The structure of self-similar stable mixed moving averages
A symmetric \(\alpha\)-stable self-similar process with stationary increments \(X_\alpha\) is considered given by the mixed moving average \[ X_{\alpha(t)}=\int_X\int_R(G(x,t+u)-G(x,u))M_{\alpha}(dx,du), \] where \(M_{\alpha}\) is a random measure with the Lebesgue control measure. It is shown that \(X_{\alpha}\) can be decomposed into a sum of three independent processes \(X^{(1)}\), \(X^{(2)}\), \(X^{(3)}\), where \(X^{(1)}\) is generated by a nonsingular dissipative flow, \(X^{(2)}\) and \(X^{(3)}\) are generated by a conservative nonsingular flow. Processes \(X^{(1)}\) and \(X^{(2)}\) are described in terms of integrals by stochastic measures. Some examples of \(X^{(3)}\) are given.
stable self-similar processes with stationary increments
scientific article; zbMATH DE number 1906072