Large deviations for the maximum of a branching random walk

Abstract

We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but in general the rate functions for lower large deviations do not.

References in corpus (4)

• Slower deviations of the branching Brownian motion and of branching random walks
• Large Deviations for the Empirical Distribution in the General Branching Random Walk
• On large deviation probabilities for empirical distribution of branching random walks: Schr{ö}der case and B{ö}ttcher case
• Large deviations for the rightmost position in a branching Brownian motion

Cited by in corpus (3)

• Maximum of Catalytic Branching Random Walk with Regularly Varying Tails
• The empirical mean position of a branching Lévy process
• Large Deviations for the Right-Most Position of a Last Progeny Modified Branching Random Walk