Hölder Regularity for Operator Scaling Stable Random Fields

Abstract. We investigate the sample paths regularity of operator scaling α-stable random fields. Such fields were introduced in [6] as anisotropic generalizations of self-similar fields and satisfy the scaling property {X(cx);x ∈ R} (fdd) = {cX(x);x ∈ R} 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 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 random α-stable random fields, with α ∈ (0, 2) and d ≥ 2, the sample paths are almost surely discontinous.