Gaussian distribution on a locally compact Abelian group

Probability distributions

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 . Let be the character group of the group . A distribution on is Gaussian if and only if itscharacteristic function can be represented in the form , where is thevalue of a character at an element , and is a continuous nonnegative function on satisfyingthe equation . A Gaussian distribution is called symmetric if . Denote by the set of Gaussian distributions on the group , and by the set of symmetric Gaussian distribution on. If , then is a continuoushomomorphic image of a Gaussian distribution in a real linear space.This space is either finite dimensional or infinite dimensional(the space of all sequences of real numbers in the producttopology). If a distribution can be embedded in a continuousone-parameter semigroup , of distributions on, then if and only if for any neighbourhood of zero in the group. Let be a connected group, and. If is not a locally connected, then is singular (with respect of a Haar distribution on ). If is a locally connected and has a finitedimension, then is either absolutely continuous orsingular. The question of the validity of a similar statement onlocally connected groups of infinite dimension is open, although onsuch groups it is possible to construct both absolutely continuousand singular Gaussian distributions. It is well known that two Gaussian distributions in a linear spaceare either mutually absolutely continuous or mutually singular. Thisalternative is true for Gaussian distributions on connected groupsof finite dimension. The following theorem is valid, which can be consideredas an analogue of Cramer's theorem on the decomposition of the normal distribution for locally compact Abelian groups. (Wikipedia).

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