Encoding multitype Galton-Watson forests and a multitype Ray-Knight theorem

Abstract

We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching process with immigration. These assumptions underlying our weak convergence arguments are easily seen to be met in the Brownian setting, and more generally an $\alpha$-stable setting for any $\alpha \in (1,2]$.

arXiv comment23 pages, 2 figures

References in corpus (5)

• Random trees and applications
• Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration
• Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth
• On scaling limits of multitype Galton-Watson trees with possibly infinite variance
• Fast Simulation of Size-Constrained Multitype Bienaymé-Galton-Watson Forests and Applications