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On operator fractional Lévy motion: integral representations and time-reversibility

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  06 June 2022

B. Cooper Boniece  and
Gustavo Didier
B. Cooper Boniece*
Affiliation:
University of Utah
Gustavo Didier*
Affiliation:
Tulane University
*
*Postal address: 155 South 1400 East, Salt Lake City, UT 84112, USA. Email address: bcboniece@math.utah.edu
**Postal address: 6823 St. Charles Avenue, New Orleans, LA 70118, USA. Email address: gdidier@tulane.edu
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Abstract

In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).

Keywords

Infinite divisibilityLévy processesoperator self-similarity

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G51: Processes with independent increments; L'evy processes 60H05: Stochastic integrals
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 493 - 535
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abry, P. and Didier, G. (2018). Wavelet eigenvalue regression for n-variate operator fractional Brownian motion. J. Multivariate Anal. 168, 75–104.CrossRefGoogle Scholar
Ali, A. B. (2014). Tempered operator stabile Verteilungen. Doctoral Thesis, University of Siegen.Google Scholar
Applebaum, D. (2009). Lévy Processes and Stochastic Calculus. Cambridge University Press.CrossRefGoogle Scholar
Bai, S. and Taqqu, M. S. (2018). How the instability of ranks under long memory affects large-sample inference. Statist. Sci. 33, 96–116.CrossRefGoogle Scholar
Barndorff-Nielsen, O. and Stelzer, R. (2011). Multivariate supOU processes. Ann. Appl. Prob. 21, 140–182.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2008). Time change, volatility, and turbulence. In Mathematical Control Theory and Finance, Springer, Berlin, Heidelberg, pp. 29–53.CrossRefGoogle Scholar
Basse, A. and Pedersen, J. (2009). Lévy driven moving averages and semimartingales. Stoch. Process. Appl. 119, 2970–2991.CrossRefGoogle Scholar
Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97–115.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10, 357–373.CrossRefGoogle Scholar
Bender, C. and Marquardt, T. (2008). Stochastic calculus for convoluted Lévy processes. Bernoulli 14, 499–518.CrossRefGoogle Scholar
Benson, D. A., Meerschaert, M. M., Baeumer, B. and Scheffler, H.-P. (2006). Aquifer operator scaling and the effect on solute mixing and dispersion. Water Resources Res. 42, W01415.CrossRefGoogle Scholar
Boniece, B. C., Didier, G. and Sabzikar, F. (2020). On fractional Lévy processes: tempering, sample path properties and stochastic integration. J. Statist. Phys. 178, 954–985.CrossRefGoogle Scholar
Boniece, B. C., Didier, G., Wendt, H. and Abry, P. (2019). On multivariate non-Gaussian scale invariance: fractional Lévy processes and wavelet estimation. In 2019 27th European Signal Processing Conference (EUSIPCO), Institute of Electrical and Electronics Engineers, Piscataway, NJ, 5 pp.Google Scholar
Boniece, B. C., Wendt, H., Didier, G. and Abry, P. (2019). Wavelet-based detection and estimation of fractional Lévy signals in high dimensions. In 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 574–578.CrossRefGoogle Scholar
Brockwell, P. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica 15, 477–494.Google Scholar
Briody, D. (2011). Big data: harnessing a game-changing asset. White paper. The Economist Intelligence Unit Ltd, London.Google Scholar
Cheng, Q. (1999). On time-reversibility of linear processes. Biometrika 86, 483–486.CrossRefGoogle Scholar
Ciuciu, P. et al. (2012). Scale-free and multifractal properties of fMRI signals during rest and task. Frontiers Physiol. 3, article no. 186.CrossRefGoogle Scholar
Cox, D. R. (1981). Statistical analysis of time series: some recent developments. Scand. J. Statist. 8, 93–115.Google Scholar
Cox, D. R. (1991). Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal. 12, 329–335.CrossRefGoogle Scholar
De Gooijer, J. (2017). Elements of Nonlinear Time Series Analysis and Forecasting. Springer, Cham.CrossRefGoogle Scholar
Didier, G., Meerschaert, M. M. and Pipiras, V. (2018). Domain and range symmetries of operator fractional Brownian fields. Stoch. Process. Appl. 128, 39–78.CrossRefGoogle Scholar
Didier, G. and Pipiras, V. (2011). Integral representations and properties of operator fractional Brownian motions. Bernoulli 17, 1–33.CrossRefGoogle Scholar
Didier, G. and Pipiras, V. (2012). Exponents, symmetry groups and classification of operator fractional Brownian motions. J. Theoret. Prob. 25, 353–395.CrossRefGoogle Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.Google Scholar
Grabchak, M. (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.CrossRefGoogle Scholar
Horn, R. and Johnson, C. (1991). Topics in Matrix Analysis. Cambridge University Press.CrossRefGoogle Scholar
Hudson, W. N. and Mason, J. D. (1982). Operator-self-similar processes in a finite-dimensional space. Trans. Amer. Math. Soc. 273, 281–297.CrossRefGoogle Scholar
Isotta, F. et al. (2014). The climate of daily precipitation in the Alps: development and analysis of a high-resolution grid dataset from pan-Alpine rain-gauge data. Internat. J. Climatol. 34, 1657–1675.CrossRefGoogle Scholar
