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 .