## Bernoulli

- Bernoulli
- Volume 24, Number 1 (2018), 772-800.

### Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

ZhiQiang Gao and Quansheng Liu

#### Abstract

Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.

#### Article information

**Source**

Bernoulli, Volume 24, Number 1 (2018), 772-800.

**Dates**

Received: September 2015

Revised: May 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142462

**Digital Object Identifier**

doi:10.3150/16-BEJ895

**Mathematical Reviews number (MathSciNet)**

MR3706776

**Zentralblatt MATH identifier**

06778347

**Keywords**

asymptotic expansion branching random walks central limit theorem convergence rate martingale approximation random environment

#### Citation

Gao, ZhiQiang; Liu, Quansheng. Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time. Bernoulli 24 (2018), no. 1, 772--800. doi:10.3150/16-BEJ895. https://projecteuclid.org/euclid.bj/1501142462