If \(P_1,\ldots , P_n\) and \(Q_1,\ldots ,Q_n\) are probability measures on \(\mathbb {R}^d\) and \(P_1*\cdots *P_n\) and \(Q_1*\cdots *Q_n\) are their respective convolutions, the Rényi divergence \(D_{\lambda }\) of order \(\lambda \in (0,1]\) satisfies \(D_{\lambda }(P_1*\cdots *P_n||Q_1*\cdots *Q_n)\le \sum _{i=1}^nD_{\lambda }(P_i||Q_i).\) When \(P_i\) belongs to the natural exponential family generated by \(Q_i\) , with the same natural parameter \(\theta \) for any \(i=1,\ldots ,n\) , the equality sign holds. The present note tackles the inverse problem, namely “does the equality \(D_{\lambda }(P_1*\cdots *P_n||Q_1*\cdots *Q_n)=\sum _{i=1}^nD_{\lambda }(P_i||Q_i)\) imply that \(P_i\) belongs to the natural exponential family generated by \(Q_i\) for every \(i=1,\ldots ,n\) ?” The answer is not always positive and depends on the set of solutions of a generalization of the celebrated Cauchy functional equation. We discuss in particular the case \(P_1=\cdots =P_n=P\) and \(Q_1=\cdots =Q_n=Q\) , with \(n=2\) and \(n=\infty \) , the latter meaning that the equality holds for all n. Our analysis is mainly devoted to P and Q concentrated on non-negative integers, and P and Q with densities with respect to the Lebesgue measure. The results cover the Kullback–Leibler divergence (KL), this being the Rényi divergence for \(\lambda = 1\) . We also show that the only f-divergences such that \(D_{f}(P^{*2}||Q^{*2})=2D_{f}(P||Q)\) , for P and Q in the same exponential family, are mixtures of KL divergence and its dual.
