On the survival probability of a branching process in a finite state i.i.d. environment
A Galton-Watson process \(\{Z_ n\}_{n=0,1,...}\) is considered in an environment with randomly chosen states 0,1,...,r-1. The environmental process \(\{\zeta_ n\}\) is assumed i.i.d., and if m(\(\zeta)\) is the offspring mean corresponding to environment \(\zeta\), the classical condition for the process to be subcritical is then E[log m(\(\zeta\) \({}_ n)]<0.\) It is shown here that if the additional condition \(E[m(\zeta_ n)\log m(\zeta_ n)]<0\) holds (the process is strongly subcritical), then the survival probability \(P(Z_ n>0)\) tends to zero essentially geometrically, \(P(Z_ n>0)^{1/n}\to E Z_ 1\). For other subcritical processes the rate of decay of \(P(Z_ n>0)\) is strictly smaller than E \(Z_ 1\).
scientific article; zbMATH DE number 4032749