Abstract
This paper studies the local structure of continuous random fields on Rd taking values in a complete separable linear metric space 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 V-valued intrinsic random functions of order k (IRFk, k=0,1,…). To this end, we focus on the special case where V is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order V-valued IRFk’s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar V-valued IRFk’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.
Highlights
The tangent process of a random field is the stochastic process obtained in the limit of the suitably normalized increments of the random field at a fixed location
Falconer proved that the tangent processes must be self-similar and have stationary increments (a.e.)
Our results provide characterizations of Gaussian operator self-similar intrinsic random function of order k (IRFk)’s, which can be viewed as infinite-dimensional versions of the k-th order fractional Brownian fields
Summary
The tangent process of a random field is the stochastic process obtained in the limit of the suitably normalized increments of the random field at a fixed location. Most if not all of the existing work, focuses on random fields taking values in Rm. In this paper, we provide a first comprehensive treatment of Hilbert space valued operator fractional Brownian fields and their higher order stationary increment counterparts – the Gaussian IRFk’s. We provide a first comprehensive treatment of Hilbert space valued operator fractional Brownian fields and their higher order stationary increment counterparts – the Gaussian IRFk’s This leads to infinite-dimensional extensions of seminal results due to Didier and Pipiras (2011); Didier et al (2017); Perrin et al (2001) among others. This treatment unifies and extends results of Bochner, Cramér, Gelfand-Vilenkin, Matheron, Neeb, Sasvári, and Berschneider. Further background and details are given in Shen et al (2020)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
