The Annals of Probability

Variations and estimators for self-similarity parameters via Malliavin calculus

Ciprian A. Tudor and Frederi G. Viens

Full-text: Open access

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∈(1/2, 2/3).

Article information

Source
Ann. Probab., Volume 37, Number 6 (2009), 2093-2134.

Dates
First available in Project Euclid: 16 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1258380783

Digital Object Identifier
doi:10.1214/09-AOP459

Mathematical Reviews number (MathSciNet)
MR2573552

Zentralblatt MATH identifier
1196.60036

Subjects
Primary: 60F05: Central limit and other weak theorems 60H05: Stochastic integrals
Secondary: 60G18: Self-similar processes 62F12: Asymptotic properties of estimators

Keywords
Multiple stochastic integral Hermite process fractional Brownian motion Rosenblatt process Malliavin calculus noncentral limit theorem quadratic variation Hurst parameter self-similarity statistical estimation

Citation

Tudor, Ciprian A.; Viens, Frederi G. Variations and estimators for self-similarity parameters via Malliavin calculus. Ann. Probab. 37 (2009), no. 6, 2093--2134. doi:10.1214/09-AOP459. https://projecteuclid.org/euclid.aop/1258380783


Export citation

References

  • [1] Beran, J. (1994). Statistics for Long-memory Processes. Monographs on Statistics and Applied Probability 61. Chapman and Hall, London.
  • [2] Breton, J.-C. and Nourdin, I. (2008). Error bounds on the nonnormal approximation of Hermite power variations of fractional Brownian motion. Electron. Comm. Probab. 13 482–493.
  • [3] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
  • [4] Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 199–227.
  • [5] Dobrushin, R. L. and Major, P. (1979). Noncentral limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
  • [6] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • [7] Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton Univ. Press, Princeton, NJ.
  • [8] Guyon, X. and León, J. (1989). Convergence en loi des H-variations d’un processus gaussien stationnaire sur R. Ann. Inst. H. Poincaré Probab. Statist. 25 265–282.
  • [9] Hariz, S. B. (2002). Limit theorems for the nonlinear functional of stationary Gaussian processes. J. Multivariate Anal. 80 191–216.
  • [10] Hu, Y. and Nualart, D. (2005). Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33 948–983.
  • [11] Lang, G. and Istas, J. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407–436.
  • [12] Mandelbrot, B. (1963). The variation of certain speculative prices. J. Bus. Econom. Statist. 36 392–417.
  • [13] McLeod, A. I. and Kipel, K. W. (1978). Preservation of the rescaled adjusted range: A reaassement of the Hurst exponent. Water Resourc. Res. 14 491–508.
  • [14] León, J. and Ludeña, C. (2007). Limits for weighted p-variations and likewise functionals of fractional diffusions with drift. Stochastic Process. Appl. 117 271–296.
  • [15] Marcus, M. B. and Rosen, J. (2007). Nonnormal CLTs for functions of the increments of Gaussian processes with conve increment’s variance. Preprint.
  • [16] Nourdin, I. (2008). Asymptotic behavior of certain weighted quadratic variation and cubic varitions of fractional Brownian motion. Ann. Probab. 36 2159–2175.
  • [17] Nourdin, I. and Nualart, D. (2007). Central limit theorems for multiple Skorohod integrals. Preprint.
  • [18] Nourdin, I. and Peccati, G. (2008). Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 1229–1256.
  • [19] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields. 145 75–118.
  • [20] Nourdin, I. and Réveillac, G. (2009). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [21] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [22] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.
  • [23] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
  • [24] Peccati, G. and Tudor, C. A. (2004). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247–262. Springer, Berlin.
  • [25] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Variables. Chapman and Hall, London.
  • [26] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122–2159.
  • [27] Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [28] Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
  • [29] Tudor, C. A. (2008). Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12 230–257.
  • [30] Üstünel, A. S. (1995). An Introduction to Analysis on Wiener Space. Lecture Notes in Math. 1610. Springer, Berlin.
  • [31] Willinger, W., Taqqu, M. and Teverovsky, V. (1999). Long range dependence and stock returns. Finance Stoch. 3 1–13.
  • [32] Willinger, W., Taqqu, M., Leland, W. E. and Wilson, D. V. (1995). Self-similarity in high speed packet traffic: Analysis and modelisation of ethernet traffic measurements. Statist. Sci. 10 67–85.