Edgeworth expansions for profiles of lattice branching random walks

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 $n\in\mathbb N_0$\mathrm{.}Bytheprofileofthebranchingrandomwalk(attime$n$) we mean the function $k\mapsto L_n(k)$. We establish the following asymptotic expansion of $L_n(k)$, as $n\to \mathrm{\infty }$: $$e^{-\phi (0)n}{L}_n(k)=\frac{e^{-\frac{1}{2}{x}_n^2(k)}}{\sqrt{2\pi {\phi }^{\mathrm{\prime }\mathrm{\prime }}(0)n}}\sum _{j=0}^r\frac{F_j(x_n(k))}{n^{j\mathrm{/}2}}+o\left(n^{-\frac{r+1}{2}}\right)a\mathrm{.}s\mathrm{.},$$ 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 $$x_n(k)=\frac{k-{\phi }^{\mathrm{\prime }}(0)n}{\sqrt{{\phi }^{\mathrm{\prime }\mathrm{\prime }}(0)n}}\mathrm{.}$$ 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 $\phi$ 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 ${\phi }^{\mathrm{\prime }}(0)$ is integer, non-integer rational, or irrational.