From convergence in distribution to uniform convergence
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 Toeplitzlike matrices, convergence in distribution is ensured by theorems of the Szegő type. Our results transfer these convergence theorems into uniform convergence statements.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 DOI:
 10.48550/arXiv.1509.01836
 arXiv:
 arXiv:1509.01836
 Bibcode:
 2015arXiv150901836B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Probability;
 Primary 60B10;
 Secondary 15B05;
 15A18;
 28A20;
 47B35
 EPrint:
 15 pages, 3 figures