The Schoenberg-Levy kernel and relationships among fractional Brownian motion, bifractional Brownian motion, and others
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Ma, Chunsheng. 2013. The Schoenberg-Levy kernel and relationships among fractional Brownian motion, bifractional Brownian motion, and others. Theory of Probability & Its Applications, vol. 57:no. 4:pp. 619-632
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-Levy 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 R-d is shown to be a quasi-helix in the sense of Kahane.
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