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Schoenberg’s Theorem and Unitarily Invariant Random Arrays

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  • Olav Kallenberg

    (Auburn University)

Abstract

Motivated by Schoenberg’s theorem in classical analysis and some recent work on the circular law, we extend the basic representations of unitarily invariant random arrays and functionals to the complex case. As in the real case, the basic building blocks are multiple Wiener–Itô integrals on tensor products of a separable Hilbert space. The complex setting leads to some unexpected simplifications.

Suggested Citation

  • Olav Kallenberg, 2012. "Schoenberg’s Theorem and Unitarily Invariant Random Arrays," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1013-1039, December.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:4:d:10.1007_s10959-010-0332-5
    DOI: 10.1007/s10959-010-0332-5
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    References listed on IDEAS

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    1. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    2. Dawid, A. P., 1978. "Extendibility of spherical matrix distributions," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 559-566, December.
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    Cited by:

    1. Yanik-Pascal Förster & Mario Kieburg & Holger Kösters, 2021. "Polynomial Ensembles and Pólya Frequency Functions," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1917-1950, December.

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