Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T10:04:27.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-normalized large deviation for supercritical branching processes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  26 July 2018

Weijuan Chu
Weijuan Chu*
Affiliation:
Hohai University
*
* Postal address: College of Science, Hohai University, No. 8 Focheng Road West, Nanjing, Jiangsu Province, 210098, P. R. China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

Keywords

Self-normalized large deviationsupercritical branching processmultitype branching process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 450 - 458
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Athreya, K. B. (1994). Large deviation rates for branching processes. I. Single type case. Ann. Appl. Prob. 4, 779–790. Google Scholar
[2]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. Google Scholar
[3]Athreya, K. B. and Vidyashankar, A. N. (1995). Large deviation rates for branching processes. II. The multitype case. Ann. Appl. Prob. 5, 566–576. Google Scholar
[4]He, H. (2016). On large deviation rates for sums associated with Galton–Watson processes. Adv. Appl. Prob. 48, 672–690. Google Scholar
[5]Jones, O. D. (2004). Large deviations for supercritical multitype branching processes. J. Appl. Prob. 41, 703–720. Google Scholar
[6]Ney, P. E. and Vidyashankar, A. N. (2003). Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Prob. 13, 475–489. Google Scholar
[7]Ney, P. E. and Vidyashankar, A. N. (2004). Local limit theory and large deviations for supercritical branching processes. Ann. Appl. Prob. 14, 1135–1166. (Correction: 16 (2006), 2272.) Google Scholar
[8]Royden, H. L. (1988). Real Analysis, 3rd edn. Macmillan, New York. Google Scholar
[9]Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Prob. 25, 285–328. Google Scholar