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