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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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