The Schoenberg--Lévy Kernel and Relationships among Fractional Brownian Motion, Bifractional Brownian Motion, and Others

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 $\bR^d$ is shown to be a quasi-helix in the sense of Kahane.