ResearchTrend.AI
  • Papers
  • Communities
  • Events
  • Blog
  • Pricing
Papers
Communities
Social Events
Terms and Conditions
Pricing
Parameter LabParameter LabTwitterGitHubLinkedInBlueskyYoutube

© 2025 ResearchTrend.AI, All rights reserved.

  1. Home
  2. Papers
  3. 2010.14715
26
4
v1v2v3v4 (latest)

Tangent fields, intrinsic stationarity, and self similarity

28 October 2020
Jinqi Shen
Stilian A. Stoev
T. Hsing
ArXiv (abs)PDFHTML
Abstract

This paper studies the local structure of continuous random fields on Rd\mathbb R^dRd taking values in a complete separable linear metric space V{\mathbb V}V. Extending the seminal work of Falconer (2002), we show that the generalized kkk-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{\mathbb V}V-valued intrinsic random functions of order kkk (IRFk_kk​,\ k=0,1,⋯k=0,1,\cdotsk=0,1,⋯). To this end, we focus on the special case where V{\mathbb V}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{\mathbb V}V-valued IRFk_kk​'s, extending the classic Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar V{\mathbb V}V-valued IRFk_kk​'s, extending some results of Dobrushin (1979) and Didier, Meerschaert, and Pipiras (2018), among others. These processes are V{\mathbb V}V-valued counterparts to kkk-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.

View on arXiv
Comments on this paper