Abstract:
We investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in an environment generated by a sequence of independent identically distributed random variables. Under fairly general assumptions on the form of the offspring generating functions of particles, we show that the probability of survival up to generation n of the process initiated at moment zero by a single particle of type i is equivalent to βin−1/2, where βi is a positive constant. This assertion essentially generalizes a number of previously known results.
Keywords:
branching processes, random environment, survival probability, change of measure.
Citation:
V. A. Vatutin, E. E. D'yakonova, “Multitype branching processes in random environment: survival probability for the critical case”, Teor. Veroyatnost. i Primenen., 62:4 (2017), 634–653; Theory Probab. Appl., 62:4 (2018), 506–521
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\by V.~A.~Vatutin, E.~E.~D'yakonova
\paper Multitype branching processes in random environment: survival probability for the critical case
\jour Teor. Veroyatnost. i Primenen.
\yr 2017
\vol 62
\issue 4
\pages 634--653
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\jour Theory Probab. Appl.
\yr 2018
\vol 62
\issue 4
\pages 506--521
\crossref{https://doi.org/10.1137/S0040585X97T988782}
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Linking options:
https://www.mathnet.ru/eng/tvp5146
https://doi.org/10.4213/tvp5146
https://www.mathnet.ru/eng/tvp/v62/i4/p634
This publication is cited in the following 13 articles:
M. Peigné, C. Pham, “The survival probability of a weakly subcritical multitype branching process in iid random environment”, Electron. J. Probab., 29:none (2024)
Ion Grama, Quansheng Liu, Erwan Pin, “A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment”, Ann. Appl. Probab., 33:2 (2023)
Ion Grama, Quansheng Liu, Erwan Pin, “Convergence in Lp for a Supercritical Multi-type Branching Process in a Random Environment”, Proc. Steklov Inst. Math., 316 (2022), 160–183
V. A. Vatutin, E. E. Dyakonova, “Multitype branching processes in random environment”, Russian Math. Surveys, 76:6 (2021), 1019–1063
Sh. K. Formanov, Sh. Yu. Jurayev, “On transient phenomena in branching random processes with discrete time”, Lobachevskii J. Math., 42:12 (2021), 2777–2784
Emile Le Page, Marc Peigné, Da Cam Pham, “Central limit theorem for a critical multitype branching process in random environments”, Tunisian J. Math., 3:4 (2021), 801
Yanqing Wang, Quansheng Liu, “Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment”, Stochastics and Quality Control, 36:2 (2021), 145
V. A. Vatutin, E. E. D'yakonova, “The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment”, Math. Notes, 107:2 (2020), 189–200
V. A. Vatutin, E. E. D'yakonova, “Properties of multitype subcritical branching processes in random environment”, Discrete Math. Appl., 31:5 (2021), 367–382
V. A. Vatutin, E. E. Dyakonova, “Branching processes in random environment with sibling dependence”, J. Math. Sci. (N.Y.), 246:4 (2020), 569–579
W. Hong, M. Liu, V. Vatutin, “Limit theorems for supercritical mbpre with linear fractional offspring distributions”, Markov Process. Relat. Fields, 25:1 (2019), 1–31
E. Le Page, M. Peigné, C. Pham, “The survival probability of a critical multi-type branching process in i.i.d. random environment”, Ann. Probab., 46:5 (2018), 2946–2972
V. Vatutin, V. Wachtel, “Multi-type subcritical branching processes in a random environment”, Adv. Appl. Probab., 50:A (2018), 281–289
Citation in format AMSBIB
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