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Short Communications
On the Representation of Infinitely Divisible Distributions on Locally Compact Abelian Groups
K. R. Parthasarathya,
V. V. Sazonovb a Calkutta
b Moscow
Abstract:
Let
$X$ be a locally compact abelian separable metric group and
$Y$ the group of characters on
$X$ be a locally compact abelian separable metric group and
$Y$. For any
$x\in X$,
$y\in Y$ let
$(x,y)$ be the value of the character
$y$ at
$x$. It is shown that the characteristic function
$\tilde\mu$ of any infinitely divisible distribution
$\mu$ on
$X$ has the form
$$
\tilde\mu(y)=\left( {x_0,y}\right)\tilde\lambda (y)\exp\left\{{\int {[(x,y)-1-ig(x,y)]dF(x)-\Phi(y)}}\right\},
$$
where
$x_0$ is an element of
$X$,
$\tilde\lambda$ is the characteristic function of the normalised Haar measure
$\lambda$ of a compact subgroup,
$g$ is a special function on
$X\times Y$ not depending on
$\mu $,
$F$ is a measure with finite mass outside every neighbourhood of the identity of
$X$ which integrates
$1-\operatorname{Re}(x,y)$ for each
$y\in Y$, and
$\Phi$ is a non-negative continuous function on
$Y$ satisfying the identity
$$
\Phi \left( {y_1 + y_2 } \right) + \Phi \left( {y_1 - y_2 } \right) = 2\left[ {\Phi \left( {y_1 } \right) + \Phi \left( {y_2 } \right)} \right],\quad y_1 ,y_2 \in Y.
$$
This is an extension of an earlier result of K. R. Parthasarathy et al. [1].
Received: 21.03.1963