Gaussian distribution on a locally compact Abelian group is a distribution on a secondcountable locally compact Abelian group which satisfies theconditions: (i) is an infinitely divisible distribution; (ii) if , where is the generalizedPoisson distribution, associated with a finite measure , and is an infinitely divisible distribution, then the measure is degenerated at zero. This definition of the Gaussian distribution for the group coincides with the classical one. The support ofa Gaussian distribution is a coset of a connected subgroup of . , for any neighbourhood of zero in the group .