Large Deviations for a Randomly Indexed Branching Process with Immigration

Abstract

Branching processes are widely used in mathematical biology. Here, we consider a supercritical continuous time branching process  called randomly indexed branching processes with immigration. Large deviation results are established for the logarithms of such processes. Our results show that when the offspring distribution belongs to the Schr\"{o}der case, the immigration distribution affects the rate function of the large deviation, while when the offspring distribution belongs to the B\"{o}ttcher case, the immigration distribution has no effect on the rate function.

Keywords: branching process with immigration, Poisson process, large deviations