The Schoenberg–Lévy kernel and relationships among fractional Brownian motion, bifractional Brownian motion, and others

C. Ma

Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 4, pp. 744-760

Résumé

Starting with a discussion about the relationship between the fractional Brownian motion and the bifractional Brownian motion on the real line, we find that a fractional Brownian motion can be decomposed as an independent sum of a bifractional Brownian motion and a trifractional Brownian motion that is defined in the paper. More generally, this type of orthogonal decomposition holds for a large class of Gaussian or elliptically contoured random functions whose covariance functions are Schoenberg–Lévy kernels on a temporal, spatial, or spatio-temporal domain. Also, many self-similar, nonstationary (Gaussian, elliptically contoured) random functions are formulated, and properties of the trifractional Brownian motion are studied. In particular, a bifractional Brownian motion in $\mathbb{R}^d$ is shown to be a quasi-helix in the sense of Kahane.

Détail

Keywords: bifractional Brownian motion; conditionally negative definite; covariance function; elliptically contoured random function; Gaussian random function; positive definite; quasi-helix; Schoenberg–Lévy kernel; self-similarity; trifractional Brownian motion; variogram.

Bibliographie

[1] Alpay D., Levanony D., “On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions”, Potential Anal., 28:2 (2008), 163–184 | DOI | MR | Zbl

[2] Bardina X., Es-Sebaiy K., “An extension of bifractional Brownian motion”, Commun. Stoch. Anal., 5:2 (2011), 333–340 | MR

[3] Berman S. M., “Gaussian processes with biconvex covariances”, J. Multivariate Anal., 8:1 (1978), 30–44 | DOI | MR | Zbl

[4] Bojdecki T., Gorostiza L. G., Talarczyk A., “Sub-fractional Brownian motion and its relation to occupation times”, Statist. Probab. Lett., 69:4 (2004), 405–419 | DOI | MR | Zbl

[5] Bojdecki T., Gorostiza L. G., Talarczyk A., “Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems”, Electron. Comm. Probab., 12 (2007), 161–172 | MR | Zbl

[6] Berg C., Christensen J. P. R., Ressel P., Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Springer, New York, 1984, 289 pp. | MR

[7] De Chávez J. R., Tudor C., “A decomposition of sub-fractional Brownian motion”, Math. Rep. (Bucur.), 11(61):1 (2009), 67–74 | MR

[8] Doob J. L., “The elementary Gaussian processes”, Ann. Math. Statist., 15 (1944), 229–282 | DOI | MR | Zbl

[9] Du J., Ma C., “Variogram matrix functions for vector random fields with second-order increments”, Math. Geosci., 44 (2012), 411–425 | DOI | MR

[10] Dzhaparidze K. O., van Zanten J. H., “A series expansion of fractional Brownian motion”, Probab. Theory Related Fields, 130:1 (2004), 39–55 | DOI | MR | Zbl

[11] El-Nouty C., “The increments of a bifractional Brownian motion”, Studia Sci. Math. Hungar., 46:4 (2009), 449–478 | MR

[12] Es-Sebaiy K., Tudor C. A., “Multidimensional bifractional Brownian motion: Itô and Tanaka formulas”, Stoch. Dyn., 7:3 (2007), 365–388 | DOI | MR | Zbl

[13] Gangolli R., “Abstract harmonic analysis and Lévy's Brownian motion of several parameters”, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, 1965/1966), v. II, Univ. of California Press, Berkeley, 1967, 13–30 | MR | Zbl

[14] Gangolli R., “Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters”, Ann. Inst. Henri Poincaré B, 3 (1967), 121–226 | MR | Zbl

[15] Genton M. G., Perrin O., Taqqu M. S., “Self-similarity and Lamperti transformation for random fields”, Stoch. Models, 23:3 (2007), 397–411 | DOI | MR | Zbl

[16] Houdré C., Villa J., “An example of infinite-dimensional quasi-helix”, Stochastic Models (Mexico City, 2002), Amer. Math. Soc., Providence, 2003, 195–201 | DOI | MR | Zbl

