Abstract
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 $(\left \vert{\sigma _n} \right \vert -1)_{n\in \mathbb{N} }$ is square-summable.
Highlights
Let (Un)n∈N be a sequence of independent, Cauchy random variables defined on a probability space (Ω, F, P), where each random variable Un has the density γn[2] fn(x, δn, γn) = πγn (x − δn)2 + γ2 . n (1.1)The distribution of Un is parametrised by the location and scale parameters δn and γn respectively
Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequencen∈N of scale parameters, with σn = 0 for all n ∈ N
Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn| − 1)n∈N is square-summable
Summary
Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn = 0 for all n ∈ N. For every Un, we may define another Cauchy random variable Wn by multiplicatively perturbing the scale parameter γn by σn = 0, so that the pair (δn)n∈N and (σnγn)n∈N determines the law of (Wn)n∈N.
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