A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk

TL;DR

This paper establishes a precise condition under which the derivative martingale in a branching random walk converges to a non-trivial limit, extending the understanding of its behavior in boundary cases.

Contribution

It provides a necessary and sufficient condition for the non-trivial limit of the derivative martingale, along with a Kesten-Stigum-like theorem for branching random walks.

Findings

Identifies the exact condition for non-trivial limit convergence.

Extends Kesten-Stigum theorem to boundary case of branching random walk.

Clarifies the law as a fixed point of a smoothing transformation.

Abstract

We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and sufficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.