Central limit theorem for a critical multi-type branching process in random environment

TL;DR

This paper establishes a central limit theorem for multi-type critical branching processes in random environments, extending known results from single-type processes and providing new insights into their asymptotic behavior.

Contribution

It introduces convergence theorems for multi-type processes, generalizing single-type results to more complex, multi-dimensional settings in random environments.

Findings

Proves convergence in distribution for normalized multi-type processes.

Extends CLT results from single-type to multi-type branching processes.

Provides theoretical foundation for understanding multi-type process fluctuations.

Abstract

Let (Z n) n$\ge$0 with Z n = (Z n (i, j)) 1$\le$i,j$\le$p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn $\sqrt$ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.