On f-Divergences Between Cauchy Distributions

We prove that all
$f$
-divergences between univariate Cauchy distributions are symmetric. Furthermore, those
$f$
-divergences can be calculated as strictly increasing scalar functions of the chi-square divergence. We report a criterion which allows one to expand
$f$
-divergences as converging series of power chi divergences, and exemplifies the technique for some
$f$
-divergences between Cauchy distributions. In contrast with the univariate case, we show that the
$f$
-divergences between multivariate Cauchy densities are in general asymmetric although symmetric when the Cauchy scale matrices coincide. Then we prove that the square roots of the Kullback-Leibler and Bhattacharyya divergences between univariate Cauchy distributions yield complete metric spaces. Finally, we show that the square root of the Kullback-Leibler divergence between univariate Cauchy distributions can be isometrically embedded into a Hilbert space.