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

Xinxin Chen

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arXiv · 1402.5864

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

Xinxin Chen

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Abstract

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 finite 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 sufficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.

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Xinxin Chen. 2014-02-24. A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk. https://arxiv.org/abs/1402.5864

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