Variations and estimators for self-similarity parameters via Malliavin calculus

Annals of Probability

Volumn 37, Issue 6, 2009, Pages 2093-2134

  Tudor, Ciprian A ^a   Viens, Frederi G ^b  

^a University Paris I Sorbonne Panthéon   (France)

^b PURDUE UNIVERSITY   (United States)

Author keywords

Fractional Brownian motion; Hermite process; Hurst parameter; Malliavin calculus; Multiple stochastic integral; Noncentral limit theorem; Quadratic variation; Rosenblatt process; Self similarity; Statistical estimation

Indexed keywords

EID: 77049086317     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOP459     Document Type: Article

References (32)

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.PrincetonUniv.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.