  We consider convex hulls of random walks whose steps belong to the domain of
attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the
convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$,
equipped with the Hausdorff distance, towards the convex hull spanned by a path
of the limit stable L\'{e}vy process. As an application, we establish
convergence of (expected) intrinsic volumes under some mild moment/structure
assumptions posed on the random walk.
