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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Abstract

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

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

Dates
First available in Project Euclid: 8 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1444308880

Digital Object Identifier
doi:10.1239/aap/1444308880

Mathematical Reviews number (MathSciNet)
MR3406606

Zentralblatt MATH identifier
1326.60121

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

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

Citation

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. https://projecteuclid.org/euclid.aap/1444308880


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