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

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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}
**