On the derivative martingale in a branching random walk

Abstract

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{Z≤x}=logx+o(logx) as x→∞. Also, we provide necessary and sufficient conditions under which EZ1{Z≤x}=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.