A Tight Bound on the Distance Between a Noncentral Chi Square and a Normal Distribution

We provide a nonasymptotic bound on the distance between a noncentral chi square distribution and a normal approximation. It improves on both the classical Berry-Esséen bound and previous distances derived specifically for this situation. First, the bound is nonasymptotic and provides an upper limit for the real distance. Second, the bound has the correct rate of decrease and even the correct leading constant when either the number of degrees of freedom or the noncentrality parameter (or both) diverge to infinity. The bound is applied to some probabilities arising in energy detection and Rician fading.