Central and non-central limit theorems for weighted power variations of fractional brownian motion

Ivan Nourdin; David Nualart; Ciprian A. Tudor

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 1055-1079
  • ISSN: 0246-0203

Abstract

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In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q≥2 of the fractional brownian motion with Hurst parameter H∈(0, 1), where q is an integer. The central limit holds for 1/2q<H≤1−1/2q, the limit being a conditionally gaussian distribution. If H<1/2q we show the convergence in L2 to a limit which only depends on the fractional brownian motion, and if H>1−1/2q we show the convergence in L2 to a stochastic integral with respect to the Hermite process of order q.

How to cite

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Nourdin, Ivan, Nualart, David, and Tudor, Ciprian A.. "Central and non-central limit theorems for weighted power variations of fractional brownian motion." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 1055-1079. <http://eudml.org/doc/241766>.

@article{Nourdin2010,
abstract = {In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q≥2 of the fractional brownian motion with Hurst parameter H∈(0, 1), where q is an integer. The central limit holds for 1/2q&lt;H≤1−1/2q, the limit being a conditionally gaussian distribution. If H&lt;1/2q we show the convergence in L2 to a limit which only depends on the fractional brownian motion, and if H&gt;1−1/2q we show the convergence in L2 to a stochastic integral with respect to the Hermite process of order q.},
author = {Nourdin, Ivan, Nualart, David, Tudor, Ciprian A.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {fractional brownian motion; central limit theorem; non-central limit theorem; Hermite process; fractional Brownian motion},
language = {eng},
number = {4},
pages = {1055-1079},
publisher = {Gauthier-Villars},
title = {Central and non-central limit theorems for weighted power variations of fractional brownian motion},
url = {http://eudml.org/doc/241766},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Nourdin, Ivan
AU - Nualart, David
AU - Tudor, Ciprian A.
TI - Central and non-central limit theorems for weighted power variations of fractional brownian motion
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 1055
EP - 1079
AB - In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q≥2 of the fractional brownian motion with Hurst parameter H∈(0, 1), where q is an integer. The central limit holds for 1/2q&lt;H≤1−1/2q, the limit being a conditionally gaussian distribution. If H&lt;1/2q we show the convergence in L2 to a limit which only depends on the fractional brownian motion, and if H&gt;1−1/2q we show the convergence in L2 to a stochastic integral with respect to the Hermite process of order q.
LA - eng
KW - fractional brownian motion; central limit theorem; non-central limit theorem; Hermite process; fractional Brownian motion
UR - http://eudml.org/doc/241766
ER -

References

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  1. [1] P. Breuer and P. Major. Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 (1983) 425–441. Zbl0518.60023MR716933
  2. [2] K. Burdzy and J. Swanson. A change of variable formula with Itô correction term. Preprint, 2008. Available at arXiv:0802.3356. Zbl1204.60044MR2722787
  3. [3] P. Cheridito and D. Nualart. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0, 1/2). Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 1049–1081. Zbl1083.60027MR2172209
  4. [4] J. M. Corcuera, D. Nualart and J. H. C. Woerner. Power variation of some integral fractional processes. Bernoulli 12 (2006) 713–735. Zbl1130.60058MR2248234
  5. [5] R. L. Dobrushin and P. Major. Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979) 27–52. Zbl0397.60034MR550122
  6. [6] L. Giraitis and D. Surgailis. CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 (1985) 191–212. Zbl0575.60024MR799146
  7. [7] M. Gradinaru and I. Nourdin. Milstein’s type scheme for fractional SDEs. Ann. Inst. H. Poincaré Probab. Statist. (2007). To appear. Available at arXiv:math/0702317. Zbl1197.60070MR2572165
  8. [8] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois. m-order integrals and Itô’s formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 781–806. Zbl1083.60045MR2144234
  9. [9] M. Gradinaru, F. Russo and P. Vallois. Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index H≥¼. Ann. Probab. 31 (2001) 1772–1820. Zbl1059.60067MR2016600
  10. [10] J. Jacod. Limit of random measures associated with the increments of a Brownian semimartingale. Preprint. Univ. Paris VI (revised version, unpublished work), 1994. 
  11. [11] J. León and C. Ludeña. Limits for weighted p-variations and likewise functionals of fractional diffusions with drift. Stochastic Proc. Appl. 117 (2006) 271–296. Zbl1110.60023MR2290877
  12. [12] A. Neuenkirch and I. Nourdin. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J. Theoret. Probab. 20 (2007) 871–899. Zbl1141.60043MR2359060
  13. [13] I. Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. In Séminaire de Probabilités XLI 181–197. Springer, Berlin, 2008. Zbl1148.60034MR2483731
  14. [14] I. Nourdin. Asymptotic behavior of some weighted quadratic and cubic variations of the fractional Brownian motion. Ann. Probab. 36 (2008) 2159–2175. Zbl1155.60010MR2478679
  15. [15] I. Nourdin and D. Nualart. Central limit theorems for multiple Skorohod integrals. J. Theoret. Probab. (2008). In revision. Available at arXiv:0707.3448. Zbl1202.60038MR2591903
  16. [16] I. Nourdin and G. Peccati. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2007) 1229–1256 (electronic). Zbl1193.60028MR2430706
  17. [17] I. Nourdin and A. Réveillac. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4. Ann. Probab. (2008). To appear. Available at arXiv:0802.3307. Zbl1200.60023MR2573556
  18. [18] D. Nualart. Malliavin Calculus and Related Topics, 2nd edition. Springer, New York, 2005. Zbl0837.60050MR1344217
  19. [19] D. Nualart. Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 (2003) 3–39. Zbl1063.60080MR2037156
  20. [20] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII 247–262. Lecture Notes in Math. 1857. Springer, Berlin, 2005. Zbl1063.60027MR2126978
  21. [21] M. Taqqu. Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1979) 53–83. Zbl0397.60028MR550123
  22. [22] C. A. Tudor. Analysis of the Rosenblatt process. ESAIM Probab. Statist. 12 230–257. Zbl1187.60028MR2374640

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