Generalized maximal branching processes on bounded sets (Q1425534)

The author analyses maximal branching processes determined as Markov chains with transient probabilities \[ P(Z_{n+1}\leq y\mid Z_n=x) = F(y)^x, \quad x,\,y\in T. \] The distribution \(F\) is concentrated on the Borel set \(T\subset \mathbb R_+\), when \(T\subset [\alpha, \beta]\), where \(0<\alpha<\beta<\infty\). For the family of processes \(\big\{Z_n^{(\lambda)}\big\}\) with \(T^{(\lambda)}\subset [1,\lambda]\) a limiting theorem is proved concerning the behavior of stationary distributions \(\Psi^{(\lambda)}\) as \(\lambda\to \infty\). Some examples are cited. The results are illustrated by the computer simulation.