On functional limit theorems for branching processes with dependent immigration

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 deter-ministic 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 immig-ration 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.