Asymptotic theory for quadratic variation of harmonizable fractional stable processes

Abstract

In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional $\al$-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a L\'evy-driven Rosenblatt random variable when the Hurst parameter satisfies $H\in (1/2,1)$ and $\al(1-H)<1/2$. This result complements the asymptotic theory for fractional stable processes investigated in e.g. \cite{BHP19,BLP17,BP17,BPT20,LP18,MOP20}.