Tangent fields, intrinsic stationarity, and self similarity (Q2119696)

The authors define \(k\)'th order tangent fields based on scaling actions, where the random field may take values in a complete and separable linear metric space. It is shown that \((1+k)\)'th order increment tangent fields are self-similar, and further, that they are almost everywhere strictly intrinsically stationary. When the random field takes values in a general Hilbert space, a general correlation theory and spectral theory for stationary and intrinsically stationary processes is developed, and a spectral characterization for intrinsic random functions of order \(k\) is obtained. Finally, the class of Gaussian operator self-similar intrinsically stationary random functions of order \(k\) is characterized. The paper concludes with several examples.