Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Edgeworth expansions for profiles of lattice branching random walks

Rudolf Grübel and Zakhar Kabluchko

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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 $n\to\infty$: \begin{equation*}\mathrm{e}^{-\varphi(0)n}L_{n}(k)=\frac{\mathrm{e}^{-\frac{1}{2}x_{n}^{2}(k)}}{\sqrt{2\pi\varphi"(0)n}}\sum_{j=0}^{r}\frac{F_{j}(x_{n}(k))}{n^{j/2}}+o(n^{-\frac{r+1}{2}})\quad \text{a.s.},\end{equation*} where $r\in\mathbb{N}_{0}$ is arbitrary, $\varphi(\beta)=\log\sum_{k\in\mathbb{Z}}\mathrm{e}^{\beta k}\mathbb{E}L_{1}(k)$ is the cumulant generating function of the intensity of the branching random walk and \begin{equation*}x_{n}(k)=\frac{k-\varphi'(0)n}{\sqrt{\varphi"(0)n}}.\end{equation*} 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 $\mathop{\operatorname{arg\,max}}_{k\in\mathbb{Z}}L_{n}(k)$ 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. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.

Résumé

Nous considérons une marche branchante sur $\mathbb{Z}$ en temps discret. Soit $L_{n}(k)$ le nombre de particules au site $k\in\mathbb{Z}$ au temps $n\in\mathbb{N}_{0}$. Nous appelons profil de la marche branchante (au temps $n$) la fonction $k\mapsto L_{n}(k)$. Nous établissons le développement asymptotique suivant pour $L_{n}(k)$, lorsque $n\to\infty$ : \begin{equation*}\mathrm{e}^{-\varphi(0)n}L_{n}(k)=\frac{\mathrm{e}^{-\frac{1}{2}x_{n}^{2}(k)}}{\sqrt{2\pi\varphi"(0)n}}\sum_{j=0}^{r}\frac{F_{j}(x_{n}(k))}{n^{j/2}}+o(n^{-\frac{r+1}{2}})\quad \text{p.s.},\end{equation*} où $r\in\mathbb{N}_{0}$ est arbitraire, $\varphi(\beta)=\log\sum_{k\in\mathbb{Z}}\mathrm{e}^{\beta k}\mathbb{E}L_{1}(k)$ est la fonction génératrice des cumulants de l’intensité de la marche branchante, et \begin{equation*}x_{n}(k)=\frac{k-\varphi'(0)n}{\sqrt{\varphi"(0)n}}.\end{equation*} Le développement est valable uniformément en $k\in\mathbb{Z}$ avec probabilité $1$ et les $F_{j}$ sont des polynômes dont les coefficients aléatoires s’expriment à l’aide des dérivées de $\varphi$ et des dérivées de la limite de la martingale de Biggins en $0$. En utilisant une déformation exponentielle, nous établissons aussi des développements plus généraux qui couvrent tout le spectre de la marche branchante à l’exception des valeurs extrêmes. Comme application de ce développement pour $r=0,1,2$ nous retrouvons de façon unifiée plusieurs résultats connus et montrons de nouveaux théorèmes limite. En particulier, nous étudions le comportement p.s. des nombres d’occupation $L_{n}(k_{n})$, où $k_{n}\in\mathbb{Z}$ dépend de $n$ de façon régulière. Nous montrons aussi un théorème limite p.s. pour le mode $\mathop{\operatorname{arg\,max}}_{k\in\mathbb{Z}}L_{n}(k)$ et la hauteur $\max_{k\in\mathbb{Z}}L_{n}(k)$ du profil. Le comportement asymptotique de ces quantités dépend de si le paramètre de la dérive $\varphi'(0)$ est entier, rationnel, ou irrationnel. D’autres applications de nos résultats aux profils d’arbres aléatoires, incluant les arbres de recherche binaires et les arbres aléatoires récursifs, seront donnés dans un autre article.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2103-2134.

Dates
Received: 30 March 2015
Revised: 12 June 2016
Accepted: 10 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773740

Digital Object Identifier
doi:10.1214/16-AIHP785

Mathematical Reviews number (MathSciNet)
MR3729649

Zentralblatt MATH identifier
06847076

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations 60F15: Strong theorems

Keywords
Branching random walk Edgeworth expansion Central limit theorem Profile Biggins martingale Random analytic function Mod-$\varphi$-convergence Height Mode

Citation

Grübel, Rudolf; Kabluchko, Zakhar. Edgeworth expansions for profiles of lattice branching random walks. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2103--2134. doi:10.1214/16-AIHP785. https://projecteuclid.org/euclid.aihp/1511773740


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