Advances in Applied Probability

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

Xinxin Chen

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We consider a branching random walk. Biggins and Kyprianou (2004) proved that, in the boundary case, the associated derivative martingale converges almost surely to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In this paper, we give a necessary and sufficient condition for the nontriviality of the limit in this boundary case.

Article information

Adv. in Appl. Probab., Volume 47, Number 3 (2015), 741-760.

First available in Project Euclid: 8 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G42: Martingales with discrete parameter

Branching random walk derivative martingale Mandelbrot's cascade random walk conditioned to stay positive


Chen, Xinxin. A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. Adv. in Appl. Probab. 47 (2015), no. 3, 741--760. doi:10.1239/aap/1444308880.

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