We consider the small value probability of a supercritical multi-type branching process with immigration (Zn,n≥0). It is well known that with a suitable normalization, ρ−nZn converges to a finite and non-negative limit, which can be written as Wt where W is a random variable and t is a...

We consider a critical branching process with immigration controlled by a sequence of random variables {ξk,k≥1}, and prove a fluctuation limit theorem when {ξk} is (N−1)-dependent, improving the result in previous literatures.