A Lamperti-type representation of continuous-state branching processes with immigration

Abstract

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.

Published in at http://dx.doi.org/10.1214/12-AOP766 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Self-similar fragmentations derived from the stable tree I: splitting at heights
• On the Height Profile of a Conditioned Galton-Watson Tree
• Changing the branching mechanism of a continuous state branching process using immigration

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