Gaussian distribution on a locally compact Abelian group

Gaussian distribution on a locally compact Abelian group is a generalization of the classical normal distribution. Roughly speaking, it extends the concept of a normal distribution from the Euclidean space to more general topological groups, i.e., when the group is $\mathbb {R} ^{n}$ , it coincides with the usual multivariate normal distribution. It was introduced by Kalyanapuram Rangachari Parthasarathy, Ranga Rao, and Srinivasa R. S. Varadhan in 1963^[1], see also ^[2]. This distribution plays an important role in the representation of characteristic functions of infinitely divisible distributions^[1], the arithmetic of probability distributions^[3], and in characterization problems of mathematical statistics on locally compact Abelian groups^[4].

Definition

Let $X$ be a second countable locally compact Abelian group, let $Y$ be its character group, and let $(x,y)$ denote the value of a character $y\in Y$ at an element $x\in X$ .

A probability distribution $\gamma$ on the group $X$ is called Gaussian if its characteristic function (Fourier transform) is of the form

${\hat {\gamma }}(y)=(x,y)\exp\{-\varphi (y)\},$

where $\varphi$ is a continuous non-negative function on $Y$ satisfying the functional equation

$\varphi (u+v)+\varphi (u-v)=2{\bigl [}\varphi (u)+\varphi (v){\bigr ]},\quad u,v\in Y.$

A Gaussian distribution is called symmetric if $x=0$ .

Properties

1. A Gaussian distribution is infinitely divisible^[1].

2. The support of a Gaussian distribution is a coset of a closed connected subgroup of $X$ ^[1].

3. A symmetric Gaussian distribution is a continuous homomorphic image of a Gaussian distribution on a linear space (either finite-dimensional $\mathbb {R} ^{n}$ or infinite-dimensional $\mathbb {R} ^{\infty }$ , the space of all sequences with product topology)^[5].

4. Let $X$ be connected. If $X$ is not locally connected, then every Gaussian distribution on $X$ is singular with respect to the Haar measure on $X$ , whereas if $X$ is locally connected and finite-dimensional, then any Gaussian distribution on $X$ is either absolutely continuous or singular with respect to the Haar measure on $X$ ^[5]. The corresponding question for infinite-dimensional locally connected groups remains open, although both types of Gaussian distributions on such groups can be constructed.

5. On finite-dimensional connected groups, any two Gaussian distributions are either mutually absolutely continuous or mutually singular^[5].

Gaussian distribution on a locally compact Abelian group

Definition

Properties

References

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