## Quasi-invariance of countable products of Cauchy measures under non-unitary dilations.(English)Zbl 1390.60149

Summary: 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 S. Kakutani [Ann. Math. (2) 49, 214–224 (1948; Zbl 0030.02303)] 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 $$(|{\sigma_n}| -1)_{n\in \mathbb{N}}$$ is square-summable.

### MSC:

 60G30 Continuity and singularity of induced measures 60G20 Generalized stochastic processes 60E07 Infinitely divisible distributions; stable distributions

Zbl 0030.02303
Full Text:

### References:

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