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

TL;DR

This paper introduces a forest model encoding multitype Galton-Watson processes with immigration, extending the Ray-Knight theorem to multitype continuous state branching processes, with convergence results in Brownian and stable settings.

Contribution

It provides new stochastic process encodings of multitype Galton-Watson forests and extends the Ray-Knight theorem to multitype continuous state branching processes with immigration.

Findings

Depth-first encodings converge to solutions of stochastic integral equations

Local times form multitype continuous state branching processes with immigration

Results apply in Brownian and $ ext{alpha}$-stable settings

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 $\alpha\in(1,2]$.