Abstract
We consider a supercritical discrete-time branching process 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 be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios
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