ON THE DERIVATIVE MARTINGALE IN A BRANCHING RANDOM WALK

We work under the Aidekon-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 EZ 1({Z <= x}) = log x + o(log x) as x -> infinity. Also, we provide necessary and sufficient conditions under which EZ 1({Z <= x}) = log x + const + o(1) as x -> infinity. 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.