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From convergence in distribution to uniform convergence

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Abstract

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of n-by-n matrices as n goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szegő type. Our results transfer these convergence theorems into uniform convergence statements.

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Acknowledgments

We are greatly indebted to the referee for bringing reference [9] to our attention. Furthermore, in the original version of this paper, we still had Proposition 3.2 accompanied by a proof. We thank R. Michael Porter and Carlos G. Pacheco (CINVESTAV, Mexico) for giving us the hint that this proposition is just Problem 127 of [15].

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Authors and Affiliations

  1. Departamento de Matemáticas, Pontificia Universidad Javeriana, 01110, Bogotá D.C., Colombia

    J. M. Bogoya

  2. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    A. Böttcher

  3. Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, 07730, Ciudad de México, Mexico

    E. A. Maximenko

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  1. J. M. Bogoya
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  2. A. Böttcher
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  3. E. A. Maximenko
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Correspondence to E. A. Maximenko.

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To Sergei Grudsky, who has left his imprint on all the three of us, on his 60th birthday.

E. A. Maximenko’s research was partially supported by project IPN-SIP 20150422 (Instituto Politécnico Nacional, Mexico).

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Bogoya, J.M., Böttcher, A. & Maximenko, E.A. From convergence in distribution to uniform convergence. Bol. Soc. Mat. Mex. 22, 695–710 (2016). https://doi.org/10.1007/s40590-016-0105-y

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  • DOI: https://doi.org/10.1007/s40590-016-0105-y

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