  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 Szeg\H{o} type. Our results transfer
these convergence theorems into uniform convergence statements.
