This article is cited in 10 papers

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

C. Ma

Department of Mathematics and Statiatics, Wichita State University

Abstract: 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.

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.

MSC: 60G22

Received: 18.05.2008
Revised: 22.02.2012

Language: English

DOI: 10.4213/tvp4477

 English version: Theory of Probability and its Applications, 2013, 57:4, 619–632

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