  We show that external randomization may enforce the convergence of test
statistics to their limiting distributions in particular cases. This results in
a sharper inference. Our approach is based on a central limit theorem for
weighted sums. We apply our method to a family of rank-based test statistics
and a family of phi-divergence test statistics and prove that, with
overwhelming probability with respect to the external randomization, the
randomized statistics converge at the rate $O(1/n)$ (up to some logarithmic
factors) to the limiting chi-square distribution in Kolmogorov metric.
