Asymptotic behavior for bi-fractional regression models via Malliavin calculus

Abstract

Let B^H1,K1 and B^H2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence

S_n:={\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}K(n^{\alpha }{B}_i^{H_{\mathrm{1}},K_{\mathrm{1}}})}(B_{i+\mathrm{1}}^{H_{\mathrm{2}},K_{\mathrm{2}}}-B_i^{H_{\mathrm{2}},K_{\mathrm{2}}}),

where K is a standard Gaussian kernel function and the bandwidth parameter αsatisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion B^H1,K1. We also give the stable convergence of the sequence S_n by using the techniques of the Malliavin calculus.

Key words： Bi-fractional Brownian motion (bi-fBm); Malliavin calculus; regression model

References

1 Alós E, Mazet O, Nualart D. Stochastic calculus with respect to Gaussian processes. Ann Probab, 2001, 29: 766-801

2 Biagini F, Hu Y, Oksendal B, Zhang T. Stochastic Calculus for Fractional Brownian Motion and Applications. London: Springer-Verlag, 2008

3 Bourguin S, Tudor C A. Asymptotic theory for fractional regression models via Malliavin calculus. J Theoret Probab, 2012, 25: 536-564

4 Chen Z. Polar functions of multiparameter bifractional Brownian Sheets. Acta Math Appl Sin Engl Ser, 2009, 25: 255-572

5 Chen Z, Li H. Polar sets of multiparameter bifractional Brownian sheets. Acta Math Sci Ser B, 2010, 30: 857-872

6 Duncan T E, Hu Y, Duncan B P. Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J Control Optim, 2000, 38: 582-612

7 Eddahbi M, Lacayo R, Sole J L, Tudor C A, Vives J. Regularity of the local time for the d-dimensional fractional Brownian motion with N-parameters. Stoch Anal Appl, 2001, 23: 383-400

8 Es-sebaiy K, Tudor C A. Multidimensional bifractional Brownian motion: Itô and Tanaka formulas. Stoch Dyn, 2007, 7: 366-388

9 Geman D, Horowitz J. Occupation densities. Ann Probab, 1980, 8: 1-67

10 Houdré C, Villa J. An example of infinite dimensional quasi-helix. Stoch Models, 2003, 336: 195-201

11 Jiang Y, Wang Y. Self-intersection local times and collision local times of bifractional Brownian motions. Sci China Ser A, 2009, 52: 1905-1919

12 Karlsen H A, Mykklebust T, Tjostheim D. Nonparametric estimation in a nonlinear cointegrated model. Ann Statistics, 2007, 35: 252-299

13 Karlsen H A, Tjostheim D. Nonparametric estimation in null recurrent time series. Ann Statistics, 2001, 29: 372-416

14 Kruk I, Russo F, Tudor C A. Wiener integrals, Malliavin calculus and covariance measure structure. J Funct Anal, 2007, 249: 92-142

15 Luan N. Hausdorff measures of the image, graph and level set of bifractional Brownian motion. Sci China Math, 2010, 53: 2973-2992

16 Nualart D. Malliavin Calculus and Related Topics. New York: Springer, 2006

17 Russo F, Tudor C A. On the bifractional Brownian motion. Stochastic Process Appl, 2006, 5: 830-856

18 Schienle M. Nonparametric Nonstationary Regression. Ph D Thesis. Mannheim: University of Mannheim, 2008

19 Shen G, Yan L. Smoothness for the collision local times of bifractional Brownian motions. Sci China Math, 2011, 54: 1859-1873

20 Tudor C A, Xiao Y. Sample path properties of bifractional brownian motion. Bernoulli, 2007, 13: 1023-1052

21 Wang Q, Phillips P C B. Asymptotic theory for the local time density estimation and nonparametric cointegrated regression. Econometric Theory, 2009, 25: 710-738

22 Wang Q, Phillips P C B. Structural nonparametric cointegrating regression. Econometrica, 2009, 77: 1901-1948

23 Wang W. On p-variation of bifractional Brownian motion. Appl Math J Chinese Univ, 2011, 26: 127-141

24 Yan L, Gao B, Liu J. Quadratic covariation and the Bouleau-Yor identity for a bi-fractional Brownian motion. Preprint, 2012

25 Yan L, Liu J, Chen C. On the collision local time of bifractional Brownian motions. Stoch Dyn, 2009, 9: 479-491

26 Yan L, Xiang J. The generalized quadratic covariation for a bi-fBm. J Nat Sci Heilongjiang Univ, 2011, 28: 20-37