On aggregation of multitype Galton-Watson branching processes with immigration

TL;DR

This paper investigates the limit behavior of aggregated multitype Galton-Watson branching processes with immigration, showing convergence to Brownian motion under certain moment conditions, with applications to autoregressive and single-type processes.

Contribution

It provides new limit theorems for the aggregation of multitype Galton-Watson processes with immigration, including both temporal and contemporaneous cases, under third order moment conditions.

Findings

Limit processes are zero mean Brownian motions with specific covariance.

Results apply to generalized integer-valued autoregressive processes.

Findings include specialized cases for single-type Galton-Watson processes.

Abstract

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.