Hölder regularity for operator scaling stable random fields

Abstract

We investigate the sample path regularity of operator scaling α -stable random fields. Such fields were introduced in [H. Biermé, M.M. Meerschaert, H.P. Scheffler, Operator scaling stable random fields, Stochastic Process. Appl. 117 (3) (2007) 312–332.] as anisotropic generalizations of self-similar fields and satisfy the scaling property { X ( c E x ) ; x ∈ R d } = ( f d d ) { c H X ( x ) ; x ∈ R d } where E is a d × d real matrix and H > 0 . In the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of R d with respect to E . In particular, the directional Hölder regularity may vary and is given by the eigenvalues of E . In the case of moving average operator scaling α -stable random fields, with α ∈ ( 0 , 2 ) and d ≥ 2 , the sample paths are almost surely discontinuous.