## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 25 (2020), paper no. 55, 22 pp.

### Exponential and Laplace approximation for occupation statistics of branching random walk

Erol A. Peköz, Adrian Röllin, and Nathan Ross

#### Abstract

We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng [4] showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots $, converges in distribution as $n$ goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation $n$. Here we study the second order fluctuations around this limit, first, by providing a rate of convergence in the Wasserstein metric that holds for all $d\geq 3$, and second, by showing that for $d\geq 7$, the weak limit of the scaled joint differences between the number of occupancy-$j$ sites and appropriate multiples of the total population size converge in the Wasserstein metric to a multivariate symmetric Laplace distribution. We also provide a rate of convergence for this latter result.

#### Article information

**Source**

Electron. J. Probab., Volume 25 (2020), paper no. 55, 22 pp.

**Dates**

Received: 4 September 2019

Accepted: 20 April 2020

First available in Project Euclid: 5 May 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1588644038

**Digital Object Identifier**

doi:10.1214/20-EJP461

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

branching random walk distributional approximation exponential distribution multivariate symmetric Laplace distribution

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Exponential and Laplace approximation for occupation statistics of branching random walk. Electron. J. Probab. 25 (2020), paper no. 55, 22 pp. doi:10.1214/20-EJP461. https://projecteuclid.org/euclid.ejp/1588644038