On the derivative martingale in a branching random walk
Abstract
We work under the A\"d\'ekon-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 E Z1\Z x\= x+o( x) as x∞. Also, we provide necessary and sufficient conditions under which E Z1\Z x\= x+ 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.