Tail asymptotics of maximums on trees in the critical case

Abstract

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, $m(s)<1$ for some $s$, proved that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e. \[\mathbb{P} [R>t]\sim{Ct^{-\alpha }} \] 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 $.