Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes
نویسندگان
چکیده
We consider the class of all the Hermite processes (Z t )t2[0;1] of order q 2 N and with Hurst index H 2 ( 1 2 ; 1). The process Z (q;H) is H-self-similar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For q = 1, Z is fBm, which is Gaussian; for q = 2, Z is the Rosenblatt process, which lives in the second Wiener chaos; for any q > 2, Z is a process in the qth Wiener chaos. We study the variations of Z for any q, by using multiple Wiener -Itô stochastic integrals and Malliavin calculus. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of their variations give rise other Hermite processes of di¤erent orders and with di¤erent Hurst parameters. We apply our results to construct a strongly consistent estimator for the self-similarity parameter H from discrete observations of Z; the asymptotics of this estimator, after appropriate normalization, are proved to be distributed like a Rosenblatt random variable (value at time 1 of a Rosenblatt process).with self-similarity parameter 1 + 2(H 1)=q. 2000 AMS Classi cation Numbers: Primary: 60G18; Secondary 60F05, 60H05, 62F12.
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