On Gaussian distributions on locally compact Abelian groups

G. M. Fel'dman

Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 3, pp. 548-563

Résumé

Let $X$ be a connected locally compact separable metric Abelian group, $\mu$ be a symmetric Gaussian distribution (G. d.) on $X$. It is proved that if $X$ is a group of finite dimension $l$, then there exist a continuous homomorphism $p:R^l\to X$ (independent of $\mu$) and a G. d. $\mathbf M$ on $R^l$ such that $\mu=p(\mathbf M)$. If $X$ is an infinite-dimensional group, then there exist a continuous homomorphism $p:R^{\infty}\to X$ (independent of $\mu$) and a G. d. $\mathbf M$ on $R^{\infty}$ such that $\mu=p(\mathbf M)$; here $R^{\infty}$ is the space of all real sequences with the topology determined by the coordinate convergence. By means of these results, the singularity of G. d.'s (with respect to the Haar measure) on not locally connected groups is proved. It is also proved that any two G. d.'s on finite-dimensional groups are either mutually absolutely continuous or singular. For infinite-dimensional groups an analogous result is established under the assumption that the group in question contains no subgroup isomorphic to the circle group $T$.