|
Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 1, Pages 140–148
(Mi tvp1976)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
Short Communications
An estimate of the convergence rate for the absorption probability
S. V. Nagaev
Abstract:
Let $\xi_1$, $\xi_2$, …be a sequence of mutually independent equally distributed random variables. Let $\mathbf M\xi_1=m$, $\mathbf M\xi_1^2=2\lambda^2$, $\mathbf M|\xi_1|^3=c_3$. Define $n_x$ as the least integer $n$ for which $\zeta_n+x\notin(a,b)$ where $\zeta_n=\sum_{i=0}^n\xi_i$ and $(a,b)$ is a finite interval of the real line. Put
$$
P(x)=P\{\zeta_{n_x}+x\ge b\},\quad x\in(a,b).
$$
The following assertion is proved: there exists an absolute constant $L$ such that
$$
\sup_{a<x<b}|P(x)-u(x)|<\frac{Lc_3}{(b-a)\lambda^2}\biggl(1+\frac{|m|}{\lambda^2}(b-a)\biggr)
$$
where $u(x)$ is the solution of the equation
$$
u''+\frac m{\lambda^2}u'=0
$$
satisfying the boundary conditions $u(a)=0$, $u(b)=1$.
Citation:
S. V. Nagaev, “An estimate of the convergence rate for the absorption probability”, Teor. Veroyatnost. i Primenen., 16:1 (1971), 140–148; Theory Probab. Appl., 16:1 (1971), 147–154
Linking options:
https://www.mathnet.ru/eng/tvp1976 https://www.mathnet.ru/eng/tvp/v16/i1/p140
|
|