论文标题
在分支随机行走中
On the derivative martingale in a branching random walk
论文作者
论文摘要
我们在a \“ıdékon-chen条件下工作，该条件确保了在线上的超临界分支随机步行中的衍生品玛格代尔几乎可以肯定地收敛到非统一的非平均随机变量，我们用$ z $表示，这表明$ \ \ mathbb {e} z \ mathbf {e} z \ mathbf {z \ mathbf {z {z z {z Z} x+o（\ log x）$作为$ x \ to \ infty $，我们提供了必要的条件，在此条件下证明分布限制定理的关键工具将衍生物果酱的收敛速率量化为$ z $。
We work under the A\"ıdékon-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 $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm const}+o(1)$ as $x\to\infty$. 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.
