Electronic Communications in Probability

Tail asymptotics of maximums on trees in the critical case

Mariusz Maślanka

Full-text: Open access

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 $.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1532678499

Digital Object Identifier
doi:10.1214/18-ECP145

Mathematical Reviews number (MathSciNet)
MR3841409

Zentralblatt MATH identifier
1394.60074

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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