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Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 2, Pages 224–240 (Mi tvp105)  

This article is cited in 3 scientific papers (total in 3 papers)

Transient phenomena for random walks with nonidential jumps having nonidetically infinite variances

A. A. Borovkov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $\zeta_1,\zeta_2,…$ be independent random variables,
$$ Z_n=\sum_{i=1}^n\zeta_i,\qquad \overline{Z}_n=\max_{k\leq n}Z_k,\qquad Z=\overline{Z}_\infty. $$
It is well known that if $\zeta_i$ are identically distributed, then $Z$ is a proper random variable when ${\mathbf{E}\zeta_i=-a<0}$, and $Z=\infty$ a.s. when $a=0$. The limiting distribution for $\overline{Z}_n$ as $n\to\infty$, $a\to 0$ (in the triangular array scheme) when $\mathbf{E}\zeta_i^2<\infty$ is well studied (see, e.g., [J. F. C. Kingman, J. Roy. Statist. Soc. Ser. B, 24 (1962), pp. 383–392], [Yu. V. Prokhorov, Litov. Math. Sb., 3 (1963), pp. 199–204 (in Russian)], and [A. A. Borovkov, Stochastic Process in Queueing Theory, Springer-Verlag, New York, 1976]).
In the present paper, we study the limiting distribution for $\overline{Z}_n$ as $\mathbf{E}\zeta_i\to 0$, $n\to\infty$, in the case when $\mathbf{E}\zeta_i^2=\infty$ and the summands $\zeta_i$ are nonidentically distributed.

Keywords: random walks, maxima of partial sums, transient phenomena, convergence to stable processes, nonidentically distributed summands, infinite variance.

DOI: https://doi.org/10.4213/tvp105

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English version:
Theory of Probability and its Applications, 2006, 50:2, 199–213

Bibliographic databases:

Received: 16.09.2004

Citation: A. A. Borovkov, “Transient phenomena for random walks with nonidential jumps having nonidetically infinite variances”, Teor. Veroyatnost. i Primenen., 50:2 (2005), 224–240; Theory Probab. Appl., 50:2 (2006), 199–213

Citation in format AMSBIB
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\by A.~A.~Borovkov
\paper Transient phenomena for random walks with nonidential jumps having nonidetically infinite variances
\jour Teor. Veroyatnost. i Primenen.
\yr 2005
\vol 50
\issue 2
\pages 224--240
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\crossref{https://doi.org/10.4213/tvp105}
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\zmath{https://zbmath.org/?q=an:1090.60040}
\elib{http://elibrary.ru/item.asp?id=9153120}
\transl
\jour Theory Probab. Appl.
\yr 2006
\vol 50
\issue 2
\pages 199--213
\crossref{https://doi.org/10.1137/S0040585X97981627}
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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. V. I. Lotov, “Ob asimptotike momentov supremuma traektorii dlya sluchainykh bluzhdanii s malym snosom”, Vestn. NGU. Ser. matem., mekh., inform., 8:1 (2008), 51–58  mathnet
    2. A. A. Borovkov, P. S. Ruzankin, “Transient phenomena for random walks in the absence of the expected value of jumps”, Siberian Math. J., 50:5 (2009), 776–797  mathnet  crossref  mathscinet  isi
    3. V. I. Vakhtel', V. V. Shneer, “General approach to the maximum of a random walk under heavy upload”, Theory Probab. Appl., 55:2 (2011), 332–341  mathnet  crossref  crossref  mathscinet  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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