On Seneta-Heyde Scaling for a stable branching random walk

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 $D_{\mathrm{\infty }}$ 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 $D_{\mathrm{\infty }}$.