The convergence of a super-critical age-dependent branching process allowing immigration at the epochs of a renewal process

Abstract

Let N( t) be the number of individuals in a super-critical age-dependent branching process allowing immigration at the epochs of a renewal process. The mean square and almost sure convergence of the random variable W( t) = e -α t N( t) are proved, where a is the Malthusian parameter of the branching process.