Operator scaling stable random fields

Abstract

A scalar valued random field { X ( x ) } x ∈ R d is called operator-scaling if for some d × d matrix E with positive real parts of the eigenvalues and some H > 0 we have { X ( c E x ) } x ∈ R d = f . d . { c H X ( x ) } x ∈ R d for all c > 0 , where = f . d . denotes equality of all finite-dimensional marginal distributions. We present a moving average and a harmonizable representation of stable operator scaling random fields by utilizing so called E -homogeneous functions φ , satisfying φ ( c E x ) = c φ ( x ) . These fields also have stationary increments and are stochastically continuous. In the Gaussian case, critical Hölder-exponents and the Hausdorff-dimension of the sample paths are also obtained.