[17] Kahane J.-P., Some Random Series of Functions, Cambridge Univ. Press, London, 1985, 305 pp. | MR | Zbl

[18] Kolmogorov A. N., “Spiral Vinera i nekotorye drugie interesnye krivye v gilbertovskom prostranstve”, Dokl. AN SSSR, 26:2 (1940), 115–118

[19] Kruk I., Russo F., Tudor C. A., “Wiener integrals, Malliavin calculus and covariance measure structure”, J. Funct. Anal., 249:1 (2007), 92–142 | DOI | MR | Zbl

[20] Lamperti J., “Semi-stable stochastic processes”, Trans. Amer. Math. Soc., 104 (1962), 62–78 | DOI | MR | Zbl

[21] Lei P., Nualart D., “A decomposition of the bifractional Brownian motion and some applications”, Statist. Probab. Lett., 79:5 (2009), 619–624 | DOI | MR | Zbl

[22] Liu J., “The law of a stochastic integral with two independent bifractional Brownian motions”, Commun. Korean Math. Soc., 26:4 (2011), 669–684 | DOI | MR | Zbl

[23] Ma C., “Nonstationary covariance functions that model space-time interactions”, Statist. Probab. Lett., 61:4 (2003), 411–419 | DOI | MR | Zbl

[24] Ma C., “Spatio-temporal variograms and covariance models”, Adv. Appl. Probab., 37:3 (2005), 706–725 | DOI | MR | Zbl

[25] Ma C., “Intrinsically stationary variograms in space and time”, Teoriya veroyatn. i ee primen., 53:1 (2008), 189–200 | MR

[26] Ma C., “Construction of non-Gaussian random fields with any given correlation structure”, J. Statist. Plann. Inference, 139:3 (2009), 780–787 | DOI | MR | Zbl

[27] Ma C., “Vector random fields with second-order moments or second-order increments”, Stoch. Anal. Appl., 29:2 (2011), 197–215 | DOI | MR | Zbl

[28] Mandelbrot B. B., Van Ness J. W., “Fractional Brownian motions, fractional noises and applications”, SIAM Rev., 10 (1968), 422–437 | DOI | MR | Zbl

[29] Nualart D., “Stochastic integration with respect to fractional Brownian motion and applications”, Stochastic Models (Mexico City, 2002), Amer. Math. Soc., Providence, 2003, 3–39 | DOI | MR | Zbl

[30] Nualart D., Ortiz-Latorre S., “An Itô–Stratonovich formula for Gaussian processes: A Riemann sums approach”, Stochastic Process. Appl., 118:10 (2008), 1803–1819 | DOI | MR | Zbl

[31] Russo F., Tudor C. A., “On bifractional Brownian motion”, Stochastic Process. Appl., 116:5 (2006), 830–856 | DOI | MR | Zbl

[32] Székely G. J., Rizzo M. L., “A new test for multivariate normality”, J. Multivariate Anal., 93 (2005), 58–80 | DOI | MR

[33] Taqqu M. S., “Fractional Brownian motion and long-range dependence”, Theory and Applications of Long-range Dependence, eds. P. Doukhan, G. Oppenheim, M. S. Taqqu, Birkhäuser, Basel, 2003, 5–38 | MR | Zbl

[34] Tudor C. A., Xiao Y., “Sample path properties of bifractional Brownian motion”, Bernoulli, 13:4 (2007), 1023–1052 | DOI | MR | Zbl

[35] Wang W., “On $p$-variation of bifractional Brownian motion”, Appl. Math. J. Chinese Univ., 26:2 (2011), 127–141 | DOI | MR | Zbl

[36] Yaglom A. M., Correlation Theory of Stationary and Related Random Functions, v. 1, Springer, New York, 1987, 526 pp. ; v. 2, 258 pp. | MR

[37] Zastavnyi V. P., “On positive definiteness of some functions”, J. Multivariate Anal., 73 (2000), 55–81 | DOI | MR | Zbl