The survival probability of a critical multi-type branching process in i.i.d. random environment

Conditioned on the generating functions of offspring distribution, we study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to $1/\sqrt{n}$.