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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]
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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 

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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