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Consider an infinite sequence $(U_n)_{n\in\mathbb{N}}$ of independent Cauchyrandom variables, defined by a sequence $(\delta_n)_{n\in\mathbb{N}}$ oflocation parameters and a sequence $(\gamma_n)_{n\in\mathbb{N}}$ of scaleparameters. Let $(W_n)_{n\in\mathbb{N}}$ be another infinite sequence ofindependent Cauchy random variables defined by the same sequence of locationparameters and the sequence $(\sigma_n\gamma_n)_{n\in\mathbb{N}}$ of scaleparameters, with $\sigma_n\neq 0$ for all $n\in\mathbb{N}$. Using a result ofKakutani on equivalence of countably infinite product measures, we show thatthe laws of $(U_n)_{n\in\mathbb{N}}$ and $(W_n)_{n\in\mathbb{N}}$ areequivalent if and only if the sequence $(\vert\sigma_n\vert-1)_{n\in\mathbb{N}}$ is square-summable.

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations