Edgeworth expansions for profiles of lattice branching random walks

TL;DR

This paper develops Edgeworth expansions for the profiles of lattice branching random walks, providing detailed asymptotic approximations and new limit theorems for occupation numbers, mode, and height of the profile.

Contribution

It introduces a comprehensive asymptotic expansion for the profile of lattice branching random walks, extending known results and covering the entire range except extreme values.

Findings

Unified asymptotic expansions for $L_n(k)$ with arbitrary order $r$

Limit theorems for occupation numbers, mode, and height

Dependence of asymptotics on the drift parameter's nature

Abstract

Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_n(k)$. We establish the following asymptotic expansion of $L_n(k)$, as $n\to\infty$: $$ e^{-\varphi(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2\pi \varphi''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right) \quad a.s., $$ where $r\in\mathbb N_0$ is arbitrary, $\varphi(\beta)=\log \sum_{k\in\mathbb Z} e^{\beta k} \mathbb E L_1(k)$ is the cumulant generating function of the intensity of the branching random walk and $$ x_n(k) = \frac{k-\varphi'(0) n}{\sqrt{\varphi''(0)n}}. $$ The expansion is valid uniformly in $k\in\mathbb Z$ with probability $1$ and the $F_j$'s are polynomials whose random coefficients can be expressed through the derivatives of $\varphi$ and the derivatives of the limit of the Biggins martingale at $0$. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for $r=0,1,2$ we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers $L_n(k_n)$, where $k_n\in\mathbb Z$ depends on $n$ in some regular way. We also prove a.s. limit theorems for the mode $\arg \max_{k\in\mathbb Z} L_n(k)$ and the height $\max_{k\in\mathbb Z} L_n(k)$ of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter $\varphi'(0)$ is integer, non-integer rational, or irrational.