Teoriya Veroyatnostei i ee Primeneniya
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 3, Pages 435–452 (Mi tvp264)

On the ratio between the maximal and total numbers of individuals in a critical branching process in a random environment

V. I. Afanasyev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $\{\xi_n\}$ be a critical branching process in a random environment. Under some restrictions on the characteristics of the process, we show that the ratio of $\sum_{i=0}^n\xi_i$ to $\max_{0\leqq i\leqq n}\xi_i$, given $\{\xi_n>0\}$, converges in distribution as $n\to\infty$ to a random variable taking on values in $(1,+\infty)$.

Keywords: branching process in a random environment, conditional random walk, conditional limit theorems.

DOI: https://doi.org/10.4213/tvp264

Full text: PDF file (1739 kB)
References: PDF file   HTML file

English version:
Theory of Probability and its Applications, 2004, 48:3, 384–399

Bibliographic databases:

Citation: V. I. Afanasyev, “On the ratio between the maximal and total numbers of individuals in a critical branching process in a random environment”, Teor. Veroyatnost. i Primenen., 48:3 (2003), 435–452; Theory Probab. Appl., 48:3 (2004), 384–399

Citation in format AMSBIB
\Bibitem{Afa03} \by V.~I.~Afanasyev \paper On the ratio between the maximal and total numbers of individuals in a critical branching process in a random environment \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 3 \pages 435--452 \mathnet{http://mi.mathnet.ru/tvp264} \crossref{https://doi.org/10.4213/tvp264} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2141344} \zmath{https://zbmath.org/?q=an:1054.60086} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 3 \pages 384--399 \crossref{https://doi.org/10.1137/S0040585X97980506} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000224300900001} 

• http://mi.mathnet.ru/eng/tvp264
• https://doi.org/10.4213/tvp264
• http://mi.mathnet.ru/eng/tvp/v48/i3/p435

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Kozlov, “On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants”, Discrete Math. Appl., 16:2 (2006), 155–174
2. V. I. Afanasyev, “On the time of attaining a high level by a transient random walk in a random environment”, Theory Probab. Appl., 61:2 (2017), 178–207
•  Number of views: This page: 303 Full text: 44 References: 32