Minimum and extremal process for a branching random walk outside the boundary case

TL;DR

This paper investigates the minimum and extremal process of a supercritical branching random walk outside the boundary case, confirming a conjecture and establishing convergence and infimum results under specific tail conditions.

Contribution

It extends the analysis of branching random walks beyond the boundary case, providing new convergence results and confirming a prior conjecture under stretched exponential tail conditions.

Findings

Confirmed the conjecture of Barral, Hu, and Madaule (2018).

Established weak convergence of the minimum and extremal process.

Proved an a.s. infimum result over all infinity rays.

Abstract

This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system.