  We prove an almost sure invariance principle for the capacity and the
cardinality of the range of a class of $\alpha$-stable random walks on the
integer lattice $\mathbb{Z}^d$ with $d/\alpha > 5/2$, and $d/\alpha >3/2$,
respectively. As a direct consequence, we conclude Khintchine's and Chung's
laws of the iterated logarithm for both processes.
