  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 parameterization 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.
