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

We consider a supercritical discrete-time branching process (Z(n)) with immigration Y in a stationary and ergodic environment xi. Let m(n) be the mean of the reproduction distribution at time n conditioned on the environment xi and W-n = Z(n)/Pi(n-1)(i=0)m(i) be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments sup(n)E[W-n(s)vertical bar xi,Y] and sup(n)E[W-n(s)vertical bar xi] for s is an element of R, and discover the exponential L-p decay rates of Wn+1-W-n as well as the rates of Z(n+1)-m(n)Z(n). Then, as an application of the moment results, we show the exponential decay rates of Z(n+1)/Z(n-)m(n) and the convergence rates of the average of ratios 1(n) n-ary sumation k=0(n-1)Z(k+1)/Z(k).