Geodesic
Subcritical branching processes in random environment with
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 671-692 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathcal{A}_{i}(n)$ in which all individuals alive at time $n$ are descendants of the immigrant, who joined the population at time $i$, and investigate the asymptotic probability of this extreme event for $n\to \infty$ when $i$ is fixed, the difference $n-i$ is fixed, or $\min (i,n-i)\to \infty$. To deduce the desired asymptotics we establish some limit theorems for random walks conditioned to be nonnegative or negative on $[0,n]$.
Keywords: branching process, random environment, conditioned random walk.
Mots-clés : immigration 
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V. A. Vatutin; E. E. D'yakonova. Subcritical branching processes in random environment with. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 671-692. http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a1/

[1] C. Smadi, V. A. Vatutin, Critical branching processes in random environment with immigration: survival of a single family, 2019, 21 pp., arXiv: 1911.00316

[2] H. Kesten, M. V. Kozlov, F. Spitzer, “A limit law for random walk in a random environment”, Compositio Math., 30:2 (1975), 145–168 | MR | Zbl

[3] 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 | DOI | DOI | MR | Zbl

[4] V. I. Afanasyev, “On the time of reaching a high level by a transient random walk in a random environment”, Theory Probab. Appl., 61:2 (2017), 178–207 | DOI | DOI | MR | Zbl

[5] V. I. Afanasyev, “On the non-recurrent random walk in a random environment”, Discrete Math. Appl., 28:3 (2018), 139–156 | DOI | DOI | MR | Zbl

[6] V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350 | DOI | DOI | MR | Zbl

[7] E. Dyakonova, Doudou Li, V. Vatutin, Mei Zhang, “Branching processes in a random environment with immigration stopped at zero”, J. Appl. Probab., 57:1 (2020), 237–249 ; (2019), 13 pp., arXiv: 1905.03535 | DOI | MR | Zbl

[8] Doudou Li, V. Vatutin, Mei Zhang, “Subcritical branching processes in random environment with immigration stopped at zero”, J. Theor. Probability, 2020, 1–23, Publ. online ; 2019, 22 pp., arXiv: 1906.09590 | DOI

[9] V. Vatutin, Xinghua Zheng, “Subcritical branching processes in a random environment without the Cramer condition”, Stochastic Process. Appl., 122:7 (2012), 2594–2609 | DOI | MR | Zbl

[10] V. Bansaye, V. Vatutin, “On the survival probability for a class of subcritical branching processes in random environment”, Bernoulli, 23:1 (2017), 58–88 | DOI | MR | Zbl

[11] V. I. Afanasyev, J. Geiger, G. Kersting, V. A. Vatutin, “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673 | DOI | MR | Zbl

[12] V. I. Afanasyev, C. Böinghoff, G. Kersting, V. A. Vatutin, “Limit theorems for weakly subcritical branching processes in random environment”, J. Theoret. Probab., 25:3 (2012), 703–732 | DOI | MR | Zbl

[13] G. Kersting, V. Vatutin, Discrete time branching processes in random environment, Math. Stat. Ser., ISTE, London; John Wiley Sons, Inc., Hoboken, NJ, 2017, xiv+273 pp. | DOI | Zbl

[14] Y. Guivarc'h, Quansheng Liu, “Propriétés asymptotiques des processus de branchement en environnement aléatoire”, C. R. Acad. Sci. Paris Sér. I Math., 332:4 (2001), 339–344 | DOI | MR | Zbl