Transition phenomena for a Galton-Watson process with immigration in a Markovian environment (Q1365726)

Suppose \(\{Z_n,\;n\geq0\}\) is a Galton-Watson branching process in a random environment, which is regenerated by immigration in the sense that if \(Z_n=0\), then \(Z_{n+1}=1\); if \(Z_n=j\geq1\), all \(j\) particles reproduce according to the probability generating function \(f_{\zeta_n}\), where \(\{\zeta_n\}\) is an irreducible aperiodic Markov chain on \(\{0,1,\dots,l-1\}\). It is assumed that \(0<f_0'(1)\leq f_k'\). Also, defining \(M=\sum_{k=0}^{l-1}\omega_kf_k'(1)\), where \(\{\omega_k\}\) is the stationary distribution of \(\{\zeta_n\}\), it is shown that as \(M\to1\) from below, \(\pi[0,\frac{e^x}{1-M}]\to x\) for \(x\in[0,1]\).