On the $n$th-Order subfractional Brownian motion

Year 2023, Volume: 52 Issue: 5, 1396 - 1407, 31.10.2023

El Omari Mohamed , Mabdaoui Mohamed

Abstract

In the present work, we introduce the $n$th-Order subfractional Brownian motion $S_H^n = \lbrace S_H^n(t),~t\geq 0\rbrace$ with Hurst index $H\in (n-1,n)$ and order $n\geq 1$; then we examine some of its basic properties: self-similarity, long-range dependence, non Markovian nature and semimartingale property. A local law of iterated logarithm for $S_H^n$ is also established.

Keywords

Gaussian self-similar process, non Markovian process, subfractional Brownian motion, semimartingale property, local law of the iterated logarithm

References

• [1] M.A. Arcones, On the law of the iterated logarithm for Gaussian processes, J. Theoret. Probab. 8 (4), 877-903, 1995.
• [2] T. Bojdecki, L. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett. 69 (4), 405-419, 2004.
• [3] T. Bojdecki, L. Gorostiza and A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems i: long-range dependence, Stochastic Process. Appl. 116 (1), 1-18, 2006.
• [4] T. Bojdecki, L. Gorostiza and A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems, Electron. Commun. Probab. 12, 161-172, 2007.
• [5] J.J. Collins and C.J. De Luca, Upright, correlated random walks: A statisticalbiomechanics approach to the human postural control system, Chaos 5 (1), 57-63, 1995.
• [6] R.M. Dudley, R. Norvaia and R. Norvaia, Concrete Functional Calculus, Springer, 2011.
• [7] M. El Omari, Notes on spherical bifractional Brownian motion, Mod. Stoch. Theory Appl. 9 (3), 339-355, 2022.
• [8] M. El Omari, An $\alpha$-order fractional Brownian motion with Hurst index $H\in (0,1)$ and $\alpha\in\R_+$, Sankhya A 85 (1), 572-599, 2023.
• [9] M. El Omari, Mixtures of higher-order fractional Brownian motions, Comm. Statist. Theory Methods 52 (12), 4200-4215, 2023.
• [10] M. El Omari, Parameter estimation for nth-order mixed fractional Brownian motion with polynomial drift, J. Korean Stat. Soc. 52, 450-461, 2023.
• [11] C.W.J. Granger, The typical spectral shape of an economic variable, Econometrica 34 (1), 150-161, 1966.
• [12] D.P. Huy, A remark on non-Markov property of a fractional Brownian motion, Vietnam J. Math. 31 (3), 237-240, 2003.
• [13] R. Jennane and R. Harba, Fractional Brownian motion: A model for image texture, EUSIPCO Signal Process. 3, 1389-1392, 1994.
• [14] W.S. Kuklinski, K. Chandra, U.E. Ruttirmann and R.L. Webber, Application of fractal texture analysis to segmentation of dental radiographs, Proc. SPIE 1092, Medical Imaging III: Image Processing 1092, 111-117, 1989.
• [15] C.M. Lee, Generalizations of l’Hôpitals rule, Proc. Amer. Math. Soc. 66 (2), 315-320, 1977.
• [16] T. Lundahl, W.J. Ohley, S.M. Kay and R. Siffert, Fractional Brownian motion: A maximum likelihood estimator and its application to image texture, IEEE Trans. Med. Imag. 5 (3), 152-161, 1986.
• [17] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (4), 422-437, 1968.
• [18] A.I. McLeod and K.W. Hipel, Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst phenomenon, Water Resour. Res. 14 (3), 491-508, 1978.
• [19] A.P. Pentland, Fractal-based description of natural scenes, IEEE Trans. Pattern Anal. Mach. Intell. 6 (6), 661-674, 1984.
• [20] E. Perrin, R. Harba, C. Berzin-Joseph, I. Iribarren, and A. Bonami, nth-order fractional Brownian motion and fractional gaussian noises, IEEE Trans. Signal Process. 49 (5), 1049-1059, 2001.
• [21] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer Science & Business Media, 1999.
• [22] G. Shen, C. Chen and L. Yan, Remarks on sub-fractional Bessel processes, Acta Math. Sci. Ser. 31 (5), 1860-1876, 2011.
• [23] C. Tudor, Some properties of the sub-fractional Brownian motion, Stochastics 79 (5), 431-448, 2007.
• [24] C. Tudor, On the Wiener integral with respect to a sub-fractional Brownian motion on an interval, J. Math. Anal. Appl. 351 (1), 456-468, 2009.
• [25] W. Willinger, M.S. Taqqu, W.E. Leland and D.V. Wilson, Self-similarity in highspeed packet traffic: analysis and modeling of ethernet traffic measurements, Statist. Sci. 10 (1), 67-85, 1995.
• [26] H. Qi and L. Yan, A law of iterated logarithm for the subfractional Brownian motion and an application, J. Inequal. Appl. 2018 (96), 1-18, 2018.
• [27] W.H. Young, On indeterminate forms, Proc. Lond. Math. Soc. 2 (1), 40-76, 1910.
• [28] M. Zabat, M. Vayer-Besançon, R. Harba, S. Bonnamy and H. Van Damme, Surface topography and mechanical properties of smectite films, Trends in Colloid and Interface Science XI, 96-102, Springer, Berlin, 1997.