Consider an infinite sequence $(U_n)_{n\in\mathbb{N}}$ of independent Cauchy random variables, defined by a sequence $(δ_n)_{n\in\mathbb{N}}$ of location parameters and a sequence $(γ_n)_{n\in\mathbb{N}}$ of scale parameters. Let $(W_n)_{n\in\mathbb{N}}$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $(σ_nγ_n)_{n\in\mathbb{N}}$ of scale parameters, with $σ_n\neq 0$ for all $n\in\mathbb{N}$. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $(U_n)_{n\in\mathbb{N}}$ and $(W_n)_{n\in\mathbb{N}}$ are equivalent if and only if the sequence $(\vert σ_n\vert-1)_{n\in\mathbb{N}}$ is square-summable.
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