Tuesday, September 29, 2009
03:30 PM in HAAS 111
Alexandra Chronopoulou
Department of Statistics, Purdue University

Variations and Hurst Index Estimation for non-Gaussian Hermite processes

Abstract

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a class of non-Gaussian self-similar processes, the Hermite processes (Z(q,H), q > 1). The process Z(q,H) has stationary, H-self-similar increments that exhibit long-memory, identical to that of the fractional Brownian motion (fBm). For q = 1, Z(1,H), is fBm, which is Gaussian; for q = 2, Z(2,H), is the Rosenblatt process, which lives in the second Wiener chaos; for any q > Z(q,H), is a process in the qth Wiener chaos. 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 to other Hermite processes of different orders and with different Hurst parameters. We also study the behavior of the variations of the Roseblatt process using longer filters. We apply our results to construct a strongly consistent estimator for the self-similarity parameter H from discrete observations of the process. The asymptotic distribution of the estimator depends explicitly on the order and the length of the filter. We compare the numerical values of the asymptotic variances for various choices of filters, including finite-difference and wavelet-based filters. This is joint work with Ciprian Tudor (Sorbonne I) and Frederi Viens (Purdue University).