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Let X be a second countable locally compact Abelian group, and let μ be a Gaussian measure in the sense of Bernstein on X. Under the assumption that the connected component of zero of X contains a finite number of elements of order 2, we prove that μ is a convolution of a Gaussian measure, the Haar distribution of a compact subgroup of X, and a signed measure supported in the subgroup of X generated by elements of order 2. We describe the support of μ on an arbitrary group X. We prove that if the connected component of zero of X has finite dimension, then the zero-one law holds for μ under the assumption that μ has no idempotent factors. [ABSTRACT FROM AUTHOR]

In this paper we prove that the unboundedness of nonconstant homogeneous stochastic processes with independent increments is a natural corollary of the Kolmogorov zero-one law. The known proof of this statement is based on a deep preliminary investigation of stochastic processes with independent increments. [ABSTRACT FROM AUTHOR]