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∈(1/2, 2/3).
"Variations and estimators for self-similarity parameters via Malliavin calculus." Ann. Probab. 37 (6) 2093 - 2134, November 2009. https://doi.org/10.1214/09-AOP459