May 2021 On the derivative martingale in a branching random walk
Dariusz Buraczewski, Alexander Iksanov, Bastien Mallein
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Ann. Probab. 49(3): 1164-1204 (May 2021). DOI: 10.1214/20-AOP1474


We work under the Aïdékon–Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that EZ1{Zx}=logx+o(logx) as x. Also, we provide necessary and sufficient conditions under which EZ1{Zx}=logx+const+o(1) as x. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of, at most, linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.


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Dariusz Buraczewski. Alexander Iksanov. Bastien Mallein. "On the derivative martingale in a branching random walk." Ann. Probab. 49 (3) 1164 - 1204, May 2021.


Received: 1 February 2020; Revised: 1 August 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1474

Primary: 60G50 , 60J80
Secondary: 60F05 , 60G42

Keywords: Branching random walk , derivative martingale , Killed random walk , rate of convergence , subharmonic function , tail behavior

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 3 • May 2021
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