A KESTEN-STIGUM TYPE THEOREM FOR A SUPERCRITICAL MULTITYPE BRANCHING PROCESS IN A RANDOM ENVIRONMENT

Consider a multitype branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton-Watson process. The famous Kesten-Stigum theorem for a supercritical multitype Galton-Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the nondegeneracy of the fundamental mar-tingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin for the sufficiency (Ann. Math. Stat. 42 (1971) 1499- 1520) and Tanny for the necessity (Stochastic Process. Appl. 28 (1988) 123- 139), but for the multitype case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by es-tablishing a criterion for the nondegeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry-Essen bound, and large deviation results.