Edgeworth expansions for profiles of lattice branching random walks

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.