Large deviation principle for the maximal positions in critical branching random walks with small drifts

We consider critical branching random walks V-(n), n >= 1 on Z(+). For fixed n, the displacement of an offspring from its parent is given by a nearest random walk with drift 2 beta/n(alpha) towards the origin and reflected at the origin. For any kappa > 2 alpha, let M-[n kappa]((n)) denote the rightmost position of the particles in [n(kappa)]-th generation. We prove that, conditioned on survival to generation [n(kappa)], M-[n kappa]((n)) satisfies a large deviation principle. (C) 2018 Elsevier B.V. All rights reserved.