Wavelet analysis of the multivariate fractional Brownian motion

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function $\sign(t) |t|^\alpha$. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation.

[1]  G. Shilov,et al.  DEFINITION AND SIMPLEST PROPERTIES OF GENERALIZED FUNCTIONS , 1964 .

[2]  B. V. Bahr,et al.  Inequalities for the $r$th Absolute Moment of a Sum of Random Variables, $1 \leqq r \leqq 2$ , 1965 .

[3]  G. Shilov,et al.  Generalized Functions, Volume 1: Properties and Operations , 1967 .

[4]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[5]  Patrick Flandrin,et al.  On the spectrum of fractional Brownian motions , 1989, IEEE Trans. Inf. Theory.

[6]  Gregory W. Wornell,et al.  A Karhunen-Loève-like expansion for 1/f processes via wavelets , 1990, IEEE Trans. Inf. Theory.

[7]  Patrick Flandrin,et al.  Wavelet analysis and synthesis of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[8]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[9]  R. Peltier,et al.  Multifractional Brownian Motion : Definition and Preliminary Results , 1995 .

[10]  E. Masry,et al.  On the Spectral Density of the Wavelet Transform of Fractional Brownian Motion , 1999 .

[11]  A. T. A. Wood,et al.  Simulation of stationary Gaussian vector fields , 1999, Stat. Comput..

[12]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[13]  Monique Pontier,et al.  Drap brownien fractionnaire , 1999 .

[14]  Jean-Marc Bardet,et al.  Wavelet Estimator of Long-Range Dependent Processes , 2000 .

[15]  J. Coeurjolly,et al.  Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths , 2001 .

[16]  A fractional multivariate long memory model for the US and the Canadian real output , 2003 .

[17]  E. Bullmore,et al.  A Resilient, Low-Frequency, Small-World Human Brain Functional Network with Highly Connected Association Cortical Hubs , 2006, The Journal of Neuroscience.

[18]  Eric Moulines,et al.  Estimators of Long-Memory: Fourier versus Wavelets , 2008, 0801.4329.

[19]  E. Bullmore,et al.  Fractal connectivity of long-memory networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Donatas Surgailis,et al.  Covariance function of vector self-similar processes , 2009, 0906.4541.

[21]  A. Carbone,et al.  Cross-correlation of long-range correlated series , 2008, 0804.2064.

[22]  Anne Philippe,et al.  Basic properties of the Multivariate Fractional Brownian Motion , 2010, 1007.0828.

[23]  G. Didier,et al.  Integral representations and properties of operator fractional Brownian motions , 2011, 1102.1822.

[24]  M. B. Rajarshi.  Statistical Inference for Stochastic Processes , 2011, International Encyclopedia of Statistical Science.

[25]  Pierre-Olivier Amblard,et al.  Identification of the Multivariate Fractional Brownian Motion , 2011, IEEE Transactions on Signal Processing.

[26]  곽순섭,et al.  Generalized Functions , 2006, Theoretical and Mathematical Physics.