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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\al-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a L\évy-driven Rosenblatt random variable when the Hurst parameter satisfies H(1/2,1)H\in (1/2,1) and \al(1H)<1/2\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}.

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