On the derivative martingale in a branching random walk

We work under the A{i}d{e}kon-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 $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+o(log x)$ as $xtoinfty$. Also, we provide necessary and sufficient conditions under which $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+{rm const}+o(1)$ as $xtoinfty$. 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.