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The Heyde theorem on \(\mathbf a\)-adic solenoids. (English) Zbl 1285.60005
Summary: We prove the following analogue of the Heyde theorem for \(\mathbf a\)-adic solenoids. Let \( \xi_1, \xi_2\) be independent random variables with values in an \({\mathbf a}\)-adic solenoid \( \varSigma_{\mathbf a}\) and with distributions \(\mu_1, \mu_2\). Let \(\alpha_j, \beta_j\) be topological automorphisms of \(\varSigma_{\mathbf a}\) such that \(\beta_1\alpha^{-1}_1 \pm \beta_2\alpha^{-1}_2\) are topological automorphisms of \(\varSigma_{\mathbf a}\), too. Assuming that the conditional distribution of the linear form \(L_2=\beta_1\xi_1 + \beta_2\xi_2\) given \(L_1=\alpha_1\xi_1 + \alpha_2\xi_2\) is symmetric, we describe the possible distributions \(\mu_1, \mu_2\).

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
62E10 Characterization and structure theory of statistical distributions
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