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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 2, Pages 255–266 (Mi tvp520)  

This article is cited in 1 scientific paper (total in 1 paper)

On the results of the asymptotic analysis in problems with boundaries

A. A. Borovkova, V. S. Korolyukb

a Novosibirsk
b Kiev

Abstract: The paper reviews the results of the asymptotic analysis in the boundary problems for random walks. Let $\xi_1,\xi_2,…$ be a sequence of independent identically distributed random variables $S_n=\sum_{k=1}^n\xi_k$ and let $g_n^-(t)<g_n^+(t)$ ($0\le t\le1$) be two functions such that $g_n^\pm(t)/b_n\to g^\pm(t)$ for some $b_n\to\infty$ uniformly on $[0,1]$. Let $\eta_g$ be the first passade time of the random trajectory $\{k/n,S_k\}$, $k=\overline{1,n}$ out of the region $g_n$ contained between the curves $x=g_n^\pm(t)$, $0\le t\le1$:
$$ \eta_g=1+\max\{k\colon g_n^-(\frac jn)<S_j<g_n^+(\frac jn),\quad j=0,1,…,k\le n\} $$
and $\chi_g$ be the value of the first jump over the boundary of $g_n$. The content of the article is the review of the results on limit theorems for the joint distributions of random variables $\eta_g$, $\chi_g$, $S_n$ as $n\to\infty$. The distributions of some other functionals of the trajectory $S_k$, $k=\overline{1,n}$ are also considered.

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English version:
Theory of Probability and its Applications, 1965, 10:2, 236–246

Bibliographic databases:


Citation: A. A. Borovkov, V. S. Korolyuk, “On the results of the asymptotic analysis in problems with boundaries”, Teor. Veroyatnost. i Primenen., 10:2 (1965), 255–266; Theory Probab. Appl., 10:2 (1965), 236–246

Citation in format AMSBIB
\Bibitem{BorKor65}
\by A.~A.~Borovkov, V.~S.~Korolyuk
\paper On the results of the asymptotic analysis in problems with boundaries
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 2
\pages 255--266
\mathnet{http://mi.mathnet.ru/tvp520}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=185638}
\zmath{https://zbmath.org/?q=an:0144.18608}
\transl
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 2
\pages 236--246
\crossref{https://doi.org/10.1137/1110028}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Borovkov, “Boundary-value problems, the invariance principle, and large deviations”, Russian Math. Surveys, 38:4 (1983), 259–290  mathnet  crossref  mathscinet  zmath  adsnasa  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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