Electronic Communications in Probability

Tail asymptotics of maximums on trees in the critical case

Mariusz Maślanka

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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 $.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 48, 11 pp.

Received: 24 May 2017
Accepted: 19 June 2018
First available in Project Euclid: 27 July 2018

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Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K05: Renewal theory

Maximum recursion stochastic fixed point equation weighted branching process branching random walk power law distributions

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Maślanka, Mariusz. Tail asymptotics of maximums on trees in the critical case. Electron. Commun. Probab. 23 (2018), paper no. 48, 11 pp. doi:10.1214/18-ECP145. https://projecteuclid.org/euclid.ecp/1532678499

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