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 Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 2, Pages 271–300 (Mi tvp173)

Limit theorems for reduced branching processes in a random environment

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $Z(n)$, $n=0,1…$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s)…$ and let $S_0=0$, $S_k=X_1+…+X_k$, $k\ge 1$, be the associated random walk with $X_i=\log f_{i-1}'(1)$, and let $\tau(n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Denoting by $Z(k,n)$ the number of particles existing in the branching process at moment $k\le n$ and having nonempty offspring at time $n$ and assuming that the associated random walk satisfies the Spitzer–Doney condition $\mathbf{P}\{S_n>0\}\to\rho\in(0,1)$, $n\to\infty$, we show (under the quenched approach) that for each fixed $m=0,\pm 1,\pm 2,…$ the distribution of $Z(\tau(n)+m,n)$ given $Z(n)>0$ converges as $n\to\infty$ to a (random) discrete distribution which is proper with probability 1. On the other hand, if $m=m(n)\to\infty$ as $n\to\infty$, then to prove a conditional limit theorem for $Z(\tau(n)+m,n)$ given $Z(n)>0$, a scaling of $Z(\tau(n)+m,n)$ is needed by a function growing to infinity as $m\to\infty$.

Keywords: branching processes in a random environment, Spitzer–Doney condition, conditional limit theorems, reduced process, most recent common ancestor.

DOI: https://doi.org/10.4213/tvp173

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English version:
Theory of Probability and its Applications, 2008, 52:2, 277–302

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Citation: V. A. Vatutin, E. E. D'yakonova, “Limit theorems for reduced branching processes in a random environment”, Teor. Veroyatnost. i Primenen., 52:2 (2007), 271–300; Theory Probab. Appl., 52:2 (2008), 277–302

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp173
• https://doi.org/10.4213/tvp173
• http://mi.mathnet.ru/eng/tvp/v52/i2/p271

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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. V. A. Vatutin, E. E. D'yakonova, “Waves in Reduced Branching Processes in a Random Environment”, Theory Probab. Appl., 53:4 (2009), 679–695
2. V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242
3. V. A. Vatutin, “The structure of decomposable reduced branching processes. I. Finitedimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662
4. V. A. Vatutin, “The structure of decomposable reduced branching processes. II. Functional limit theorems”, Theory Probab. Appl., 60:1 (2016), 103–119
5. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
6. Smadi C. Vatutin V.A., “Reduced Two-Type Decomposable Critical Branching Processes With Possibly Infinite Variance”, Markov Process. Relat. Fields, 22:2 (2016), 311–358
7. M. Liu, V. A. Vatutin, “Reduced critical branching processes for small populations”, Theory Probab. Appl., 63:4 (2019), 648–656
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