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 α \alpha -stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a Lévy-driven Rosenblatt random variable when the Hurst parameter satisfies H ∈ ( 1 / 2 , 1 ) H\in (1/2,1) and α ( 1 − H ) > 1 / 2 \alpha (1-H)>1/2 . This result complements the asymptotic theory for fractional stable processes.