Functional CLT for the range of stable random walks

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3\alpha / 2$.