  Consider an infinite sequence $(U_n)_{n\in\mathbb{N}}$ of independent Cauchy
random variables, defined by a sequence $(\delta_n)_{n\in\mathbb{N}}$ of
location parameters and a sequence $(\gamma_n)_{n\in\mathbb{N}}$ 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 $(\sigma_n\gamma_n)_{n\in\mathbb{N}}$ of scale
parameters, with $\sigma_n\neq 0$ 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 $(\vert
\sigma_n\vert-1)_{n\in\mathbb{N}}$ is square-summable.
