A necessary and su cient condition for the non-trivial limit of the derivative martingale in a branching random walk

Xinxin Chen (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a fi nite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and su fficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.

Mots clés: random walk conditioned to stay positive.; Mandelbrot's cascade; random walk conditioned to stay positive; Branching random walk; derivative martingale

Domaines: Probabilités [math.PR]

Xinxin Chen. A necessary and su cient condition for the non-trivial limit of the derivative martingale in a branching random walk. Advances in Applied Probability, 2015, 47 (3), pp.741-760. ⟨hal-00951159⟩