Multiparameter Shimizu equation (Q1368812)

The author investigates the set of solutions of the integral equation \[ \log\widehat\mu(y)= \int^1_0 \log\widehat\mu(c^By)\kappa(dc) \] in the class of probability measures \(\mu\) in \(\mathbb{R}^d\). Here \(\widehat\mu\) is the characteristic function of \(\mu\), \(B\) is a matrix, \(\kappa\) is bounded Borel measure on \([0,1]\). In particular, the problem of characterization of distributions of independent identically distributed random vectors \(X_1,\dots, X_n\), \[ X_1\overset{\text{distr}} = C_1X_1+\cdots+ C_nX_n \] with diagonal matrices \(C_1,\dots, C_n\) is solved with the help of some properties of the mentioned integral equation. The main results are generalizations of the corresponding results for the one-dimensional case investigated by \textit{R. Shimizu} [Sankhyā, Ser. A 40, 319-332 (1978; Zbl 0422.62013)].