Shen, Jinqi; Stoev, Stilian; Hsing, Tailen Tangent fields, intrinsic stationarity, and self similarity. (English) Zbl 1507.60043 Electron. J. Probab. 27, Paper No. 34, 56 p. (2022). The authors define \(k\)’th order tangent fields based on scaling actions, where the random field may take values in a complete and separable linear metric space. It is shown that \((1+k)\)’th order increment tangent fields are self-similar, and further, that they are almost everywhere strictly intrinsically stationary. When the random field takes values in a general Hilbert space, a general correlation theory and spectral theory for stationary and intrinsically stationary processes is developed, and a spectral characterization for intrinsic random functions of order \(k\) is obtained. Finally, the class of Gaussian operator self-similar intrinsically stationary random functions of order \(k\) is characterized. The paper concludes with several examples. Reviewer: Alexander Lindner (Ulm) MSC: 60G10 Stationary stochastic processes 60G12 General second-order stochastic processes 60G18 Self-similar stochastic processes 60G22 Fractional processes, including fractional Brownian motion Keywords:tangent field of higher order; intrinstic random function of order \(k\); intrinstic stationarity; IRFk; self-similarity; spectral theory Software:fda (R) PDFBibTeX XMLCite \textit{J. Shen} et al., Electron. J. Probab. 27, Paper No. 34, 56 p. (2022; Zbl 1507.60043) Full Text: DOI arXiv References: [1] P. Abry and G. Didier. Wavelet estimation for operator fractional Brownian motion. Bernoulli, 24(2):895-928, 2018. · Zbl 1432.60045 [2] P. Abry, H. Wendt, S. Jaffard, and G. Didier. Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis. Comptes Rendus Physique, 20(5):489 - 501, 2019. [3] P. Amblard and J. Coeurjolly. Identification of the multivariate fractional Brownian motion. 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