A generalisation of the fractional Brownian field based on nonEuclidean norms
Abstract
We explore a generalisation of the Lévy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all selfsimilar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several nonEuclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial $p$th mean body and the polar projection transforms.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2523
 Bibcode:
 2014arXiv1410.2523M
 Keywords:

 Mathematics  Probability;
 Mathematics  Metric Geometry;
 60G22;
 60G55;
 60G60;
 52A21
 EPrint:
 28 pages, To appear in J. Math. Anal. Appl