Minima in branching random walks

Description

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbfEM_n$ to within O(1) and prove exponential tail bounds for $\\mathbfP{|M_n-\\mathbfEM_n|>x}$, 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 $\\mathbf EM_n$ when the branching random walk has bounded branching and step size.

Abstract

International audience

Additional details