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

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  21 March 2016

Xinxin Chen
Xinxin Chen*
Affiliation:
Université Paris VI
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 Place Jussieu, 75005 Paris, France. Email address: [email protected]
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Abstract

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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.

Keywords

Branching random walkderivative martingaleMandelbrot's cascaderandom walk conditioned to stay positive

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 741 - 760
Copyright
Copyright © Applied Probability Trust 2015 

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