Startseite Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants
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Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants

  • Konstantin Yu. Denisov EMAIL logo
Veröffentlicht/Copyright: 12. Oktober 2022
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Abstract

We consider local probabilities of lower deviations for branching process Zn=Xn,1++Xn,Zn1 in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1++ξn has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)< if h<h<0 for some h<1. Under these assumptions, we find the asymptotic representation of local probabilities P(Zn= exp(θn) ) for θ[ θ1,θ2 ](μ;μ) for some non-negative μ.


Note

Originally published in Diskretnaya Matematika (2020) 32,№3, 24–37 (in Russian).


Funding statement: This work was supported by the Russian Science Foundation (project 19-11-001115) in Steklov Mathematical Institute of Russian Academy of Sciences.

Acknowledgment

The author is grateful to A. V. Shklyaev for constant attention and useful discussions and also to the anonimous reviewer for a number of valuable corrections.

References

[1] Kozlov M. V., “On large deviations of branching processes in a random environment: geometric distribution of descendants”, Discrete Math. Appl., 16:2 (2006), 155–174.10.1515/156939206777344593Suche in Google Scholar

[2] Kozlov M. V., “On large deviations of strictly subcritical branching processes in a random environment with geometric distribution of progeny”, Theory Probab. Appl., 54:3 (2010), 424–446.10.1137/S0040585X97984292Suche in Google Scholar

[3] Bansaye V., Berestycki J., “Large deviations for branching processes in random environment”, Markov Proc. Rel. Fields, 15:3 (2009), 493–524.Suche in Google Scholar

[4] Buraczewski D., Dyszewski P., “Precise large deviation estimates for branching process in random environment”, 2017, arXiv: 1706.03874.Suche in Google Scholar

[5] Shklyaev A. V., “Large deviations of branching process in a random environment. II”, Discrete Math. Appl., 31:6 (2021), 431–447.10.1515/dma-2021-0039Suche in Google Scholar

[6] Bansaye V., Böinghoff C., “Lower large deviations for supercritical branching processes in random environment”, Proc. Steklov Institute of Mathematics, 282:1 (2013), 15–34.10.1134/S0081543813060035Suche in Google Scholar

[7] Borovkov A. A., Asymptotic Analysis of Random Walking. Rapidly Decreasing Distributions of Increments, M.: Fizmatlit, 2013 (in Russian), 448 pp.Suche in Google Scholar

[8] Petrov V. V., “On the probabilities of large deviations for sums of independent random variables”, Theory Probab. Appl., 10:2 (1965), 287–298.10.1137/1110033Suche in Google Scholar

[9] Agresti A., “On the extinction times of varying and random environment branching processes”, J. Appl. Prob., 12:1 (1975), 39–46.10.2307/3212405Suche in Google Scholar

Received: 2020-05-28
Published Online: 2022-10-12
Published in Print: 2022-10-26

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 30.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dma-2022-0026/pdf
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