On a limit theorem for non-stationary branching processes

Abstract

The purpose of this paper is to give a limit theorem for a certain class of discrete-time multi-type non-stationary branching processes (multi-type varying environment Galton-Watson processes). Let ${X_N} = {}^t({X_{N,1,}}{X_{N,2,}} \cdots,{X_{N,d}}), N = 0,1,2, \cdots$ be a discrete-time branching process with d types. The discrete-time branching process ${X_N}$ is determined by its generating functions ${\phi _{n,N,i}}(z).$ For $i \in \left\{ {1,2, \cdots,d} \right\},$ define ${e_i} = {}^t({e_{i,1}},{e_{i,2,}} \cdots,{e_{i,d}}) \in {Z_ + }^d by {e_{i,i}} = 1, and {e_{i,j}} = 0, if i \ne j.$ Then

$$ {\phi _{n,N,i}}(z)\mathop = \limits^d \sum {\left( {\prod\limits_{i = 1}^d {z_i^{ai}} } \right)} Prob\left\{ {{X_N} = \left. a \right| {X_n} = {e_i}} \right\}, n \in {Z_ + }, N \in {Z_ + }, n \leqslant N,i = 1,2, \cdots,d, z \in {C^d} $$

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