A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. (September 2015)

• Record Type: Journal Article
• Authors: Chen, Xinxin
• Is Part Of: Advances in applied probability. Volume 47:Number 3(2015)
• Page Start: 741
• Page End: 760
• Publication Date: 2015-09
• Subjects: Branching random walk, -- derivative martingale, -- Mandelbrot's cascade, -- random walk conditioned to stay positive; 60J80, -- 60G42; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1239/aap/1444308880
• Languages: English
• ISSNs: 0001-8678

Abstract

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.