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On the Limit Behavior of Compositions of Measures in the Plane and Space of Lobachevsky

1962; Society for Industrial and Applied Mathematics; Volume: 7; Issue: 2 Linguagem: Inglês

10.1137/1107018

1095-7219

V. N. Tutubalin,

Advanced Differential Equations and Dynamical Systems

Let O be the origin in $L_2 ({\text{or}}\ L_3 ),\theta _x $ a translation taking O into $x,\mu _1 $ and $\mu _2 $ measures satisfying the conditions $\mu _1 (\Gamma ) = \mu _1 (h\Gamma )$ and $\mu _2 (\Gamma ) = \mu _2 (h\Gamma )$ for each $h,hO = O$. The measure $\mu _1 * \mu _2 (\Gamma ) = \int {\mu _1 } (\theta _x^{ - 1} \Gamma )\mu _2 (dx)$ does not depend on the choice of $\theta _x $. Let $\eta = \rho (O,x)$, ($\rho $ is the invariant metric), $\mu ^n = \mu \underbrace { * \cdots * }_n\mu ,F_\mu (y) = \mu \{ x:\eta < y\} $. If the measures $\mu $ and $\nu$ satisfy the condition $a_1 (\mu ) = a_1 (\nu ), a_2 (\mu ) = a_2 (\nu )$ (where $a_1 $ and $a_2 $ are defined for $L_3 $ in § 1 and for $L_2 $ in § 3), one can show that $\sup _{0 \leqq y < \infty } | {F_{\mu ^n } (y) - F_{\nu ^n } (y)} |_{n \to \infty } \to 0$. The measure $\nu$ is called infinity divisible if one can find for each $\varepsilon > 0$ the measures $\mu _1^{(\varepsilon )} , \cdots ,\mu _{n_\varepsilon }^{(\varepsilon )} $ such that $\nu = \mu _1^{(\varepsilon )} * \cdots * \mu _{n_\varepsilon }^{(\varepsilon )} $ and $\mu _i^{(\varepsilon )} (x:\eta > \varepsilon ) \leqq \varepsilon , i = 1,2, \cdots , n_\varepsilon $. If the measure $\mu $ satisfies the condition $a_1 (\mu ) \leqq a_2 (\mu ) - a_1^2 (\mu )$ for $L_3 $ and $2a_1 (\mu ) \leqq a_2 (\mu ) - a_1^2 (\mu )$ for $L_2 $, one can find an infinitely divisible measure $\nu $ such that $\sup _{0 \leqq y < \infty } | {F_{\mu ^n } (y) - F_{\nu ^n } (y)} |_{n \to \infty } \to 0$.