On functional limit theorems for branching processes with dependent immigration

Abstract

In this paper we consider a triangular array of branching
processes with non-stationary immigration.We prove a weak convergence
of properly normalized branching processes with immigration to deterministic
function under assumptions that immigration satisfies some mixing
conditions, the offspring mean tends to its critical value 1 and immigration
mean and variance controlled by regularly varying functions. Moreover,
we obtain a fluctuation limit theorem for branching process with immigration
when immigration generated by a sequence of m-dependent random
variables. In this case the limiting process is a time-changed Wiener
process. Our results extend the previous known results in the literature.