Home Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment
Article
Licensed
Unlicensed Requires Authentication

Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment

  • Valeriy Ivanovich Afanasyev EMAIL logo
Published/Copyright: November 17, 2021
Become an author with De Gruyter Brill

Abstract

We consider a strongly supercritical branching process in random environment with immigration stopped at a distant time 𝑛. The offspring reproduction law in each generation is assumed to be geometric. The process is considered under the condition of its extinction after time 𝑛. Two limit theorems for this process are proved: the first one is for the time interval from 0 till 𝑛, and the second one is for the time interval from 𝑛 till + .

MSC 2010: 60J80; 60J85

Award Identifier / Grant number: 19-11-00111

Funding statement: This work is supported by the Russian Science Foundation under grant 19-11-00111.

References

[1] V. I. Afanasyev, About time of reaching a high level by a random walk in a random environment, Theory Probab. Appl. 57 (2013), no. 4, 547–567. 10.1137/S0040585X97986175Search in Google Scholar

[2] V. I. Afanasyev, On the time of attaining a high level by a transient random walk in a random environment, Theory Probab. Appl. 61 (2017), no. 2, 178–207. 10.1137/S0040585X97T988101Search in Google Scholar

[3] V. I. Afanasyev, On the non-recurrent random walk in a random environment, Discrete Math. Appl. 28 (2018), no. 3, 139–156. 10.1515/dma-2018-0014Search in Google Scholar

[4] V. I. Afanasyev, A critical branching process with immigration in random environment, Stochastic Process. Appl. 139 (2021), 110–138. 10.1016/j.spa.2021.05.001Search in Google Scholar

[5] K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren Math. Wiss. 196, Springer, New York, 1972. 10.1007/978-3-642-65371-1Search in Google Scholar

[6] V. Bansaye, Cell contamination and branching processes in a random environment with immigration, Adv. in Appl. Probab. 41 (2009), no. 4, 1059–1081. 10.1239/aap/1261669586Search in Google Scholar

[7] C. Böinghoff, Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions, Stochastic Process. Appl. 124 (2014), no. 11, 3553–3577. 10.1016/j.spa.2014.05.009Search in Google Scholar

[8] E. Dyakonova, D. Li, V. Vatutin and M. Zhang, Branching processes in a random environment with immigration stopped at zero, J. Appl. Probab. 57 (2020), no. 1, 237–249. 10.1017/jpr.2019.94Search in Google Scholar

[9] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compos. Math. 30 (1975), 145–168. Search in Google Scholar

[10] C. Smadi and V. Vatutin, Critical branching processes in random environment with immigration: Survival of a single family, Extremes 24 (2021), no. 3, 433–460. 10.1007/s10687-021-00413-7Search in Google Scholar

[11] Y. Wang and Q. Liu, Limit theorems for a supercritical branching process with immigration in a random environment, Sci. China Math. 60 (2017), no. 12, 2481–2502. 10.1007/s11425-016-9017-7Search in Google Scholar

Received: 2021-08-26
Accepted: 2021-11-03
Published Online: 2021-11-17
Published in Print: 2022-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 21.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/eqc-2021-0036/html
Scroll to top button