  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 $D_\infty$.
