Exact Rényi and Kullback–Leibler Divergences Between Multivariate $t$-Distributions

In this letter, we propose a closed-form expression of the Rényi divergence (RD) of order $\beta$ between two zero-mean real multivariate $t$-distributions (MTDs). Such distribution has been deployed in several signal and image processing applications where heavy-tailed distribution is well-suited. Based on the computation of the multiple integral involved in the RD, the expression of the divergence is provided without resorting to the conventional time-consuming Monte Carlo (MC) integration technique. In addition, the Kullback–Leibler divergence (KLD) is deduced from RD. Finally, a comparison is made between the MC method and the numerical value of the RD expression to show how the former gives close approximations to the latter.