Geodesic
Independent linear statistics on the two-dimensional torus
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 3-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $X=\mathbf T^2$ be the two-dimensional torus, $\mathrm{Aut}(\mathbf T^2)$ be the group of topological automorphisms of $\mathbf T^2$, $\Gamma(\mathbf T^2)$ be the set of Gaussian distributions on $\mathbf T^2$, and $\xi_1$, $\xi_2$ be independent random variables taking on values in $\mathbf T^2$ and having distributions $\mu_j$ with the nonvanishing characteristic functions. Consider $\delta\in\mathrm{Aut}(\mathbf T^2)$ and assume that the linear forms $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\delta\xi_2$ are independent. We give the description of possible distributions $\mu_j$ depending on $\delta$. In particular we give the complete description of $\delta$ such that the independence of $L_1$ and $L_2$ implies that $\mu_1,\mu_2\in\Gamma(\mathbf T^2)$. It turned out that the corresponding Gaussian distributions are either degenerate or concentrated on shifts of the same dense in $\mathbf T^2$ one-parameter subgroup.
Keywords: independent linear statistics, two-dimensional torus, topological automorphism. 
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M. V. Mironyuk; G. M. Feldman. Independent linear statistics on the two-dimensional torus. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 3-20. http://geodesic.mathdoc.fr/item/TVP_2007_52_1_a0/

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