On a characterization theorem for a Cauchy distribution on Abelian groups
This paper generalizes the following classical characterization of Cauchy distributions on \(\mathbb{R}\) to second countable, locally compact Abelian groups \(G\): If \(X_1,\dots, X_n\) are i.i.d. random variables with distribution \(\mu\) such that \(nX_1\) and \(X_1+\cdots+ X_n\) are identically distributed for two values \(n_1\), \(n_2\) of \(n\) with \(\ln n_1/\ln n_2\not\in \mathbb{Q}\), then \(\mu\) is a Cauchy distribution. For the generalization, \(\mu\in M^1(G)\) is called a Cauchy distribution if \(\widehat \mu(y)= y(x)\cdot \exp(- \sqrt{\varphi(y)})\) for some \(x\in G\) and some nonnegative quadratic form \(\varphi\) on the dual group \(\widehat G\). It is proved that, up to a technical additional term concerning Haar measures of compact subgroups of \(G\), the characterization above remains valid if and only if the connected component \(C\) of the identity in \(G\) has \(\dim C\leq 1\), and if there is no element in \(C\) of order \(p\) for some prime \(p\).
scientific article; zbMATH DE number 923492