  We give a comprehensive theoretical characterization of a nonparametric
estimator for the $L_2^2$ divergence between two continuous distributions. We
first bound the rate of convergence of our estimator, showing that it is
$\sqrt{n}$-consistent provided the densities are sufficiently smooth. In this
smooth regime, we then show that our estimator is asymptotically normal,
construct asymptotic confidence intervals, and establish a Berry-Ess\'{e}en
style inequality characterizing the rate of convergence to normality. We also
show that this estimator is minimax optimal.
