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 2β∕nα towards the origin and reflected at the origin. For any κ>2α, let M[nκ](n) denote the rightmost position of the particles in [nκ]-th generation. We prove that, conditioned on survival to generation [nκ], M[nκ](n) satisfies a large deviation principle.