Gaussian distributions in the sense of Bernstein on locally compact abelian groups (Q653025)

In a series of investigations the validity of the classical Kac-Bernstein characterization of Gaussian laws was investigated for abelian locally compact groups (e.g., by \textit{L. A. Corwin} [J. Funct. Anal. 5, 412-427 (1970; Zbl 0192.48802); ibid. 6, 481--505 (1970; Zbl 0202.40603)], \textit{H. Heyer} and \textit{C. Rall} [Math. Z. 128, 343--361 (1972; Zbl 0246.60003)], \textit{A. L. Rukhin} [Math. Notes 6(1969), 638--641 (1970); translation from Mat. Zametki 6, 301--307 (1969; Zbl 0195.20301); Proc. Steklov Inst. Math. 111 (1970), 59--129 (1972); translation from Trudy Mat. Inst. Steklov 111, 52--109 (1970; Zbl 0283.62007)] and the author [Theory Probab. Appl. 31, No. 1, 40--49 (1987); translation from Teor. Veroyatn. Primen. 31, No. 1, 47--58 (1986; Zbl 0678.60007)]). Recall (Theorem of Kac-Bernstein) that on vector spaces independent identically distributed random variables \(\xi_1, \xi_2\) are Gaussian distributed iff \(\xi_1-\xi_2\) and \(\xi_1+\xi_2\) are independent. For abelian groups \(X\) such a property characterizes Gaussian laws convoluted with idempotents if certain additional conditions are satisfied, e.g., if \(x\mapsto x^2\) is an automorphism of \(X\), or if \(X_0\) contains at least one element of order 2. (\(\mu\) is called idempotent if \(\mu^2 =\mu\star\varepsilon_x\).) In the present publication (without proofs) the author extends this class of groups admitting a finite number of elements of order 2 in \(X_0\). In that case the laws satisfying the Kac-Bernstein condition are representable as \(\gamma\star\delta\star m_K\), with Gaussian \(\gamma\), a normalized Haar measure \(m_K\) on a compact subgroup and a signed measure \(\delta\) satisfying \(\delta^2=\varepsilon_0\). Finally the author extends zero-one-laws for Gaussian distributions to locally compact groups with finite dimensional connected component \(X_0\).