## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 895-928.

### Wavelet estimation for operator fractional Brownian motion

#### Abstract

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. Instead we consider the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the eigenvectors under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 895-928.

Dates
Revised: September 2015
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980882

Digital Object Identifier
doi:10.3150/15-BEJ790

Mathematical Reviews number (MathSciNet)
MR3706780

Zentralblatt MATH identifier
06778351

#### Citation

Abry, Patrice; Didier, Gustavo. Wavelet estimation for operator fractional Brownian motion. Bernoulli 24 (2018), no. 2, 895--928. doi:10.3150/15-BEJ790. https://projecteuclid.org/euclid.bj/1505980882

#### References

• [1] Abry, P., Baraniuk, R., Flandrin, P., Riedi, R. and Veitch, D. (2002). Multiscale network traffic analysis, modeling, and inference using wavelets, multifractals, and cascades. IEEE Signal Process. Mag. 3 (19) 28–46.
• [2] Abry, P. and Didier, G. (2017). Supplement to “Wavelet estimation for operator fractional Brownian motion.” DOI:10.3150/15-BEJ790SUPP.
• [3] Achard, S. and Gannaz, I. (2016). Multivariate wavelet Whittle estimation in long-range dependence. J. Time Series Anal. 37 (4) 476–512.
• [4] Amblard, P., Coeurjolly, J.-F., Lavancier, F. and Philippe, A. (2012). Basic properties of the multivariate fractional Brownian motion. Bulletin de la Société Mathématique de France, Séminaires et Congrés 28 65–87.
• [5] Amblard, P.-O. and Coeurjolly, J.-F. (2011). Identification of the multivariate fractional Brownian motion. IEEE Trans. Signal Process. 59 (11) 5152–5168.
• [6] Bardet, J.-M. (2000). Testing for the presence of self-similarity of Gaussian time series having stationary increments. J. Time Series Anal. 21 (5) 497–515.
• [7] Bardet, J.-M. (2002). Statistical study of the wavelet analysis of fractional Brownian motion. IEEE Trans. Inform. Theory 48 (4) 991–999.
• [8] Becker-Kern, P. and Pap, G. (2008). Parameter estimation of selfsimilarity exponents. J. Multivariate Anal. 99 117–140.
• [9] Chung, C.-F. (2002). Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes. Econometric Theory 18 51–78.
• [10] Coeurjolly, J.-F., Amblard, P.-O. and Achard, S. (2013). Wavelet analysis of the multivariate fractional Brownian motion. ESAIM Probab. Stat. 17 592–604.
• [11] Cohen, S., Meerschaert, M.M. and Rosiński, J. (2010). Modeling and simulation with operator scaling. Stochastic Process. Appl. 120 2390–2411.
• [12] Cramér, H. and Leadbetter, M.R. (1967). Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Mineola, NY: Dover Publications.
• [13] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, PA: SIAM.
• [14] Davidson, J. and De Jong, R. (2000). The Functional Central Limit Theorem and weak convergence to stochastic integrals. Econometric Theory 16 643–666.
• [15] Davidson, J. and Hashimzade, N. (2008). Alternative frequency and time domain versions of fractional Brownian motion. Econometric Theory 24 256–293.
• [16] Delbeke, L. and Abry, P. (2000). Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion. Stochastic Process. Appl. 86 (2) 177–182.
• [17] Delgado, R. (2007). A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic. Stochastic Process. Appl. 117 188–201.
• [18] Didier, G., Helgason, H. and Abry, P. (2015). Demixing multivariate-operator self-similar processes. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 1–5. Brisbane, Australia.
• [19] Didier, G. and Pipiras, V. (2011). Integral representations and properties of operator fractional Brownian motions. Bernoulli 17 1–33.
• [20] Didier, G. and Pipiras, V. (2012). Exponents, symmetry groups and classification of operator fractional Brownian motions. J. Theoret. Probab. 25 353–395.
• [21] Dolado, J.J. and Marmol, F. (2004). Asymptotic inference results for multivariate long-memory processes. Econom. J. 7 168–190.
• [22] Flandrin, P. (1992). Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inform. Theory 38 910–917.
• [23] Fox, R. and Taqqu, M.S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517–532.
• [24] Frisch, U. (1995). Turbulence. Cambridge: Cambridge Univ. Press. The legacy of A.N. Kolmogorov.
• [25] Helgason, H., Pipiras, V. and Abry, P. (2011). Fast and exact synthesis of stationary multivariate Gaussian time series using circulant embedding. Signal Process. 91 1123–1133.