Jacod, J. and Protter, P. (1988). Time reversal on Lévy processes. Ann. Prob. 16, 620–641.CrossRefGoogle Scholar
Kabluchko, Z. and Stoev, S. (2016). Stochastic integral representations and classification of sum- and max-infinitely divisible processes. Bernoulli 22, 107–142.CrossRefGoogle Scholar
Kallenberg, O. (2006). Foundations of Modern Probability. Springer, Cham.Google Scholar
Kolmogorov, A. N. (1941). Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kremer, D. and Scheffler, H.-P. (2017). Multivariate stochastic integrals with respect to independently scattered random measures on $ \delta$ -rings. Preprint. Available at https://arxiv.org/abs/1711.00890.Google Scholar
Kremer, D. and Scheffler, H.-P. (2019). Operator-stable and operator-self-similar random fields. Stoch. Process. Appl. 129, 4082–4107.CrossRefGoogle Scholar
Kuśmierz, Ł., Chechkin, A., Gudowska-Nowak, E. and Bier, M. (2016). Breaking microscopic reversibility with Lévy flights. Europhys. Lett. 114, 60009.CrossRefGoogle Scholar
Lacaux, C. and Loubes, J.-M. (2007). Hurst exponent estimation of fractional Lévy motion. ALEA 3, 143–164.Google Scholar
Laha, R. and Rohatgi, V. (1981). Operator self similar stochastic processes in $ \mathbb{R}^d$ . Stoch. Process. Appl. 12, 73–84.CrossRefGoogle Scholar
Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993). On the self-similar nature of Ethernet traffic. ACM SIGCOMM Computer Commun. Rev. 23, 183–193.CrossRefGoogle Scholar
Maejima, M. and Mason, J. (1994). Operator-self-similar stable processes. Stoch. Process. Appl. 54, 139–163.CrossRefGoogle Scholar
Magdziarz, M. and Weron, A. (2011). Ergodic properties of anomalous diffusion processes. Ann. Phys. 326, 2431–2443.CrossRefGoogle Scholar
Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.CrossRefGoogle Scholar
Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099–1126.CrossRefGoogle Scholar
Marquardt, T. (2007). Multivariate fractionally integrated CARMA processes. J. Multivariate Anal. 98, 1705–1725.CrossRefGoogle Scholar
Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA processes. Stoch. Process. Appl. 117, 96–120.CrossRefGoogle Scholar
Maruyama, G. (1970). Infinitely divisible processes. Theory Prob. Appl. 15, 1–22.CrossRefGoogle Scholar
Mason, J. and Xiao, Y. (2002). Sample path properties of operator-self-similiar Gaussian random fields. Theory Prob. Appl. 46, 58–78.CrossRefGoogle Scholar
Meerschaert, M. M. and Scalas, E. (2006). Coupled continuous time random walks in finance. Physica A 370, 114–118.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. John Wiley, New York.Google Scholar
Moser, M. and Stelzer, R. (2013). Functional regular variation of Lévy-driven multivariate mixed moving average processes. Extremes 16, 351–382.CrossRefGoogle Scholar
Paxson, V. and Floyd, S. (1995). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Networking 3, 226–244.CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2017). Long-Range Dependence and Self-Similarity. Cambridge University Press.CrossRefGoogle Scholar
Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.CrossRefGoogle Scholar
Rosenblatt, M. (2000). Gaussian and Non-Gaussian Linear Time Series and Random Fields. Springer, New York.CrossRefGoogle Scholar
Rosiński, J. (1989). On path properties of certain infinitely divisible processes. Stoch. Process. Appl. 33, 73–87.CrossRefGoogle Scholar
Rosiński, J. (2007). Tempering stable processes. Stoch. Process. Appl. 117, 677–707.CrossRefGoogle Scholar
Rosiński, J. (2018). Representations and isomorphism identities for infinitely divisible processes. Ann. Prob. 46, 3229–3274.CrossRefGoogle Scholar
Rozanov, Y. A. (1967). Stationary Random Processes. Holden-Day, San Francisco.Google Scholar
Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Springer, Cham.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sato, K.-I. (2006). Additive processes and stochastic integrals. Illinois J. Math. 50, 825–851.CrossRefGoogle Scholar
Sornette, D. (2006). Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools. Springer, Berlin, Heidelberg.Google Scholar
Suciu, N. (2010). Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. Phys. Rev. E 81, 056301.CrossRefGoogle ScholarPubMed
Taqqu, M. S. (2003). Fractional Brownian motion and long-range dependence. In Theory and Applications of Long-Range Dependence, eds P. Doukhan, G. Oppenheim and M. S. Taqqu, Birkhäuser, Boston, pp. 5–38.Google Scholar
Tikanmäki, H. and Mishura, Y. (2011). Fractional Lévy processes as a result of compact interval integral transformation. Stoch. Anal. Appl. 29, 1081–1101.CrossRefGoogle Scholar
Weiss, G. (1975). Time-reversibility of linear stochastic processes. J. Appl. Prob. 12, 831–836.CrossRefGoogle Scholar
Willinger, W. et al. (2002). Scaling phenomena in the Internet: critically examining criticality. Proc. Nat. Acad. Sci. USA 99, 2573–2580.CrossRefGoogle Scholar
Willinger, W., Taqqu, M. S. and Teverovsky, V. (1999). Stock market prices and long-range dependence. Finance Stoch. 3, 1–13.CrossRefGoogle Scholar
Xu, Y. et al. (2016). The switch in a genetic toggle system with Lévy noise. Sci. Reports 6, 31505.Google Scholar
Zhang, S., Lin, Z. and Zhang, X. (2015). A least squares estimator for Lévy-driven moving averages based on discrete time observations. Commun. Statist. Theory Meth. 44, 1111–1129.CrossRefGoogle Scholar