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A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator

Author

Listed:
  • G. Gaines

    (Texas Tech University)

  • K. Kaphle

    (University of Maine at Fort Kent)

  • F. Ruymgaart

    (Texas Tech University)

Abstract

The problem of a random Hermitian perturbation of a multiple isolated eigenvalue of a Hermitian operator is considered. It is shown that the combined multiplicities of the perturbed eigenvalues converge in probability to the multiplicity of the eigenvalue of the target operator. Also the asymptotic distribution of a certain average of these eigenvalues, centered at the target, is obtained. As a tool differentiation of analytic functions of operators is employed in conjunction with an ensuing “delta-method”. The result is of a probabilistic rather than statistical nature.

Suggested Citation

  • G. Gaines & K. Kaphle & F. Ruymgaart, 2014. "A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1112-1123, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0482-3
    DOI: 10.1007/s10959-013-0482-3
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    References listed on IDEAS

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    1. D. S. Gilliam & T. Hohage & X. Ji & F. Ruymgaart, 2009. "The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2009, pages 1-17, April.
    2. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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