Monotone decreasing distance between distributions of sums of unfair coins and a fair coin (Q2744581)

It is observed that for several distance functions like the Kullback-Leibler divergence, the inequality \(d(\mu_1* \mu_2,\lambda)\leq d(\mu_1,\lambda)\) holds for all probability measures \(\mu_1\), \(\mu_2\) and the uniform distribution \(\lambda\) on the cyclic group \(\mathbb{Z}_n\) with \(n\in\mathbb{N}\). As a consequence the authors claim that the uniform distribution on \(\mathbb{Z}_n\) can be written as \(\lambda_1= \mu_1* \mu_2\) only for \(\lambda= \mu_1\) or \(\lambda= \mu_2\). However, as \(\mathbb{Z}_6\simeq \mathbb{Z}_2\times \mathbb{Z}_3\) easily leads to a counterexample, this paper has to be considered with great care!