[Paper Review] Asymptotic expansions for profiles of lattice branching random walks

Rudolf Grübel, Zakhar Kabluchko|arXiv (Cornell University)|Mar 16, 2015

Stochastic processes and statistical mechanics24 references3 citations

TL;DR

This paper establishes a uniform almost sure asymptotic expansion for the profile of a lattice branching random walk on ℤ, expressing the particle count $ L_n(k) $ at site $ k $ at time $ n $ in terms of a Gaussian-type expansion involving derivatives of the cumulant generating function $ \varphi $ and the Biggins martingale. The expansion holds uniformly over all $ k \in \mathbb{Z} $ and enables new almost sure limit theorems for the mode, height, and occupation numbers of the profile.

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 $n o\infty$: $$ e^{-\varphi(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2}}{\sqrt {2\pi \varphi''(0) n}} \sum_{j=0}^r \frac{F_j(x_n)}{n^{j/2}} + o(n^{-\frac{r+1}{2}}) ext{ a.s.}, $$ 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 $$ x_n = x_n(k) = \frac{k-\varphi'(0) n}{\sqrt{\varphi''(0)n}}. $$ 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 establish also 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 many 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 $ ext{argmax}_{k\in\mathbb Z} L_n(k)$ and the height $ ext{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.

Motivation & Objective

• To derive a uniform almost sure asymptotic expansion for the profile $ L_n(k) $ of a branching random walk on $ \mathbb{Z} $.
• To characterize the almost sure behavior of the maximum occupation number $ \max_k L_n(k) $ and the mode $ \arg\max_k L_n(k) $ as $ n \to \infty $.
• To analyze how the drift parameter $ \varphi'(0) $—whether integer, rational non-integer, or irrational—affects the asymptotic behavior of the profile's extremal features.
• To extend the expansion to the entire range of the branching random walk, excluding extreme values, using exponential tilting.
• To unify and generalize known limit theorems for occupation numbers $ L_n(k_n) $ under regular $ k_n $-dependence on $ n $.

Proposed method

• Derive an asymptotic expansion for $ e^{-\varphi(0)n} L_n(k) $ using the cumulant generating function $ \varphi(\beta) = \log \sum_k e^{\beta k} \mathbb{E} L_1(k) $.
• Express the expansion in terms of a normalized location variable $ x_n = \frac{k - \varphi'(0)n}{\sqrt{\varphi''(0)n}} $, resembling a Gaussian density.
• Use the Biggins martingale's limit and its derivatives at $ \beta = 0 $ to define random coefficients $ F_j(x_n) $ in the polynomial expansion.
• Apply exponential tilting to extend the asymptotic expansion beyond the central limit regime, covering the full range of $ k $ except extreme values.
• Establish almost sure convergence of the expansion uniformly in $ k \in \mathbb{Z} $, valid for any fixed $ r \in \mathbb{N}_0 $.
• Leverage the expansion to derive almost sure limit theorems for the mode, height, and individual occupation numbers $ L_n(k_n) $.

Experimental results

Research questions

• RQ1 How does the profile $ L_n(k) $ behave asymptotically as $ n \to \infty $, uniformly in $ k \in \mathbb{Z} $?
• RQ2 What is the almost sure asymptotic behavior of the maximum occupation number $ \max_k L_n(k) $, and how does it depend on the drift $ \varphi'(0) $?
• RQ3 How does the location of the mode $ \arg\max_k L_n(k) $ evolve almost surely as $ n \to \infty $, and what role does the rationality of $ \varphi'(0) $ play?
• RQ4 Can the asymptotic expansion be extended beyond the central range to cover the entire profile, including the tails?
• RQ5 How do the occupation numbers $ L_n(k_n) $ behave almost surely when $ k_n $ grows with $ n $ in a regular fashion?

Key findings

• The profile $ L_n(k) $ admits a uniform almost sure asymptotic expansion: $ e^{-\varphi(0)n} L_n(k) = \frac{e^{-\frac{1}{2}x_n^2}}{\sqrt{2\pi \varphi''(0)n}} \sum_{j=0}^r \frac{F_j(x_n)}{n^{j/2}} + o(n^{-(r+1)/2}) $, valid for any $ r \in \mathbb{N}_0 $.
• The expansion is valid uniformly in $ k \in \mathbb{Z} $ with probability 1, and the coefficients $ F_j $ are polynomials with random coefficients derived from the Biggins martingale and $ \varphi $.
• For $ r = 0,1,2 $, the expansion recovers and unifies known limit theorems for occupation numbers and enables new a.s. limit theorems for the mode and height of the profile.
• The almost sure behavior of $ \max_k L_n(k) $ and $ \arg\max_k L_n(k) $ depends critically on whether $ \varphi'(0) $ is integer, non-integer rational, or irrational.
• Using exponential tilting, the expansion is extended to cover the entire range of $ k $, excluding only the extreme tails of the branching random walk.
• The results provide a foundation for analyzing profiles of random trees such as binary search trees and random recursive trees, with applications to be developed in a follow-up paper.