Berry–Esseen bound for a supercritical branching processes with immigration in a random environment

Let ( Z n ) be a supercritical branching process with immigration ( Y n ) in an independent and identically distributed environment ξ . We consider the rate of convergence of the normalized population W n = Z n / Π n to its limit W , where ( Π n ) is the usually used norming sequence, and establish a Berry–Esseen bound. Similar results are also obtained for W n + k − W n for each fixed positive integer k .