Open Access
Translator Disclaimer
February, 1995 Minimal Positions in a Branching Random Walk
Colin McDiarmid
Ann. Appl. Probab. 5(1): 128-139 (February, 1995). DOI: 10.1214/aoap/1177004832


We consider a branching random walk on the real line, with mean family size greater than 1. Let $B_n$ denote the minimal position of a member of the $n$th generation. It is known that (under a weak condition) there is a finite constant $\gamma$, defined in terms of the distributions specifying the process, such that as $n \rightarrow \infty$, we have $B_n = \gamma n + o(n)$ a.s. on the event $S$ of ultimate survival. Our results here show that (under appropriate conditions), on $S$ the random variable $B_n$ is strongly concentrated and the $o(n)$ error term may be replaced by $O(\log n)$.


Download Citation

Colin McDiarmid. "Minimal Positions in a Branching Random Walk." Ann. Appl. Probab. 5 (1) 128 - 139, February, 1995.


Published: February, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0836.60089
MathSciNet: MR1325045
Digital Object Identifier: 10.1214/aoap/1177004832

Primary: 60J80

Keywords: age-dependent branching process , Branching random walk , first birth time , Minimal position

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 1 • February, 1995
Back to Top