Lp convergence and large deviations for supercritical multi-type branching processes in random environments

Considerad-typesupercriticalbranchingprocessZni =(Zni(1),···,Zni(d)),n≥0,inanindependentand identically distributed random environment ξ = (ξ0, ξ1, . . .), starting with one initial particle of type i, whose offspring distributions of generation n depend on the environment ξn at time n. In [1] we have established a Kesten-Stigum type theorem for Zni , which implies that for any 1 ≤ i,j ≤ d, Zni (j)/EξZni (j) → Wi in probability as n → +∞, where Eξ denotes the conditional expectation given the environment ξ, and Wi is a non-negative and finite random variable for which a criterion for non-degeneracy is obtained. Here we present the following results established in [2]: a necessary and sufficient condition for the convergence in Lp of the normalized population size Zni (j)/EξZni (j), a theorem giving its exponential convergence rate, and similar results for the associated fundamental martingale (Wni ). We also present a result on the precise large deviations for the total population size ∥Zn∥1 := 􏰀nj=1 Zn(j) of generation n recently established in [3], whose proof uses the Lp convergence and a similar large deviation result on products of random matrices proved in [4].