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Wigner and Wishart ensembles for sparse Vinberg models

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Abstract

Vinberg cones and the ambient vector spaces are important in modern statistics of sparse models. The aim of this paper is to study eigenvalue distributions of Gaussian, Wigner and covariance matrices related to growing Vinberg matrices. For Gaussian or Wigner ensembles, we give an explicit formula for the limiting distribution. For Wishart ensembles defined naturally on Vinberg cones, their limiting Stieltjes transforms, support and atom at 0 are described explicitly in terms of the Lambert–Tsallis functions, which are defined by using the Tsallis q-exponential functions.

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Acknowledgements

The authors would like to express their sincere gratitude to Professor H. Ishi for strong encouragement and invaluable comments on this work. The authors are very grateful to Professor J. Najim for significant methodological and bibliographical indications for this work and to Professor C. Bordenave for discussions on Theorem 3. The authors thank the Scientific Committee of the CIRM Luminy 2020 Conference “Mathematical Methods of Modern Statistics” (MMMS 2) for giving the possibility to present this work. We thank numerous participants of MMMS 2 for their comments and remarks. We thank the referees and the associated editor for suggestions and corrections that improved significantly the paper. This research was carried out while the first author spent the winter semester of 2018 at Laboratoire de Mathématiques LAREMA under the support of Grant-in-Aid for JSPS fellows (2018J00379).

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  1. The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

    Hideto Nakashima

  2. Laboratoire de Mathématiques LAREMA, Université d’Angers, 2 boulevard Lavoisier, 49045, Angers Cedex 01, France

    Piotr Graczyk

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  1. Hideto Nakashima
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Correspondence to Hideto Nakashima.

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Nakashima, H., Graczyk, P. Wigner and Wishart ensembles for sparse Vinberg models. Ann Inst Stat Math 74, 399–433 (2022). https://doi.org/10.1007/s10463-021-00800-8

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