Minima in branching random walks

TL;DR

This paper analyzes the minimum position in branching random walks, providing precise expectations and tail bounds, thereby characterizing the behavior of the minimum under broad conditions and extending previous work.

Contribution

It offers exact estimates for the expected minimum and exponential tail bounds, advancing understanding of the minimum's behavior in general branching random walks.

Findings

Calculated E}M_n to within O(1)

Proved exponential tail bounds for Pig{|}M_n-E}M_n|>x

Characterized the behavior of E}M_n in bounded branching and step size cases

Abstract

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\mathbf {E}M_n$ when the branching random walk has bounded branching and step size.