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 Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 1, Pages 21–38 (Mi tvp2959)

Reduced branching processes in random environment: the critical case

V. A. Vatutin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $Z_n$ be the number of particles at time $n=0,1,2,…$ in a branching process in random environment, $Z_0=1$, and let $Z_{m,n}$ be the number of such particles in the process at time $m\in[0,n]$, each of which has a nonempty offspring at time $n$. It is shown that if the offspring generating functions $f_k(s)$ of the particles of the $k$th generation are independent and identically distributed for all $k=0,1,2,…$ with $E\log f'_k(1)=0$ and $\sigma^2=E(\log f'_k(1))^2\in(0,\infty)$, then, under certain additional restrictions, the sequence of conditional processes
$$\{\frac1{\sigma\sqrt{n}} \log Z_{[nt],n}, t\in[0,1]|Z_n>0\}$$
converges, as $n\to\infty$, in distribution in Skorokhod topology to the process $\{\inf_{t\le u\le 1}W^+(u), t\in[0,1]\}$, where $\{W_+(t), t\in [0,1]\}$ is the Brownian meander.

Keywords: critical branching process in random environment, reduced process, functional limit theorem, random walk.

DOI: https://doi.org/10.4213/tvp2959

Full text: PDF file (1421 kB)

English version:
Theory of Probability and its Applications, 2003, 47:1, 99–113

Bibliographic databases:

Citation: V. A. Vatutin, “Reduced branching processes in random environment: the critical case”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 21–38; Theory Probab. Appl., 47:1 (2003), 99–113

Citation in format AMSBIB
\Bibitem{Vat02} \by V.~A.~Vatutin \paper Reduced branching processes in random environment: the critical case \jour Teor. Veroyatnost. i Primenen. \yr 2002 \vol 47 \issue 1 \pages 21--38 \mathnet{http://mi.mathnet.ru/tvp2959} \crossref{https://doi.org/10.4213/tvp2959} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1978693} \zmath{https://zbmath.org/?q=an:1039.60077} \transl \jour Theory Probab. Appl. \yr 2003 \vol 47 \issue 1 \pages 99--113 \crossref{https://doi.org/S0040585X97979421} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000183800400008} 

• http://mi.mathnet.ru/eng/tvp2959
• https://doi.org/10.4213/tvp2959
• http://mi.mathnet.ru/eng/tvp/v47/i1/p21

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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. Vladimir V., Elena D., “Reduced branching processes in random environment”, Mathematics and Computer Science II - Algorithms, Trees, Combinatorics and Probabilities, Trends in Mathematics, 2002, 455–467
2. V. I. Afanasyev, “On the ratio between the maximal and total numbers of individuals in a critical branching process in a random environment”, Theory Probab. Appl., 48:3 (2004), 384–399
3. V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. I: limit theorems”, Theory Probab. Appl., 48:2 (2004), 314–336
4. V. A. Vatutin, E. E. D'yakonova, “Limit theorems for reduced branching processes in a random environment”, Theory Probab. Appl., 52:2 (2008), 277–302
5. V. A. Vatutin, E. E. D'yakonova, “Waves in Reduced Branching Processes in a Random Environment”, Theory Probab. Appl., 53:4 (2009), 679–695
6. V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242
7. V. A. Vatutin, “The structure of decomposable reduced branching processes. I. Finitedimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662
8. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
9. V. A. Vatutin, E. E. D'yakonova, “How many families survive for a long time?”, Theory Probab. Appl., 61:4 (2017), 692–711
10. Vatutin V. Dyakonova E., “Path to Survival For the Critical Branching Processes in a Random Environment”, J. Appl. Probab., 54:2 (2017), 588–602