New Upper bounds for KL-divergence Based on Integral Norms

In this paper, some new upper bounds for KL-divergence based on $L^1, L^2$ and $L^\infty$ norms of density functions are discussed. Our findings unveil that the convergence in KL-divergence sense sandwiches between the convergence of density functions in terms of $L^1$ and $L^2$ norms. Furthermore, we endeavor to apply our newly derived upper bounds to the analysis of the rate theorem of the entropic conditional central limit theorem.