We prove that the $f$-divergences between univariate Cauchy distributions are all symmetric, and can be expressed as strictly increasing scalar functions of the symmetric chi-squared divergence. We report the corresponding scalar functions for the total variation distance, the Kullback-Leibler divergence, the squared Hellinger divergence, and the Jensen-Shannon divergence among others. Next, we give conditions to expand the $f$-divergences as converging infinite series of higher-order power chi divergences, and illustrate the criterion for converging Taylor series expressing the $f$-divergences between Cauchy distributions. We then show that the symmetric property of $f$-divergences holds for multivariate location-scale families with prescribed matrix scales provided that the standard density is even which includes the cases of the multivariate normal and Cauchy families. However, the $f$-divergences between multivariate Cauchy densities with different scale matrices are shown asymmetric. Finally, we present several metrizations of $f$-divergences between univariate Cauchy distributions and further report geometric embedding properties of the Kullback-Leibler divergence.

翻译：我们证明,在未变差的Cafarite Cavous 分配之间,美元差异是完全对称的,可以表述为严格增加对称质量差差分的缩放功能。我们报告总变差距离、Kullback-Leiber差异、平方加仑差和Jensen-Shannon差异等相应的缩放函数。接下来,我们为扩大美元差异提供了条件,将美元差异扩大为一系列无穷无穷的更高级力量差异,并展示了表示宽度分配之间美元差异的调和泰勒系列的调和性功能。我们随后报告,美元-维值的对称属性对于具有规定矩阵尺度的多变数地点规模家庭来说,条件是标准密度甚至包括多变正常家庭和宽度家庭的情况。然而,多种变差的Caugustir 密度和不同规模矩阵之间的美元差异是不对称的。最后,我们提出了数个正位分布差异化的基数。