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Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 1, Pages 22–46 (Mi tvp144)  

This article is cited in 22 scientific papers (total in 22 papers)

Branching processes in random environment and “bottlenecks” in evolution of populations

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A branching process $Z(n)$, $n=0,1…$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $f_0(s),f_1(s),…$ . Let $S_0=0$, $S_0=0$, $S_k=\log f'_0(1)+…+\log f'_{k-1}(1)$, $k\ge 1$, be the associated random walk and let $\tau (n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Assuming that the random walk satisfies the Spitzer condition $n^{-1}\sum_{k=1}^{n}P\{S_k>0\}\to\rho\in(0,1)$, $n\to\infty$, we show (under the quenched approach) that for each fixed $t\in (0,1]$ and $m=0,\pm 1,\pm 2…$ the distribution of $Z(\tau(nt)+m)$ given $Z(n)>0$ converges as $n\to\infty $ to a (random) discrete distribution. Thus, in contrast to fixed points of the form $nt$, where the size of the population is large (even exponentially large, see [V. A. Vatutin and E. E. Dyakonova, Theory Probab. Appl., 49 (2005), pp. 275–308]), the size of the population at (random) points of sequential minima of the associated random walk becomes drastically small and, therefore, the branching process passes through a number of bottlenecks at such moments. As a corollary of our results we find (under the quenched approach) the distribution of the local time of the first excursion of a simple random walk in a random environment, provided this excursion attains a high level.

Keywords: branching processes in a random environment, Spitzer condition, conditional limit theorems, change of measure, random walk in a random environment, local time.

DOI: https://doi.org/10.4213/tvp144

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English version:
Theory of Probability and its Applications, 2007, 51:1, 189–210

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Received: 07.07.2005

Citation: V. A. Vatutin, E. E. D'yakonova, “Branching processes in random environment and “bottlenecks” in evolution of populations”, Teor. Veroyatnost. i Primenen., 51:1 (2006), 22–46; Theory Probab. Appl., 51:1 (2007), 189–210

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. E. E. D'yakonova, “Critical multitype branching processes in a random environment”, Discrete Math. Appl., 17:6 (2007), 587–606  mathnet  crossref  crossref  mathscinet  zmath  elib
    2. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. 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
    4. V. A. Vatutin, V. I. Vakhtel', “Sudden extinction of the critical branching process in random environment”, Theory Probab. Appl., 54:3 (2010), 466–484  mathnet  crossref  crossref  mathscinet  isi
    5. Loginov K.K., “Matematicheskaya model dinamiki populyatsii, razvivayuscheisya v nestatsionarnoi srede”, Vestn. Omskogo un-ta, 2009, no. 2, 54–57  mathscinet  zmath
    6. V. A. Vatutin, E. E. Dyakonova, “Asymptotic properties of multitype critical branching processes evolving in a random environment”, Discrete Math. Appl., 20:2 (2010), 157–177  mathnet  crossref  crossref  mathscinet  elib
    7. N. V. Pertsev, B. Yu. Pichugin, K. K. Loginov, “Statisticheskoe modelirovanie dinamiki populyatsii, razvivayuschikhsya v usloviyakh vozdeistviya toksichnykh veschestv”, Sib. zhurn. industr. matem., 14:2 (2011), 84–94  mathnet  mathscinet
    8. E. E. D'yakonova, “Multitype Galton–Watson branching processes in Markovian random environment”, Theory Probab. Appl., 56:3 (2011), 508–517  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. Loginov K.K., Pertsev N.V., “Primenenie $\Phi$-vetvyaschikhsya protsessov dlya issledovaniya dinamiki populyatsii v usloviyakh ogranichennogo kolichestva pischevykh resursov”, Vestn. Omskogo un-ta, 2011, no. 2, 24–28  elib
    10. V. A. Vatutin, “Total Population Size in Critical Branching Processes in a Random Environment”, Math. Notes, 91:1 (2012), 12–21  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. V. A. Vatutin, Q. Liu, “Critical branching process with two types of particles evolving in asynchronous random environments”, Theory Probab. Appl., 57:2 (2013), 279–305  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. E. E. D'yakonova, “Multitype branching processes evolving in a Markovian environment”, Discrete Math. Appl., 22:5-6 (2012), 639–664  mathnet  crossref  crossref  mathscinet  elib
    13. V. I. Afanasyev, “About time of reaching a high level by a random walk in a random environment”, Theory Probab. Appl., 57:4 (2013), 547–567  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    14. 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
    15. E. E. Dyakonova, “Branching processes in a Markov random environment”, Discrete Math. Appl., 24:6 (2014), 327–343  mathnet  crossref  crossref  mathscinet  elib  elib
    16. Boeinghoff Ch., “Limit Theorems For Strongly and Intermediately Supercritical Branching Processes in Random Environment With Linear Fractional Offspring Distributions”, Stoch. Process. Their Appl., 124:11 (2014), 3553–3577  crossref  mathscinet  zmath  isi  scopus
    17. Afanasyev V.I. Boeinghoff Ch. Kersting G. Vatutin V.A., “Conditional Limit Theorems For Intermediately Subcritical Branching Processes in Random Environment”, Ann. Inst. Henri Poincare-Probab. Stat., 50:2 (2014), 602–627  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    18. E. E. D'yakonova, “Limit theorem for multitype critical branching process evolving in random environment”, Discrete Math. Appl., 25:3 (2015), 137–147  mathnet  crossref  crossref  mathscinet  isi  elib
    19. Berestycki J., Brunet E., Harris J.W., Harris S.C., Roberts M.I., “Growth Rates of the Population in a Branching Brownian Motion With An Inhomogeneous Breeding Potential”, Stoch. Process. Their Appl., 125:5 (2015), 2096–2145  crossref  mathscinet  zmath  isi  elib  scopus
    20. 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
    21. V. A. Vatutin, E. E. D'yakonova, “How many families survive for a long time?”, Theory Probab. Appl., 61:4 (2017), 692–711  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    22. A. A. Imomov, A. Kh. Meiliev, “Ob asimptoticheskoi strukture nekriticheskikh markovskikh vetvyaschikhsya sluchainykh protsessov s nepreryvnym vremenem”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 69, 22–36  mathnet  crossref  elib
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