## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 5, Number 1 (1995), 128-139.

### Minimal Positions in a Branching Random Walk

#### Abstract

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 $O(\log n)$.

#### Article information

**Source**

Ann. Appl. Probab., Volume 5, Number 1 (1995), 128-139.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004832

**Digital Object Identifier**

doi:10.1214/aoap/1177004832

**Mathematical Reviews number (MathSciNet)**

MR1325045

**Zentralblatt MATH identifier**

0836.60089

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Branching random walk minimal position age-dependent branching process first birth time

#### Citation

McDiarmid, Colin. Minimal Positions in a Branching Random Walk. Ann. Appl. Probab. 5 (1995), no. 1, 128--139. doi:10.1214/aoap/1177004832. https://projecteuclid.org/euclid.aoap/1177004832