Abstract
We consider the small value probability of supercritical continuous state branching processes with immigration. From Pinsky (1972) it is known that under regularity condition on the branching mechanism and immigration mechanism, the normalized population size converges to a non-degenerate finite and positive limit W as t tends to infinity. We provide sharp estimate on asymptotic behavior of ℙ(W ⩽ ɛ) as ɛ → 0+ by studying the Laplace transform of W. Without immigration, we also give a simpler proof for the small value probability in the non-subordinator case via the prolific backbone decomposition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berestycki J, Kyprianou A E, Murillo-Salas A. The prolific backbone for supercritical superprocesses. Stochastic Process Appl, 2011, 121: 1315–1331
Bertoin J, Fontbona J, Martinez S. On prolific individuals in a supercritical continuous state branching process. J Appl Probab, 2008, 45: 714–726
Bingham N H. Continuous branching processes and spectral positivity. Stochastic Process Appl, 1976, 4: 217–242
Bingham N H. On the limit of a supercritical branching process. J Appl Probab, 1988, 25: 215–228
Bingham N H, Goldie C M, Teugels J L. Regular Variation. Cambridge: Cambridge Univ Press, 1987
Bingham N H, Teugels J L. Duality for regularly varying functions. Quart J Math Oxford, 1975, 26: 333–353
Biggins J D, Bingham N H. Large deviation in the supercritical branching process. Adv Appl Prob, 1993, 25: 757–772
Biggins J D, Bingham N H. Near-constancy of phenomena in branching processes. Math Proc Camb Phil Soc, 1991, 110: 545–558
Chu WJ, Li WV, Ren Y-X. Small value probabilities for supercritical branching processes with immigration. Bernoulli, in press
Dubuc S, Problémes relatifs àlitération de fonctions suggérés par les processus en cascade. Ann Inst Fourier (Grenoble), 1971, 21: 171–251
Duquesne T, Winkel M. Growth of Lévy trees. Probab Theory Relat Fields, 2007, 139: 313–371
Fleischmann K, Wachtel V. Lower deviation probabilities for supercritical Galton-Watson processes. Ann Inst H Poincaré Probab Statist, 2007, 43: 233–255
Fleischmann K, Wachtel V. On the left tail asymtotics for the limit law of supercritical Galton-Watson processes in the Böttcher case. Ann Inst H Poincaré Probab Statist, 2009, 45: 201–225
Grey D R. Asymptotic Behaviour of continuous time, continuous state-space branching processes. J Appl Prob, 1974, 11: 669–677
Harris T E. Branching processes. Ann Math Statist, 1948, 19: 474–494
Karlin S, McGregor J. Embeddability of discrete time simple branching processes into continuous time branching processes. Trans Amer Math Soc, 1968, 132: 115–136
Karlin S, McGregor J. Embedding iterates of analytic functions with two fixed points into continuous groups. Trans Amer Math Soc, 1968, 132: 137–145
Kyprianou A E. Introductory Lectures on Fluctuations of Lévy Processes with Applications. New York: Springer-Verlag, 2006
Pinsky M A. Limit theorems for continuous state branching process with immigration. Bull Amer Math Soc, 1972, 78: 242–244
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chu, W., Li, W.V. & Ren, Y. Small value probabilities for continuous state branching processes with immigration. Sci. China Math. 55, 2259–2271 (2012). https://doi.org/10.1007/s11425-012-4522-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4522-8