Extreme positions of regularly varying branching random walk in random and time-inhomogeneous environment

TL;DR

This paper studies the extreme positions in a supercritical branching random walk with regularly varying displacements in a random environment, revealing the limiting behavior of maximum positions and point processes under certain conditions.

Contribution

It provides an explicit description of the limit point process and demonstrates how the environment influences the asymptotic behavior of extremes in the model.

Findings

Normalized maximum converges to a scale-mixture of Fréchet distribution.

Limit point process is a randomly scaled scale-decorated Poisson process.

Environmental influence affects the joint asymptotic behavior of extreme positions.

Abstract

In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Frech\'{e}t random variable. Furthermore, we derive the weak limit of the point processes composed of appropriately scaled positions at the $n$-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes. Hence, an analog of the predictions by Brunet and Derrida (2011) holds. We have obtained an explicit description of the limit point process. This description captures the influence of the environment in the joint asymptotic behavior of the extreme positions. We show that the law of the clusters in the limit depends on the time-reversed environment. The asymptotic distribution of the normalized rightmost position is derived as a consequence. This approach (based on weak convergence of extremal processes) can not be adapted when the genealogical structure is given through a supercritical Branching Process in a time-Inhomogeneous Environment (BPIE) (due to lack of structural regularity in the genealogical structure). We provide a simple example where the point processes do not converge weakly. The tightness of the point processes holds in this example though the point processes do not (weakly) converge (due to having different subsequential weak limits). This phenomenon is not yet known in the literature.