Limit theorems for a supercritical Poisson random indexed branching process

Abstract

Let {Z_n, n = 0, 1, 2, . . .} be a supercritical branching process, {N_t, t ≥ 0} be a Poisson process independent of {Z_n, n = 0, 1, 2, . . .}, then {Z_Nt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log Z_Nt.