Gaussian distribution on a locally compact Abelian group

Apr 22, 2026 - 00:21
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Gaussian distribution on a locally compact Abelian group

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3. A symmetric Gaussian distribution is a continuous homomorphic image of a Gaussian distribution on a linear space (either finite-dimensional \mathbb{R}^n or infinite-dimensional \mathbb{R}^\infty, the space of all sequences with product topology)G. M. Feldman. ''Gaussian Distributions on Locally Compact Abelian Groups''. Theory of Probability and Its Applications, 23 (1979), pp. 529–542. doi:10.1137/112306.
3. A symmetric Gaussian distribution is a continuous homomorphic image of a Gaussian distribution on a linear space (either finite-dimensional \mathbb{R}^n or infinite-dimensional \mathbb{R}^\infty, the space of all sequences with product topology)G. M. Feldman. ''Gaussian Distributions on Locally Compact Abelian Groups''. Theory of Probability and Its Applications, 23 (1979), pp. 529–542. doi:10.1137/112306.


4. Let X be connected. If X is not [[locally connected space|locally connected]], then every Gaussian distribution on X is [[singular measure|singular]] with respect to the [[Haar measure]] on X, whereas if X is locally connected and finite-dimensional, then any Gaussian distribution on X is either [[Absolute continuity|absolutely continuous]] or singular. The corresponding question for infinite-dimensional locally connected groups remains open, although both types of Gaussian distributions can be constructed.
4. Let X be connected. If X is not [[locally connected space|locally connected]], then every Gaussian distribution on X is [[singular measure|singular]] with respect to the [[Haar measure]] on X, whereas if X is locally connected and finite-dimensional, then any Gaussian distribution on X is either [[Absolute continuity|absolutely continuous]] or singular with respect to the Haar measure on X. The corresponding question for infinite-dimensional locally connected groups remains open, although both types of Gaussian distributions on such groups can be constructed.


5. On finite-dimensional connected groups, any two Gaussian distributions are either mutually absolutely continuous or mutually singular.
5. On finite-dimensional connected groups, any two Gaussian distributions are either mutually absolutely continuous or mutually singular.
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