Variations and estimators for the selfsimilarity order through Malliavin calculus

Abstract

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in (1/2,2/3)$.

References in corpus (4)

• Central limit theorems for sequences of multiple stochastic integrals
• Analysis of the Rosenblatt process
• Renormalized self-intersection local time for fractional Brownian motion
• Central limit theorems for multiple Skorohod integrals

Cited by in corpus (4)

• A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter
• Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes
• Discretizing the fractional Levy area
• Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion