Gaussian distribution on a locally compact Abelian group

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

(i) $\gamma$ is an infinitely divisible distribution;

(ii) if $\gamma =e(F)*\nu$, where $e(F)$ is the generalized Poisson distribution, associated with a finite measure $F$, and $\nu$ is an infinitely divisible distribution, then the measure $F$ is degenerated at zero.

This definition of the Gaussian distribution for the group $X={ℝ}^n$ coincides with the classical one. The support of a Gaussian distribution is a coset of a connected subgroup of $X$.

Let $Y$ be the character group of the group $X$. A distribution $\gamma$ on $X$ is Gaussian () if and only if its characteristic function can be represented in the form

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

where $(x,y)$ is the value of a character $y\in Y$ at an element $x\in X$, and $\phi (y)$ is a continuous nonnegative function on $Y$ satisfying the equation $\phi (u+v)+\phi (u-v)=2[\phi (u)+\phi (v)],u,v\in Y$.

A Gaussian distribution is called symmetric if $x=0$. Denote by $\mathrm{\Gamma }(X)$ the set of Gaussian distributions on the group $X$, and by ${\mathrm{\Gamma }}^s(X)$ the set of symmetric Gaussian distribution on $X$. If $\gamma \in {\mathrm{\Gamma }}^s(X)$, then $\gamma$ 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) ().

If a distribution $\gamma$ can be embedded in a continuous one-parameter semigroup ${\gamma }_t,t\ge 0$, of distributions on $X$, then $\gamma \in \mathrm{\Gamma }(X)$ if and only if

$\underset{t\to 0}{\lim }\frac{{\gamma }_t(X\mathrm{\setminus }U)}{t}=0$ for any neighbourhood of zero $U$ in the group $X$ ().

Let $X$ be a connected group, and $\gamma \in \mathrm{\Gamma }(X)$. If $X$ is not a locally connected, then $\gamma$ is singular (with respect of a Haar distribution on $X$) (). If $X$ is a locally connected and has a finite dimension, then $\gamma$ 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 ().

The following theorem is valid (), 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

Let $\xi$ be a random variable with values in a locally compact Abelian group $X$ with a Gaussian distribution, and let $\xi ={\xi }_1+{\xi }_2$, where ${\xi }_1$ and ${\xi }_2$ are independent random variables with values in $X$. The random variables ${\xi }_1$ and ${\xi }_2$ are Gaussian if and only if the group $X$ contains no subgroup topologically isomorphic to the circle group, i.e. the multiplicative group of complex numbers whose modulus is equal to 1.