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.
Categories: math.PR math.ST stat.TH