×

Remarks on confidence intervals for self-similarity parameter of a subfractional Brownian motion. (English) Zbl 1235.62119

Summary: We first present two convergence results for the second-order quadratic variations of asubfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self-similarity parameter associated with one-dimensional subfractional Brownian motions.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F25 Parametric tolerance and confidence regions
60G22 Fractional processes, including fractional Brownian motion
60F05 Central limit and other weak theorems
60G18 Self-similar stochastic processes

Keywords:

convergence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Alòs, O. Mazet, and D. Nualart, “Stochastic calculus with respect to Gaussian processes,” The Annals of Probability, vol. 29, no. 2, pp. 766-801, 2001. · Zbl 1015.60047 · doi:10.1214/aop/1008956692
[2] D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer, New York, NY, USA, 2nd edition, 2006. · Zbl 1099.60003
[3] T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, “Sub-fractional Brownian motion and its relation to occupation times,” Statistics & Probability Letters, vol. 69, no. 4, pp. 405-419, 2004. · Zbl 1076.60027 · doi:10.1016/j.spl.2004.06.035
[4] T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, “Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems,” Electronic Communications in Probability, vol. 12, pp. 161-172, 2007. · Zbl 1128.60025
[5] J. Liu and L. Yan, “Remarks on asymptotic behavior of weighted quadratic variation of subfractional Brownian motion,” Journal of the Korean Statistical Society. In press. · Zbl 1296.60143 · doi:10.1016/j.jkss.2011.08.002
[6] J. Liu and L. Yan, “A note on some approximation schemes associated to stochastic differential equations driven by sub-fractional Brownian motion,” Preprint.
[7] J. Liu, “Variations and estimators for self-similarity parameter of sub-fractional Brownian motion via Malliavin calculus,” Preprint. · Zbl 1369.60023
[8] C. Tudor, “Some properties of the sub-fractional Brownian motion,” Stochastics, vol. 79, no. 5, pp. 431-448, 2007. · Zbl 1124.60038 · doi:10.1080/17442500601100331
[9] C. Tudor, “Multiple sub-fractional integrals and some approximations,” Applicable Analysis, vol. 87, no. 3, pp. 311-323, 2008. · Zbl 1140.60030 · doi:10.1080/00036810801927971
[10] C. Tudor, “Inner product spaces of integrands associated to subfractional Brownian motion,” Statistics & Probability Letters, vol. 78, no. 14, pp. 2201-2209, 2008. · Zbl 1283.60082 · doi:10.1016/j.spl.2008.01.087
[11] C. Tudor, “Some aspects of stochastic calculus for the sub-fractional Brownian motion,” Analele Universităţii Bucure\csti. Matematică, vol. 57, no. 2, pp. 199-230, 2008. · Zbl 1174.60024
[12] C. Tudor, “On the Wiener integral with respect to a sub-fractional Brownian motion on an interval,” Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 456-468, 2009. · Zbl 1154.60041 · doi:10.1016/j.jmaa.2008.10.041
[13] L. Yan and G. Shen, “On the collision local time of sub-fractional Brownian motions,” Statistics & Probability Letters, vol. 80, no. 5-6, pp. 296-308, 2010. · Zbl 1185.60040 · doi:10.1016/j.spl.2009.11.003
[14] L. Yan, G. Shen, and K. He, “Itô’s formula for a sub-fractional Brownian motion,” Communications on Stochastic Analysis, vol. 5, no. 1, pp. 135-159, 2011. · Zbl 1331.60068
[15] A. Chronopoulou, C. A. Tudor, and F. G. Viens, “Application of Malliavin calculus to long-memory parameter estimation for non-Gaussian processes,” Comptes Rendus Mathématique, vol. 347, no. 11-12, pp. 663-666, 2009. · Zbl 1163.62065 · doi:10.1016/j.crma.2009.03.026
[16] A. Chronopoulou and F. Viens, “Hurst estimation for self-similar processes with long memory,” Interdisciplinary Mathematical Sciences, vol. 8, pp. 91-117, 2009. · Zbl 1196.62112
[17] C. A. Tudor and F. Viens, “Variations of the fractional Brownian motion via Malliavin calculus,” submitted to Australian Journal of Mathematics.
[18] C. A. Tudor and F. G. Viens, “Variations and estimators for self-similarity parameters via Malliavin calculus,” The Annals of Probability, vol. 37, no. 6, pp. 2093-2134, 2009. · Zbl 1196.60036 · doi:10.1214/09-AOP459
[19] J. C. Breton, I. Nourdin, and G. Peccati, “Exact confidence intervals for the Hurst parameter of a fractional Brownian motion,” Electronic Journal of Statistics, vol. 3, pp. 416-425, 2009. · Zbl 1326.62065 · doi:10.1214/09-EJS366
[20] I. Nourdin and F. G. Viens, “Density formula and concentration inequalities with Malliavin calculus,” Electronic Journal of Probability, vol. 14, pp. 2287-2309, 2009. · Zbl 1192.60066
[21] I. Nourdin and G. Peccati, “Stein’s method on Wiener chaos,” Probability Theory and Related Fields, vol. 145, no. 1-2, pp. 75-118, 2009. · Zbl 1175.60053 · doi:10.1007/s00440-008-0162-x
[22] A. Begyn, “Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes,” Bernoulli, vol. 13, no. 3, pp. 712-753, 2007. · Zbl 1143.60030 · doi:10.3150/07-BEJ5112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.