Operator-stable and operator-self-similar random fields

Multi operator-stable random measures and fields

“…Moreover, if µ = µ s is still constant but operator-stable (say with exponent B), we also get back the setting of [11]. Merely note that, in contrast to [11], µ has to be symmetric (instead of strictly) operator-stable in our framework. However, τ = τ s does not depend on s ∈ S either and (2.14) becomes…”

“…Recall Remark 2.3 and that µ s ∼ [0, 0, ϕ(s, •)] with ϕ(s, •) being symmetric for every s ∈ S. Then (a) and (b) follow immediately from Proposition 3.3 and section 5 of [10], respectively. Also recall that µ s is full for every s ∈ S. Hence, using (2.13) and Lemma 1.3.11 in [14], a slight refinement of Proposition 2.6 (a) in [11] gives part (c). The details are left to the reader.…”

“…In particular, if we assume that µ = µ s is α-stable for fixed 0 < α < 2, our construction of random measures covers the considerations in [2], [13] and [19]. Moreover, if µ = µ s is still constant but operator-stable (say with exponent B), we also get back the setting of [11]. Merely note that, in contrast to [11], µ has to be symmetric (instead of strictly) operator-stable in our framework.…”

“…At the same time we refer the reader to [8], [11] and [14] concerning more details about operator-stable distributions which will play a crucial role throughout the paper.…”

“…For a comprehensive introduction see [14]. Based on the recently developed theory of multivariate independently scattered random measures (short: ISRMs) and their integrals (see [10]), moving-average and harmonizable representations of (E, D)-operator-self-similar random fields with operator-stable marginals are presented in [11].…”