Tangent fields, intrinsic stationarity, and self-similarity
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This paper studies the local structure of continuous random fields on ℝ^d taking values in a complete separable linear metric space 𝕍. Extending the seminal work of Falconer (2002), we show that the generalized k-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron (1973). These results motivate the further study of the structure of 𝕍-valued intrinsic random functions of order k (IRF_k, k=0,1,⋯). To this end, we focus on the special case where 𝕍 is a Hilbert space. Building on the work of Sasvari (2009) and Berschneider (2012), we establish the spectral characterization of all second order 𝕍-valued IRF_k's, extending the classic Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar 𝕍-valued IRF_k's, extending some results of Dobrushin (1979) and Didier, Meerschaert, and Pipiras (2018), among others. These processes are 𝕍-valued counterparts to k-order 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 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.
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