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On Seneta-Heyde Scaling for a stable branching random walk (1610.03575v1)
Published 12 Oct 2016 in math.PR
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 $D_\infty$.