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On large deviation rates for sums associated with Galton‒Watson processes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  19 September 2016

Hui He
Hui He*
Affiliation:
Beijing Normal University
*
* Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China. Email address: hehui@bnu.edu.cn
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Abstract

Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Keywords

Galton‒Watson processdomain of attractionstable distributionslowly varying functionlarge deviationLotka‒Nagaev estimatorSchröder constant

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 672 - 690
Copyright
Copyright © Applied Probability Trust 2016 

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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.CrossRefGoogle Scholar
[2] Athreya, K. B. and Vidyashankar, A. N. (1997)..Large deviation rates for supercritical and critical branching processes. In Classical and Modern Branching Processes (IMA Vol. Math. Appl. 84),Springer,New York, pp.1–18.Google Scholar
[3] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1989).Regular Variation.Cambridge University Press.Google Scholar
[4] Chu, W.,Li, W. V. and Ren, Y.-X. (2014).Small value probabilities for supercritical branching processes with immigration.Bernoulli 20,377–393.Google Scholar
[5] Cline, D. B. H. and Hsing, T. (1998).Large deviation probabilities for sums of random variables with heavy or subexponential tails. Tech. Rep., Texas A&M University.Google Scholar
[6] Denisov, D.,Dieker, A. B. and Shneer, V. (2008).Large deviations for random walks under subexponentiality: the big-jump domain.Ann. Prob. 36,1946–1991.CrossRefGoogle Scholar
[7] Dubuc, S. (1971).Problèmes relatifs à l'itération de fonctions suggérés par les processus en cascade.Ann. Inst. Fourier (Grenoble) 21,171–251.Google Scholar
[8] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[9] Fleischmann, K. and Wachtel, V. (2007).Lower deviation probabilities for superciritcal Galton‒Watson processes.Ann. Inst. H. Poincaré Prob. Statist. 43,233–255.Google Scholar
[10] Fleischmann, K. and Wachtel, V. (2008).Large deviations for sums indexed by the generations of a Galton‒Watson process.Prob. Theory Relat. Fields 141,445–470.Google Scholar
[11] Jacob, C. and Peccoud, J. (1996).Inference on the initial size of a supercritical branching processes from migrating binomial observations.C. R. Acad. Sci. Paris I 322,875–880.Google Scholar
[12] Jacob, C. and Peccoud, J. (1998).Estimation of the parameters of a branching process from migrating binomial observations.Adv. Appl. Prob. 30,948–967.CrossRefGoogle Scholar
[13] Nagaev, A. V. (1967).On estimating the expected number of direct descendants of a particle in a branching process.Theory Prob. Appl. 12,314–320.Google Scholar
[14] Nagaev, S. V. (1979).Large deviations of sums of independent random variables.Ann. Prob. 7,745–789.Google Scholar
[15] 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
[16] Ney, P. E. and Vidyashankar, A. N. (2004).Local limit theory and large deviations for superciritcal branching processes.Ann. Appl. Prob. 14,1135–1166.Google Scholar
[17] Piau, D. (2004).Immortal branching Markov processes: averaging properties and PCR applications.Ann. Prob. 32,337–364.CrossRefGoogle Scholar
[18] Pruitt, W. E. (1981).The growth of random walks and Lévy processes.Ann. Prob. 9,948–956.Google Scholar
[19] Rozovskiĭ, L. V. (1998).Probabilities of large deviations of sums of independent random variables with a common distribution function from the domain of attraction of an asymmetric stable law.Theory Prob. Appl. 42,454–482.CrossRefGoogle Scholar
[20] Samorodnitsky, G. and Taqqu, M. S. (1994).Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance.Chapman & Hall,New York.Google Scholar