New classes of self-similar symmetric stable random fields
Two new classes of symmetric stable self-similar random fields with stationary increments are constructed in the paper: \[ X_ p(t) = \int_{R^ n} \bigl[ p(t - x)^{H - n/ \alpha} - p(x)^{H - n/ \alpha} \bigr] M(dx), \quad t \in R^ n, \tag{1} \] and \[ Y_ \gamma (t) = \int_{R^ n} \bigl( e^{i(t,x)} - 1 \bigr) \| x \|^{-H - n/ \alpha} Z_ \gamma (dx), \quad t \in R^ n, \tag{2} \] where \(p:R^ n \to R_ +\) is an appropriately chosen function and \(M\) is the \(\text{S} \alpha \text{S}\) random measure with \(n\)-dimensional Lebesgue measure as its control measure and \(Z_ \gamma\) is the complex-valued (isotropic) \(\text{S} \alpha \text{S}\) random measure with control measure \(m_ \gamma = \gamma \times \text{Leb}_ +\), \(\gamma\) is finite Borel measure on \(S_ n = \{x \in R^ n: \| x \| = 1\}\), \(\text{Leb}_ +\) is finite Lebesgue measure on \(R_ +\). (Symbol S\(\alpha\)S denotes symmetric \(\alpha\)-stable process, \(0 < \alpha \leq 2\).) It is examined how the choice of the norm affects the finite-dimensional distributions. The processes which are obtained by projecting the random fields on a one-dimensional subspace are studied. The asymptotic dependence structure of the moving average and (1) and (2) type random fields are characterized.
symmetric stable self-similar random fields
scientific article; zbMATH DE number 637647
scientific article