Branching random walks with regularly varying perturbations
Krzysztof Kowalski
TL;DR
This work analyzes a perturbed branching random walk where particle positions receive i.i.d. perturbations $X_v(\theta)=\frac{1}{\theta}\log\frac{Y_v}{E_v}$, linking the rightmost position $R^*_n(\theta)$ to the weighted sum $Y_n(\theta)$ via $\theta R^*_n(\theta) \stackrel{d}{=} \log Y_n(\theta) - \log E$. Under regularly varying tails for $\mu$ with index $\gamma\in(0,1)$, and finite-mean/variance-type moment conditions, the paper completes the picture for the regime $\theta>\theta_0$ (where $\theta_0$ is defined by $\nu(\theta_0)=\theta_0\nu'(\theta_0)$) and derives weak centered asymptotics alongside almost sure convergence results. The main contributions are (i) a.s. convergence of $R^*_n/n$ in the below and above boundary cases, (ii) three distinct weak limit theorems with centering of the form $\alpha n + c\log n$ and explicit limiting objects $H_\theta$, $H_{\theta_0}$, and $Z$ that depend on additive/derivative martingales and stable limits, and (iii) detailed representations for the limiting distributions, including stable components and log-Laplace structures. These results extend prior finite-mean, boundary-case analyses to the heavy-tailed setting and provide comprehensive asymptotics across all parameter regimes, with precise constants and distributional descriptions.
Abstract
We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take form $\frac 1 θ\log\frac X E$, where $θ$ is a positive parameter, $X$ has arbitrary distribution $μ$ and $E$ is exponential with parameter 1, independent of $X$. Working under finite mean assumption for $μ$, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when $θ$ does not exceed certain critical parameter $θ_0$. This paper complements their work by providing weak centered asymptotics for the case when $θ> θ_0$ and extending the results to $μ$ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form $αn + c \log n$, with constants $α$, $c$ depending on the ratio of $θ$ and $θ_0$. We describe the limiting distribution and provide explicitly the constants appearing in the centering.