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