A Variational Characterization of Renyi Divergences

Atar, Chowdhary, and Dupuis have recently exhibited a variational formula for exponential integrals of bounded measurable functions in terms of Renyi divergences. We show that a variational characterization of the Renyi divergences between two probability distributions on a measurable space in terms of relative entropies, when combined with the elementary variational formula for exponential integrals of bounded measurable functions in terms of relative entropy, yields the variational formula of Atar, Chowdhary, and Dupuis as a corollary. We then develop an analogous variational characterization of the Renyi divergence rates between two stationary finite state Markov chains in terms of relative entropy rates. When combined with Varadhan's variational characterization of the spectral radius of square matrices with nonnegative entries in terms of relative entropy, this yields an analog of the variational formula of Atar, Chowdary, and Dupuis in the framework of stationary finite state Markov chains.