Limit theorems for a supercritical Poisson random indexed branching process

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 N t , 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 N t .