Tangent fields, intrinsic stationarity, and self-similarity (with a supplement on Matheron Theory)

Abstract

This paper studies the local structure of continuous random fields on $\mathbb R^d$ taking values in a complete separable linear metric space ${\mathbb V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of ${\mathbb V}$-valued intrinsic random functions of order $k$ (IRF$_k$,\ $k=0,1,\cdots$). To this end, we focus on the special case where ${\mathbb V}$ is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order ${\mathbb V}$-valued IRF$_k$'s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar ${\mathbb V}$-valued IRF$_k$'s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valued versions of the general $k$-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics. In a technical Supplement of independent interest, we provide a unified treatment of the Matheron spectral theory for second-order stationary and intrinsically stationary processes taking values in a separable Hilbert space. We give the proofs of the Bochner-Neeb and Bochner-Schwartz theorems.