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Universality of Limiting Spectral Distribution Under Projective Criteria

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High Dimensional Probability VIII

Part of the book series: Progress in Probability ((PRPR,volume 74))

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Abstract

This paper has double scope. In the first part we study the limiting empirical spectral distribution of a n × n symmetric matrix with dependent entries. For a class of generalized martingales we show that the asymptotic behavior of the empirical spectral distribution depends only on the covariance structure. Applications are given to strongly mixing random fields. The technique is based on a blend of blocking procedure, martingale techniques and multivariate Lindeberg’s method. This means that, for this class, the study of the limiting spectral distribution is reduced to the Gaussian case. The second part of the paper contains a survey of several old and new asymptotic results for the empirical spectral distribution for Gaussian processes, which can be combined with our universality results.

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Acknowledgements

This paper was supported in part by a Charles Phelps Taft Memorial Fund grant, and the NSF grant DMS-1811373. We are grateful to the referee for carefully reading our manuscript and for helpful comments that improved its presentation.

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Authors and Affiliations

  1. Université Paris-Est, LAMA and CNRS UMR 8050, Marne-la-Vallée, France

    Florence Merlevède

  2. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA

    Magda Peligrad

Authors
  1. Florence Merlevède
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  2. Magda Peligrad
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Correspondence to Florence Merlevède .

Editor information

Editors and Affiliations

  1. MAP 5, Université Paris Descartes, Paris, France

    Nathael Gozlan

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Rafał Latała

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France

    Karim Lounici

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Mokshay Madiman

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Merlevède, F., Peligrad, M. (2019). Universality of Limiting Spectral Distribution Under Projective Criteria. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_13

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