We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law (1 < α < 2). We show that this sequence is not tight in the Skorokhod J₁ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod M₁ topology to an α-stable Lévy motion. This example leads to a weaker notion of weak convergence. As...

We introduce and study a class of random walks defined on the integer lattice ${\mathbb{Z}}^d$-a discrete space and time counterpart of the symmetric α-stable process in ${ℝ}^d$. When 0 < α <2 any coordinate axis in ${\mathbb{Z}}^d$, d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.