• [26] Helgason, H., Pipiras, V. and Abry, P. (2011). Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant embedding. Signal Process. 91 1741–1758.
• [27] Hudson, W.N. and Mason, J.D. (1982). Operator-self-similar processes in a finite-dimensional space. Trans. Amer. Math. Soc. 273 281–297.
• [28] Ivanov, P., Amaral, L.N., Goldberger, A., Havlin, S., Rosenblum, M., Struzik, Z. and Stanley, H. (1999). Multifractality in human heartbeat dynamics. Nature 399 461–465.
• [29] Kechagias, S. and Pipiras, V. (2015). Definitions and representations of multivariate long-range dependent time series. J. Time Series Anal. 36 1–25.
• [30] Konstantopoulos, T. and Lin, S.-J. (1996). Fractional Brownian approximations of queueing networks. In Stochastic Networks (New York, 1995). Lecture Notes in Statist. 117 257–273. New York: Springer.
• [31] Laha, R.G. and Rohatgi, V.K. (1982). Operator self-similar stochastic processes in $\textbf{R}_{d}$. Stochastic Process. Appl. 12 73–84.
• [32] Maejima, M. and Mason, J.D. (1994). Operator-self-similar stable processes. Stochastic Process. Appl. 54 139–163.
• [33] Majewski, K. (2003). Large deviations for multi-dimensional reflected fractional Brownian motion. Stoch. Stoch. Rep. 75 233–257.
• [34] Majewski, K. (2005). Fractional Brownian heavy traffic approximations of multiclass feedforward queueing networks. Queueing Syst. 50 199–230.
• [35] Mallat, S. (1998). A Wavelet Tour of Signal Processing. San Diego, CA: Academic Press.
• [36] Marinucci, D. and Robinson, P.M. (2000). Weak convergence of multivariate fractional processes. Stochastic Process. Appl. 86 103–120.
• [37] Meerschaert, M. and Scheffler, H.-P. (2003). Portfolio modeling with heavy-tailed random vectors. In Handbook of Heavy-Tailed Distributions in Finance (S.T. Rachev, ed.) 595–640. Amsterdam: Elsevier Science B.V.
• [38] Meerschaert, M.M. and Scheffler, H.-P. (1999). Moment estimator for random vectors with heavy tails. J. Multivariate Anal. 71 145–159.
• [39] Moulines, E., Roueff, F. and Taqqu, M.S. (2007). Central limit theorem for the log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context. Fractals 15 301–313.
• [40] Moulines, E., Roueff, F. and Taqqu, M.S. (2007). On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter. J. Time Series Anal. 28 155–187.
• [41] Moulines, E., Roueff, F. and Taqqu, M.S. (2008). A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series. Ann. Statist. 36 1925–1956.
• [42] Nielsen, F.S. (2011). Local Whittle estimation of multi-variate fractionally integrated processes. J. Time Series Anal. 32 (3) 317–335.
• [43] Robinson, P.M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630–1661.
• [44] Robinson, P.M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048–1072.
• [45] Robinson, P.M. (2008). Multiple local Whittle estimation in stationary systems. Ann. Statist. 36 (5) 2508–2530.
• [46] Shimotsu, K. (2007). Gaussian semiparametric estimation of multivariate fractionally integrated processes. J. Econometrics 137 277–310.
• [47] Taqqu, M.S. (2003). Fractional Brownian motion and long-range dependence. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M.S. Taqqu, eds.) 5–38. Boston, MA: Birkhäuser.
• [48] Veitch, D. and Abry, P. (1999). A wavelet-based joint estimator of the parameters of long-range dependence. IEEE Trans. Inform. Theory 45 878–897.
• [49] Veitch, D., Abry, P. and Taqqu, M.S. (2003). On the automatic selection of the onset of scaling. Fractals 11 (4) 377–390.
• [50] Vignat, C. (2012). A generalized Isserlis theorem for location mixtures of Gaussian random vectors. Statist. Probab. Lett. 82 67–71.
• [51] Wendt, H., Scherrer, A., Abry, P. and Achard, S. (2009). Testing fractal connectivity in multivariate long memory processes. In IEEE International Conference on Acoustics, Speech and Signal Processing 2913–2916. Taipei, Taiwan.
• [52] Wornell, G. and Oppenheim, A. (1992). Estimation of fractal signals from noisy measurements using wavelets. IEEE Trans. Signal Process. 40 611–623.

#### Supplemental materials

• Supplement to “Wavelet estimation for operator fractional Brownian motion”. In Section B of the supplementary file Abry and Didier [2], we provide several additional auxiliary results and proofs. In Section C, the performance of the estimators is established under the assumption that only discrete observations are available, instead of a full sample path.