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The Semicircle Law for Matrices with Dependent Entries

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Limit Theorems in Probability, Statistics and Number Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 42))

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Abstract

We investigate the spectral distribution of random matrix ensembles with correlated entries. The matrices considered are symmetric, have real-valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions and sufficiently strong decay of correlations the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law. The present note improves an earlier result (see [Friesen and Löwe, J. Theor. Probab., 2011]) by the authors using similar techniques.

2010 Mathematics Subject Classification. 60F05.

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Author information

Authors and Affiliations

  1. Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik, Einsteinstraße 62, 48149, Münster, Germany

    Olga Friesen & Matthias Löwe

Authors
  1. Olga Friesen
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  2. Matthias Löwe
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Corresponding author

Correspondence to Matthias Löwe .

Editor information

Editors and Affiliations

  1. Mathematics Faculty, Ruhr-University Bochum, Universitätstr. 150, Bochum, 44780, Germany

    Peter Eichelsbacher

  2. Mathematics Faculty, University of Bielefeld, Universitätsstr. 25, Bielefeld, 33615, Germany

    Guido Elsner

  3. Mathematics Faculty, University of Bielefeld, Universitätsstr. 25, Bielefeld, 33615, Germany

    Holger Kösters

  4. Institute for Mathematical Statistics, University of Münster, Einsteinstrasse 62, Münster, 48149, Germany

    Matthias Löwe

  5. Mathematics Institute, University of München, Theresienstr. 39, München, 80333, Germany

    Franz Merkl

  6. Centre for Mathematics, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85747, Germany

    Silke Rolles

Additional information

Dedicated to Friedrich Götze on the occasion of his sixtieth birthday

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Friesen, O., Löwe, M. (2013). The Semicircle Law for Matrices with Dependent Entries. In: Eichelsbacher, P., Elsner, G., Kösters, H., Löwe, M., Merkl, F., Rolles, S. (eds) Limit Theorems in Probability, Statistics and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36068-8_13

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