On convergence properties of infinitesimal generators of scaled multi-type CBI processes

It is a common method for proving weak convergence of a sequence of time-homogeneous Markov processes towards a time-homogeneous Markov process first to show convergence of the corresponding infinitesimal generators and then to check some additional conditions. The aim of the present paper is to investigate convergence properties of discrete infinitesimal generators of appropriately scaled random step functions formed from a multi-type continuous state and continuous time branching process with immigration. We also present a convergence result for usual infinitesimal generators of the branching processes in question appropriately normalized.