Abstract We give an equivalence-singularity criterion for infinite products of Cauchy measures under simultaneous shifts of the location and scale parameters. Our result is an extension of Lie and Sullivan’s result giving an equivalence-singularity criterion under dilations of scale parameters. Our proof utilizes McCullagh’s parametrization of the Cauchy distributions and maximal invariant, and a closed-form formula of the Kullback–Leibler divergence between two Cauchy measures given by Chyzak and Nielsen.

中文翻译：

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柯西测度无穷积的等价判据

摘要 我们给出了在位置和尺度参数同时移动的情况下柯西测度的无穷积的等价奇点判据。我们的结果是 Lie 和 Sullivan 的结果的扩展，给出了尺度参数膨胀下的等价奇异性标准。我们的证明利用了麦卡拉对柯西分布和最大不变量的参数化，以及由 Chyzak 和 Nielsen 给出的两个柯西测度之间的 Kullback-Leibler 散度的封闭式公式。