., 24:1 (1979), 18–33; Theory Probab. Appl., 24:1 (1979), 16–33
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https://www.mathnet.ru/eng/tvp948 https://www.mathnet.ru/eng/tvp/v24/i1/p18
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