CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

We are interested in the convergence rates of the submartingale Wn =Z/Πto its limit W, where(Π) is the usually used norming sequence and(Z) is a supercritical branching process with immigration(Y) in a stationary and ergodic environment ξ. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i) W-Wwith suitable normalization converges to the normal law N(0, 1), and similar results also hold for W-Wfor each fixed k∈N~*;(ii) for a branching process with immigration in a finite state random environment, if Whas a finite exponential moment, then so does W, and the decay rate of P(|W-W|> ε) is supergeometric;(iii) there are normalizing constants an(ξ)(that we calculate explicitly) such that a(ξ)(W-W) converges in law to a mixture of the Gaussian law.