Asymptotic variance of the self-intersections of stable random walks using Darboux–Wiener theory

We present a Darboux–Wiener type lemma as a powerful alternative to the classical Tauberian theorem when monotonicity is not known a priori. We apply it to obtain the exact asymptotics of the variance of the self-intersections of a one-dimensional stable random walk. Finally we prove a functional central limit theorem for stable random walk in random scenery conjectured in [1].