Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants
-
K. Yu. Denisov
Abstract
We consider local probabilities of lower deviations for branching process Zn = Xn,1 + ⋯ + Xn,Zn−1 in random environment η. We assume that η is a sequence of independent identically distributed variables and for fixed η the variables Xi,j are independent and have geometric distributions. We suppose that steps ξi of the associated random walk Sn = ξ1 + ⋯ + ξn has positive mean and satisfies left-side Cramér condition: E exp(hξi) < ∞ if h− < h < 0 for some h− < − 1. Under these assumptions we find the asymptotic of the local probabilities P(Zn = ⌊exp(θn)⌋), n → ∞, for θ ∈ (max(m−, 0); m(− 1)) and for θ in a neighbourhood of m(− 1), where m− and m(− 1) are some constants.
Originally published in Diskretnaya Matematika (2022) 34, №4, 14–27 (in Russian).
Funding statement: This work was supported by the Russian Science Foundation under grant №19-11-00111, https://rscf.ru/project/19-11-00111/.
References
[1] Kozlov M. V., “On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants”, Discrete Math. Appl., 16:2 (2006), 155–174.10.1163/156939206777344593Search 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/S0040585X97984292Search in Google Scholar
[3] Bansaye V., Berestycki J., “Large deviations for branching processes in random environment”, Markov Proc. Relat. Fields, 15:3 (2009), 493–524.Search in Google Scholar
[4] Buraczewski D., Dyszewski P., Precise large deviation estimates for branching process in random environment, 2017, arXiv: 1706.03874.Search 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-2021-0039Search in Google Scholar
[6] Bansaye V., Boinghoff C., “Lower large deviations for supercritical branching processes in random environment”, Proc. Steklov Inst. Math., 282:1 (2013), 15–34.10.1134/S0081543813060035Search in Google Scholar
[7] Borovkov A. A., Asymptotic Analysis of Random Walks: Light-Tailed Distributions, Encycl. Math. Appl., 176, Cambridge: Cambridge Univ. Press, 2020, 450 pp.10.1017/9781139871303Search in Google Scholar
[8] Agresti A., “On the extinction times of varying and random environment branching processes”, J. Appl. Prob., 12:1 (1975), 39–46.10.2307/3212405Search in Google Scholar
[9] Denisov K. Yu., “Asymptotics of the local lower deviation probabilities for a branching process in a random environment under geometric distributions of the numbers of descendants”, Discrete Math. Appl., 32:5 (2022), 313–323.10.1515/dma-2022-0026Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On large deviations of the moment of attaining far level by the random walk in a random environment
- Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants
- Cloning operations and graph diameter
- Relations between energy complexity measures of Boolean networks and positive sensitivity of Boolean functions
- On the maximal size of tree in a random forest
- Deciding multiaffinity of polynomials over a finite field
- Probability that given vertices belong to the same connected component of random equiprobable mapping
Articles in the same Issue
- Frontmatter
- On large deviations of the moment of attaining far level by the random walk in a random environment
- Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants
- Cloning operations and graph diameter
- Relations between energy complexity measures of Boolean networks and positive sensitivity of Boolean functions
- On the maximal size of tree in a random forest
- Deciding multiaffinity of polynomials over a finite field
- Probability that given vertices belong to the same connected component of random equiprobable mapping
