Mathematical Research Letters

Volume 9, Issue 5, September 2002 pp. 607-620.

Affinely infinitely divisible distributions and the embedding problem

Authors: S.G. Dani and Klaus Schmidt
Author institution: Tata Institute of Fundamental Research, and Erwin Schrödinger Institute

Summary: Let $A$ be a locally compact abelian group and let $\mu$ be a probability measure on $A$. A probability measure $\lambda$ on $A$ is an {\it affine $k$-th root} of $\mu$ if there exists a continuous automorphism $\rho $ of $A$ such that $\rho^k=I$ (the identity transformation) and $\lambda *\rho(\lambda) *\rho^2 (\lambda)* \cdots *\rho^{k-1}(\lambda)=\mu$, and $\mu$ is {\it affinely infinitely divisible} if it has affine $k$-th roots for all orders. Clearly every infinitely divisible probability measure is affinely infinitely divisible. In this paper we prove the converse for connected abelian Lie groups: Every affinely infinitely divisible probability measure on a connected abelian Lie group $A$ is infinitely divisible. If $G$ is a locally compact group, $A$ a closed abelian subgroup of $G$, and $\mu$ a probability measure on $G$ which is supported on $A$ and infinitely divisible on $G$, we give sufficient conditions which ensure that $\mu$ is infinitely divisible on $A$.

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