Variations and hurst index estimation for a rosenblatt process using longer filters

Alexandra Chronopoulou, Frederi G. Viens, Ciprian A. Tudor

Research output: Contribution to journalArticlepeer-review


The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called “non-central limit theorems”. It shares the same covariance as fractional Brownian motion. We study the asymptotic distribution of the quadratic variations of the Rosenblatt process based on long filters, including filters based on high-order finite-difference and wavelet-based schemes. We find exact formulas for the limiting distributions, which we then use to devise strongly consistent estimators of the self-similarity parameter H. Unlike the case of fractional Brownian motion, no matter now high the filter orders are, the estimators are never asymptotically normal, converging instead in the mean square to the observed value of the Rosenblatt process at time 1.

Original languageEnglish (US)
Pages (from-to)1393-1435
Number of pages43
JournalElectronic Journal of Statistics
StatePublished - 2009
Externally publishedYes


  • Fractionalbrownian motion
  • Malliavin calculus
  • Multiple wiener integral
  • Non-central limit theorem
  • Parameter estimation
  • Quadratic variation
  • Rosenblatt process
  • Self-similarity

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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