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

This article is cited in 7 scientific papers (total in 7 papers)

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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Received: 27.03.2006

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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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. V. A. Vatutin, “The structure of decomposable reduced branching processes. I. Finitedimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. V. A. Vatutin, “The structure of decomposable reduced branching processes. II. Functional limit theorems”, Theory Probab. Appl., 60:1 (2016), 103–119  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathscinet  zmath  isi
    7. M. Liu, V. A. Vatutin, “Reduced critical branching processes for small populations”, Theory Probab. Appl., 63:4 (2019), 648–656  mathnet  crossref  crossref  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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