## Journal of Applied Probability

- J. Appl. Probab.
- Volume 49, Number 3 (2012), 652-670.

### Tightness for maxima of generalized branching random walks

#### Abstract

We study generalized branching random walks on the real line **R** that
allow time dependence and local dependence between siblings. Specifically,
starting from one particle at time 0, the system evolves such that each
particle lives for one unit amount of time, gives birth independently to a
random number of offspring according to some branching law, and dies. The
offspring from a single particle are assumed to move to new locations on
**R** according to some joint displacement distribution; the branching laws
and displacement distributions depend on time. At time *n*,
*F*_{n}(·) is used to denote the distribution
function of the position of the rightmost particle in generation *n*.
Under appropriate tail assumptions on the branching laws and offspring
displacement distributions, we prove that
*F*_{n}(· - Med(*F*_{n})) is
tight in *n*, where Med(*F*_{n}) is the median of
*F*_{n}. The main part of the argument is to demonstrate
the exponential decay of the right tail
1 - *F*_{n}(· - Med(*F*_{n})).

#### Article information

**Source**

J. Appl. Probab., Volume 49, Number 3 (2012), 652-670.

**Dates**

First available in Project Euclid: 6 September 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1346955324

**Digital Object Identifier**

doi:10.1239/jap/1346955324

**Mathematical Reviews number (MathSciNet)**

MR3012090

**Zentralblatt MATH identifier**

1261.60097

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

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

**Keywords**

Branching random walk recursion tightness

#### Citation

Fang, Ming. Tightness for maxima of generalized branching random walks. J. Appl. Probab. 49 (2012), no. 3, 652--670. doi:10.1239/jap/1346955324. https://projecteuclid.org/euclid.jap/1346955324