On a multitype Galton-Watson process with state-dependent immigration (Q1586590)

Let \(Z_n\), \(n=0,1,\ldots ,\) be a multitype Galton-Watson process with \(k\) types of particles that allows immigration in general if and only if the previous generation is empty. Assume that each time an immigration occurs the group of immigrating particles consists of only one particle and the type of the particle is \(1\). This is a multitype analog of \textit{J. H. Foster}'s process [Ann. Math. Stat. 42, 1773-1776 (1971; Zbl 0245.60063)]. The limit behaviour of such sort of processes with fixed Perron root \(\rho\) of mean matrix has been studied by \textit{K. V. Mitov} and \textit{N. M. Yanev} [J. Appl. Probab. 21, 22-39 (1984; Zbl 0532.60079)]. Under some conditions they proved the existence of a stationary distribution for \(Z_n\) with the Perron root \(\rho\). Let \((\nu_1,\ldots ,\nu_k)\) be a random vector with stationary distribution and \(\rho <1\). Let \(F(s)\) be a probability generating function of offspring distribution for \(Z_n\), and let it be of the form \(F(s)=1-\frac{M(1-s)}{1+\gamma(1-s)}\), where \(1, s, \gamma\) are vectors and \(M\) is a matrix. The main result of this paper is \[ {\mathbf P} \left\{-\frac{\log\nu_i}{\log(1-\rho)}\leq x_i, i=1,\ldots ,k \right\}\longrightarrow \min_{i=1,\ldots ,k}x_i, \] as \(\rho \to 1\), for \(x_i\in [0,1]\) if some additional conditions for \(M, s\) and \(\gamma\) are satisfied.