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We consider a supercritical discrete-time branching processwith immigration  Y  in a stationary and ergodic environment  ξ . Let  m nbe the mean of the reproduction distribution at time  n  conditioned on the environment  ξ  andbe the natural submartingale of the model. We show sufficient conditions for the boundedness of the momentsandforand discover the exponential  L pdecay rates ofas well as the rates ofThen, as an application of the moment results, we show the exponential decay rates ofand the convergence rates of the average of ratios

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