Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process

The purpose of this paper is the estimation of the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  C. Tudor.  Analysis of the Rosenblatt process , 2006, math/0606602.

[3]  M. Taqqu.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[4]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[5]  Murad S. Taqqu,et al.  Theory and applications of long-range dependence , 2003 .

[6]  Giovanni Peccati,et al.  Central limit theorems for sequences of multiple stochastic integrals , 2005 .

[7]  N. T. Kottegoda,et al.  Stochastic Modelling of Riverflow Time Series , 1977 .

[8]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[9]  F. Viens,et al.  Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes , 2008, 0807.1208.

[10]  M. Taqqu,et al.  Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series , 1986 .

[11]  M. Taqqu,et al.  Whittle estimator for finite-variance non-Gaussian time series with long memory , 1999 .

[12]  David Nualart,et al.  Central limit theorems for multiple stochastic integrals and Malliavin calculus , 2007 .

[13]  C. Tudor.  Analysis of Variations for Self-similar Processes , 2013 .

[14]  P. Major,et al.  Central limit theorems for non-linear functionals of Gaussian fields , 1983 .

[15]  M. Taqqu.  A representation for self-similar processes , 1978 .

[16]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[17]  M. Taqqu,et al.  Regularization and integral representations of Hermite processes , 2010 .

[18]  Murad S. Taqqu,et al.  Multiple stochastic integrals with dependent integrators , 1987 .

[19]  Patrice Abry,et al.  Wavelet-based synthesis of the Rosenblatt process , 2006, Signal Process..

[20]  F. Viens,et al.  Variations and estimators for the selfsimilarity order through Malliavin calculus , 2007, 0709.3896.

[21]  Jean-Marc Bardet,et al.  A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter , 2008, 0811.2664.

[22]  Murad S. Taqqu,et al.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[23]  Liudas Giraitis,et al.  Nonstationarity-Extended Local Whittle Estimation , 2006 .

[24]  D. Surgailis,et al.  A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate , 1990 .

[25]  Y. Sinai.  Self-Similar Probability Distributions , 1976 .

[26]  M. Taqqu,et al.  Properties and numerical evaluation of the Rosenblatt distribution , 2013, 1307.5990.

[27]  G. Peccati,et al.  Normal Approximations with Malliavin Calculus: From Stein's Method to Universality , 2012 .

[28]  R. Dobrushin,et al.  Non-central limit theorems for non-linear functional of Gaussian fields , 1979 .

[29]  F. Viens,et al.  Variations and estimators for selfsimilarity parameters via Malliavin calculus , 2009 .

[30]  G. Peccati,et al.  Gaussian Limits for Vector-valued Multiple Stochastic Integrals , 2005 .

[31]  D. Nualart,et al.  Central and non-central limit theorems for weighted power variations of fractional brownian motion , 2007, 0710.5639.

[32]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.