On Rényi Divergence Measures for Continuous Alphabet Sources

The idea of ‘probabilistic distances’ (also called divergences), which in some sense assess how ‘close’ two probability distributions are from one another, has been widely employed in probability, statistics, information theory, and related fields. Of particular importance due to their generality and applicability are the Rényi divergence measures. While the closely related concept of Rényi entropy of a probability distribution has been studied extensively, and closed-form expressions for the most common univariate and multivariate continuous distributions have been obtained and compiled [57, 45, 62], the literature currently lacks the corresponding compilation for continuous Rényi divergences. The present thesis addresses this issue for the analytically tractable cases. Closed-form expressions for Kullback-Leibler divergences are also derived and compiled, as they can be seen as an extension by continuity of the Rényi divergences. Additionally, we establish a connection between Rényi divergence and the variance of the log-likelihood ratio of two distributions, which extends the work of Song [57] on the relation between Rényi entropy and the log-likelihood function, and which becomes practically useful in light of the Rényi divergence expressions we have derived. Lastly, we consider the Rényi divergence rate between two stationary Gaussian processes.