Electronic Journal of Probability

Large deviations for the empirical distribution in the branching random walk

Oren Louidor and Will Perkins

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

branching random walk large deviations

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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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