Central Limit Theorem on Symmetric Kullback-Leibler (KL) Divergence

In this paper we provide an asymptotic theory for the symmetric version of the Kullback–Leibler (KL) divergence. We define a estimator for this divergence and study its asymptotic properties. In particular, we prove Law of Large Numbers (LLN) and the convergence to the normal law in the Central Limit Theorem (CLT) using this estimator.