From convergence in distribution to uniform convergence
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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.
KeywordsConvergence in distribution Quantile function Toeplitz matrix Eigenvalue asymptotics
Mathematics Subject ClassificationPrimary 60B10 Secondary 15B05 15A18 28A20 47B35
We are greatly indebted to the referee for bringing reference  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 .
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