The Minimal Position of a Stable Branching Random Walk

TL;DR

This paper investigates the asymptotic behavior of the minimal position in a boundary case branching random walk with stable law attraction, establishing integral tests and laws of iterated logarithm.

Contribution

It introduces new integral tests and laws of iterated logarithm for the minimal position in a stable branching random walk boundary case.

Findings

Established an integral test for the lower limit of M_n - (1/α) log n.

Proved a law of iterated logarithm for the upper limit of M_n - (1 + 1/α) log n.

Analyzed the minimal position in a branching random walk with stable law attraction.

Abstract

In this paper, a branching random walk $(V(x))$ in the boundary case is studied, where the associated one dimensional random walk is in the domain of attraction of an $\alpha-$stable law with $1<\alpha<2$. Let $M_n$ be the minimal position of $(V(x))$ at generation $n$. We established an integral test to describe the lower limit of $M_n-\frac{1}{\alpha}\log n$ and a law of iterated logarithm for the upper limit of $M_n-(1+\frac{1}{\alpha})\log n$.