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Teoriya Veroyatnostei i ee Primeneniya, 1968, Volume 13, Issue 1, Pages 160–164
(Mi tvp829)
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This article is cited in 2 scientific papers (total in 2 papers)
Short Communications
An estimation of a convergence rate for the absorption probability in case of a null expectation
S. V. Nagaev Novosibirsk
Abstract:
Let $\xi_1,\xi_2,\dots$ be a sequence of mutually independent equally distributed random variables with a distribution function $F_\lambda(x)$ depending on a parameter $\lambda$. Let $\mathbf M\xi_1^2=2\lambda^2$ and $\mathbf M\xi_1=0$. Define $n_x$ as the least integer for which $\zeta_n+x\notin(a,b)$, where $\zeta_n=\sum_{i=1}^n\xi_i$ and $(a,b)$ is a finite interval of the real line. Put
$$
P_\lambda(x)=\mathbf P\{\zeta_{n_x}+x\ge b\},\quad x\in(a,b),
$$
and
$$
c_{3\lambda}=\mathbf M|\xi_1|^3.
$$
The following assertion is proved: there exists an absolute constant $L$ such that
$$
\sup_{a<x<b}\biggl|P_\lambda(x)-\frac{x-a}{b-a}\biggr|<L\frac{c_{3\lambda}}{\lambda^2(b-a)}.
$$
Received: 07.03.1967
Citation:
S. V. Nagaev, “An estimation of a convergence rate for the absorption probability in case of a null expectation”, Teor. Veroyatnost. i Primenen., 13:1 (1968), 160–164; Theory Probab. Appl., 13:1 (1968), 160–164
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https://www.mathnet.ru/eng/tvp829 https://www.mathnet.ru/eng/tvp/v13/i1/p160
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Abstract page: | 171 | Full-text PDF : | 62 |
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