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Gaussian distribution on a locally compact Abelian group

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Gaussian distribution on a locally compact Abelian group is a distribution on a second countable locally compact Abelian group which satisfies the conditions:

(i) is an infinitely divisible distribution;

(ii) if , where is the generalized Poisson 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 of a Gaussian distribution is a coset of a connected subgroup of .

Let be the character group of the group . A distribution on is Gaussian ([1]) if and only if its characteristic function can be represented in the form

,

where is the value of a character at an element , and is a continuous nonnegative function on satisfying the 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 continuous homomorphic 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 product topology) ([2][3]).

If a distribution can be embedded in a continuous one-parameter semigroup , of distributions on , then if and only if

for any neighbourhood of zero in the group ([4]).

Let be a connected group, and . If is not a locally connected, then is singular (with respect of a Haar distribution on ) ([2][3]). If is a locally connected and has a finite dimension, then is either absolutely continuous or singular. The question of the validity of a similar statement on locally connected groups of infinite dimension is open, although on such groups it is possible to construct both absolutely continuous and singular Gaussian distributions.

It is well known that two Gaussian distributions in a linear space are either mutually absolutely continuous or mutually singular. This alternative is true for Gaussian distributions on connected groups of finite dimension ([2][3]).

The following theorem is valid ([5]), which can be considered as an analogue of Cramer's theorem on the decomposition of the normal distribution for locally compact Abelian groups.

Cramer's theorem on the decomposition of the Gaussian distribution for locally compact Abelian groups

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Let be a random variable with values in a locally compact Abelian group with a Gaussian distribution, and let , where and are independent random variables with values in . The random variables and are Gaussian if and only if the group contains no subgroup topologically isomorphic to the circle group, i.e. the multiplicative group of complex numbers whose modulus is equal to 1.

References

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