ℕ-measure for continuous state branching processes and its application

Abstract

In this paper, we first give a direct construction of the ℕ-measure of a continuous state branching process. Then we prove, with the help of this ℕ-measure, that any continuous state branching process with immigration can be constructed as the independent sum of a continuous state branching process (without immigration), and two immigration parts (jump immigration and continuum immigration). As an application of this construction of a continuous state branching process with immigration, we give a proof of a necessary and sufficient condition, first stated without proof by M. A. Pinsky [Bull. Amer. Math. Soc., 1972, 78: 242–244], for a continuous state branching process with immigration to a proper almost sure limit. As another application of the ℕ-measure, we give a “conceptual” proof of an L log L criterion for a continuous state branching process without immigration to have an L ¹-limit first proved by D. R. Grey [J. Appl. Prob., 1974, 11: 669–677].

Keywords

Continuous state branching processes / spine decomposition / immigration

References

[1] Durrett R. Probability Theory and Examples, 1996, 2nd ed., Pacific Grove: Duxbury Press.

[2] Dynkin E. B., Kuznetsov S. E. ℕ-measures for branching Markov exit systems and their applications to differential equations. Probab Theory Relat Fields, 2004, 130, 135-150

[3] Grey D. R. Asymptotic Behaviour of continuous time, continuous state-space branching processes. J Appl Prob, 1974, 11, 669-677

[4] Kyprianou A. E. Introductory Lectures on Fluctuations of Lévy Processes with Applications, 2006, Berlin: Springer.

[5] Kyprianou A E, Liu R -L, Murillo-Salas A, Ren Y -X. Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann Inst H Poincaré Probab Statist (to appear)

[6] Liu R. -L., Ren Y. -X., Song R. Llog L criterion for a class of superdiffusions. J Appl Probab, 2009, 46, 479-496

[7] Lyons R. Athreya K. B., Jagers P. A simple path to Biggins’ martingale convergence theorem. Classical and Modern Branching Processes. Mathematics and Its Applications, IMA Vol 84, 1997, New York: Springer 217-222

[8] Lyons R., Pemantle R., Peres Y. Conceptual proofs of Llog L criteria for mean behaviour of branching processes. Ann Probab, 1995, 23, 1125-1138

[9] Pinsky M. A. Limit theorems for continuous state branching process with immigration. Bull Amer Math Soc, 1972, 78, 242-244

[10] Sato K. -I. Lévy processes and infinitely divisible distributions, 1999, Cambridge: Cambridge University Press.