On the controlled Galton-Watson process with random control function (Q1781636)

The controlled Galton-Watson process with random control function is defined as follows: \(Z_{0}:=N,\;Z_{n+1}:=\sum_{j=1}^{\varphi_{n Z_{n}}}X_{nj},\;n=0,1,\ldots\), where \(\{X_{nj},\;n=0,1,\ldots,\;j=1,2,\ldots \}\) and \(\{\varphi_{nk},\;n,k=0,1,\;\ldots \}\) are mutually independent collections of nonnegative integer-valued random variables; \(X_{nj}\) are identically distributed and, for fixed \(k\), \(\varphi_{nk}\) are identically distributed too; \(N\) is a positive integer. The authors investigate some basic transition properties between states of the homogeneous Markov chain \(Z_n,\;n=0,1,\ldots\) In particular, it is shown that under certain conditions the set of communicating states coincides with \(\mathbb N \cup \{0\}\). Set \(Y_n:=\sum_{i=0}^{n} Z_{i}, n=0,1,\ldots\) The authors point out representations of \(\mathbb{E}Y_n\), \(\mathbb{D}Y_n\) and \(\text{cov}(Y_n, Z_n)\) in terms of the first and second moments of \(X_{nj}\) and \(\varphi_{nk}\). Also they prove a strong limit theorem for \(Y_n\).