Lower deviations for supercritical branching processes with immigration

doi:10.1007/s11464-021-0922-9

Frontiers of Mathematics in China

2021, Vol. 16

Issue (2): 567-594 https://doi.org/10.1007/s11464-021-0922-9

本期目录

Lower deviations for supercritical branching processes with immigration

Qi SUN¹, Mei ZHANG²(

)

¹. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
². School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China

全文: PDF(338 KB)

Abstract：

For a supercritical branching processes with immigration ${Z n}$ ; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of $P (Z n = k n)$ with $k n = o (m n)$ as $n → ∞$ . We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.

Key words： Supercritical branching processes lower deviations immigration

收稿日期: 2020-07-28 出版日期: 2021-06-01

Corresponding Author(s): Mei ZHANG

引用本文:

. [J]. Frontiers of Mathematics in China, 2021, 16(2): 567-594.
Qi SUN, Mei ZHANG. Lower deviations for supercritical branching processes with immigration. Front. Math. China, 2021, 16(2): 567-594.

链接本文:

https://academic.hep.com.cn/fmc/CN/10.1007/s11464-021-0922-9
https://academic.hep.com.cn/fmc/CN/Y2021/V16/I2/567

1	S Asmussen, H Hering. Branching Processes.Boston: Birkhäuser-Verlag, 1983 https://doi.org/10.1007/978-1-4615-8155-0
2	K B Athreya, P E Ney. The local limit theorem and some related aspects of supercritical branching processes. Trans Amer Math Soc, 1970, 152(1): 233–251 https://doi.org/10.1090/S0002-9947-1970-0268971-X
3	K B Athreya, P E Ney. Branching Processes.Berlin-New York: Springer-Verlag, 1972 https://doi.org/10.1007/978-3-642-65371-1
4	J D Biggins, N H Bingham. Large deviations in the supercritical branching process. Adv in Appl Probab, 1993, 25(4): 757–772 https://doi.org/10.2307/1427790
5	W J Chu, W B Li, Y X Ren. Small value probabilities for supercritical branching processes with immigration. Bernoulli, 2014, 20(1): 377–393 https://doi.org/10.3150/12-BEJ490
6	S Dubuc, E Seneta. The local limit theorem for the Galton-Watson process. Ann Probab, 1976, 4:490–496 https://doi.org/10.1214/aop/1176996100
7	K Fleischmann, V Wachtel. Lower deviation probabilities for supercritical Galton- Watson processes. Ann Inst Henri Poincaré Probab Stat, 2007, 43: 233–255 https://doi.org/10.1016/j.anihpb.2006.03.001
8	A G Pakes. On supercritical Galton-Watson processes allowing immigration. J Appl Probab, 1974, 11: 814–817 https://doi.org/10.1017/S0021900200118248
9	V V Petrov. Sums of Independent Random Variables.Berlin-New York: Springer- Verlag, 1975 https://doi.org/10.1007/978-3-642-65809-9
10	E Seneta. On the supercritical Galton-Watson process with immigration. Math Biosci, 1970, 7: 9–14 https://doi.org/10.1016/0025-5564(70)90038-6
11	Q, Sun M Zhang. Harmonic moments and large deviations for supercritical branching processes with immigration. Front Math China, 2017, 12(5): 1201–1220 https://doi.org/10.1007/s11464-017-0642-3

Viewed

Full text

Abstract

Cited

Shared

Discussed