Precise Asymptotics in Limit Theorems for a Supercritical Branching Process with Immigration in a Random Environment

Let (Z_n) be a supercritical branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on log Z_n and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm. By similar approach and with the help of a change of measure, we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations.