Growth Rates for the Branching Random Walk in Random Environments-1, The Case of a Supercritical Region


Let (Z(n), n ³ 0) denote a branching random walk in stationary ergodic environments starting with a single ancestor at the origin. Let Zn(x) denote the number of particles living to the left of x. In this paper we study the large deviation problem associated with the functional Zn(x). It turns out that thi problem is closely related to the non-degeneracy of the limit of the martingale sequence (Wn(q) º (Pn (q))-1 An (q)) where An (q) = Sr exp(qzr,n) and Pn (q) = E(An (q)) and (zr,n : r) are the positions of the nth generation population. For this reason we obtain necessary and sufficient conditions for the non-degeneracy of the limit W (q) thus extending the classical Kesten-Stigum Theorem to the case of branching random walk in random environments.1

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