The Minimal Position of a Stable Branching Random Walk

In this paper, a branching random walk $(V(x))$ in the boundary case is studied, where the associated one dimensional random walk is in the domain of attraction of an $\alpha -$stable law with $1<\alpha <2$. Let $M_n$ be the minimal position of $(V(x))$ at generation $n$. We established an integral test to describe the lower limit of $M_n-\frac{1}{\alpha }\log n$ and a law of iterated logarithm for the upper limit of $M_n-(1+\frac{1}{\alpha })\log n$.