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 Z_n(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 Z_n(x). It turns out that thi problem is closely related to the non-degeneracy of the limit of the martingale sequence (W_n(q) º (P_n (q))^-1 A_n (q)) where A_n (q) = S_r exp(qz_r,n) and P_n (q) = E(A_n (q)) and (z_r,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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