Tail asymptotics of maximums on trees in the critical case

Electronic Communications in Probability

Mariusz Maślanka¹ 

关键词: Maximum recursion;    stochastic fixed point equation;    weighted branching process;    branching random walk;    power law distributions;   

DOI  :  10.1214/18-ECP145

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

We consider the endogenous solution to the stochastic recursion \[ X\,{\buildrel \mathit {d}\over =}\,\bigvee _{i=1}^{N}{A_iX_i}\vee{B} ,\] where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in{\mathbb {N}} }$ are random nonnegative numbers and $X_i$ are independent copies of $X$, independent also of $N$, $B$, $\{A_i\}_{i\in \mathbb{N} }$. The properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E} \left [\sum _{i=1}^{N}{A_i^s}\right ]$. Recently, Jelenković and Olvera-Cravioto assuming, inter alia, $m(s)t]\sim{Ct^{-\alpha }} \] for some $\alpha >0$ and $C>0$. In this paper we prove an analogous result when $m(s)=1$ has unique solution $\alpha >0$ and $m(s)>1$ for all $s\not =\alpha $.

【 授权许可】

CC BY