Moments and asymptotic properties for supercritical branching processes with immigration in random environments

We consider a supercritical discrete-time branching process (Zn) with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and Wn=Zn/∏i=0n−1mi be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments supnE[Wns|ξ,Y] and supnE[Wns|ξ] for s∈R, and discover the exponential Lp decay rates of Wn+1−Wn as well as the rates of Zn+1−mnZn. Then, as an application of the moment results, we show the exponential decay rates of Zn+1/Zn−mn and the convergence rates of the average of ratios 1n∑k=0n−1Zk+1/Zk.