Convergence in LP for a Supercritical Multi-type Branching Process in a Random Environment

Consider a d-type supercritical branching process Z(n)(i) = (Z(n)(i)(1), ...,Z(n)(i)(d)), n >= 0, in an independent and identically distributed random environment xi = (xi(0), xi(1), ...), starting with one initial particle of type i. In a previous paper we have established a Kesten-Stigum type theorem for Z(n)(i), which implies that for any 1 <= i, j <= d, Z(n)(i)(j)/E(xi)Z(n)(i)(j) -> W-i in probability as n -> +infinity, where E(xi)Z(n)(i)(j) is the conditional expectation of Z(n)(i)(j) given the environment xi and W-i is a non-negative and finite random variable. The goal of this paper is to obtain a necessary and sufficient condition for the convergence in L-p of Z(n)(i)(j)/E(xi)Z(n)(i)(j), and to prove that the convergence rate is exponential. To this end, we first establish the corresponding results for the fundamental martingale (W-n(i)) associated to the branching process (Z(n)(i)).