Diffusion approximation of a sequence of critical branching processes with dependent immigration

This paper aims to investigate a diffusion approximation of a sequence of critical branching processes with strictly stationary and ergodic immigration.

Under fairly general assumptions on the dependence structure of the immigration sequence, we establish that the scaled and properly normalized sequence of branching processes with immigration converges in distribution to the diffusion process in the Skorokhod topology.

Our result extends the functional limit theorem of Wei and Winnicki (1989) to the case where the immigration sequence is dependent.