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Function valued random fields: tangents, intrinsic stationarity, self-similarity

By Stilian Stoev

Appears in collection : Heavy Tails, Long-Range Dependence, and Beyond / Queues lourdes, dépendance de long terme et  au-delà

We study random fields taking values in a separable Hilbert space H. First, we focus on their local structure and establish a counterpart to Falconer's characterization of tangent fields. That is, we show (under general conditions) that the tangent fields to a H-valued process are self-similar and almost all of them have stationary increments. We go a bit further and study higher-order tangent fields. This leads naturally to the study of self-similar intrinsic random functions (IRF) taking values in a Hilbert space. To this end, we begin by extending Matheron's theory of scalar-valued IRFs and provide the spectral representation of H-valued IRFs. We then use this theory to characterize large classes of operator self-similar H-valued IRF processes, which in the Gaussian case can be viewed as the H-valued counterparts to fractional Brownian fields. These general results may find applications to the study of long-range dependence for random fields taking values in a Hilbert space as well as to modeling function-valued spatial data.

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Citation data

  • DOI 10.24350/CIRM.V.19938103
  • Cite this video Stoev Stilian (7/5/22). Function valued random fields: tangents, intrinsic stationarity, self-similarity. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19938103
  • URL https://dx.doi.org/10.24350/CIRM.V.19938103

Bibliography

  • SHEN, Jinqi, STOEV, Stilian, et HSING, Tailen. Tangent fields, intrinsic stationarity, and self-similarity (with a supplement on Matheron Theory). arXiv preprint arXiv:2010.14715, 2020. - https://doi.org/10.48550/arXiv.2010.14715

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