Electronic Journal of Probability

Large deviations for the empirical distribution in the branching random walk

Oren Louidor and Will Perkins

Full-text: Open access

Abstract

We consider the branching random walk $(Z_n)_{n \geq 0}$ on $\mathbb{R}$ where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. For a large class of measurable sets $A \subseteq \mathbb{R}$, it is well known that $\overline{Z}_n(\sqrt{n} A) \to \nu(A)$ almost surely as $n \to \infty$, where $\overline{Z}_n$ is the particles empirical distribution at generation $n$ and $\nu$ is the standard Gaussian measure on $\mathbb{R}$. We therefore analyze the rate at which $\mathbb{P}(\overline{Z}_n(\sqrt{n}A) > \nu(A) + \epsilon)$ and $\mathbb{P}(\overline{Z}_n(\sqrt{n}A) < \nu(A) - \epsilon)$ go to zero for any $\epsilon > 0$. We show that the decay is doubly exponential in either $n$ or $\sqrt{n}$, depending on $A$ and $\epsilon$ and find the leading coefficient in the top exponent. To the best of our knowledge, this is the first time such large deviation probabilities are treated in this model.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 18, 19 pp.

Dates
Accepted: 24 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067124

Digital Object Identifier
doi:10.1214/EJP.v20-2147

Mathematical Reviews number (MathSciNet)
MR3317160

Zentralblatt MATH identifier
1321.60049

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

Keywords
branching random walk large deviations

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Louidor, Oren; Perkins, Will. Large deviations for the empirical distribution in the branching random walk. Electron. J. Probab. 20 (2015), paper no. 18, 19 pp. doi:10.1214/EJP.v20-2147. https://projecteuclid.org/euclid.ejp/1465067124


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