On Seneta-Heyde Scaling for a stable branching random walk

TL;DR

This paper investigates the asymptotic behavior of martingales in a boundary case of a stable branching random walk, demonstrating convergence results related to the derivative and additive martingales under stable law attraction.

Contribution

It establishes the convergence of the derivative martingale and characterizes the scaling limit of the additive martingale in a stable domain of attraction.

Findings

Derivative martingale $D_n$ converges to a non-trivial limit $D_ fty$.

Scaled additive martingale $n^{1/\alpha}W_n$ converges to a multiple of $D_\infty$.

Results extend understanding of branching random walks in the domain of attraction of stable laws.

Abstract

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an $\alpha$-stable law with $1<\alpha<2$. We prove that the derivative martingale $D_n$ converges to a non-trivial limit $D_\infty$ under some regular conditions. We also study the additive martingale $W_n$, and prove $n^\frac{1}{\alpha}W_n$ converges in probability to a constant multiple of $D_\infty$.