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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

Citations:

Zbl 0030.02303
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References:

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