an:04082704
Zbl 0662.62008
Dunau, Jean-Louis; S??nateur, Henri
Une caract??risation du type de la loi de Cauchy-conforme sur \({\mathbb{R}}^ n\). (A characterization of the type of the conformal Cauchy law on \({\mathbb{R}}^ n)\)
FR
Probab. Theory Relat. Fields 77, No. 1, 129-135 (1988).
00166223
1988
j
62E10
type of probability measures; characterization; transformations; similarities; translations; conformal Cauchy distribution; invariance; non-atomic measure
Let \(\mu\) be a measure defined on \(R^ n\) and consider the set of the images of \(\mu\) under similarities and translations of \(R^ n\). The authors term this set as the type of \(\mu\) and show that the conformal Cauchy distribution defined by the density
\[
f(x)=c/(1+\| x\|^ 2)^ n,\quad x\in R^ n,
\]
is characterized by the invariance of the type of a non-atomic measure \(\mu\) on \(R^ n\) under inversions of \(R^ n\).
E.Xekalaki