Estimation of the Parameters of a Branching pro - Cess from Migrating

This paper considers a branchingprocess generated by an oospring distribution F with mean m < 1 and variance 2 < 1 and such that, at each generation n, there is an observed-migration, according to a binomial law B N bef n pv n which depends on the total population size N bef n. The-migration is deened as an emigration, an immigration, or a null migration, depending on the value of which is assumed constant throughout the diierent generations. The process with-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with-migration is supercritical, we generalize for the migrating observed process the results relative to the Galton-Watson supercritical case and that concern the asymptotic behaviour of the process and the estimation of m and 2 , as n ! 1. Moreover an asymptotic conndence interval of the initial population size is given.