On the preservation of determinacy under convolution

Abstract:

In 1959 Devinatz remarked that if $\mu *\nu$ is a determinate measure on the real line, then so are $\mu$ and $\nu$. It is shown here how this follows from a theorem of M. Riesz, and also how it can be extended to $d$ dimensions. Recently Diaconis raised the question whether the converse is true. We answer this in the negative by producing a determinate measure $\nu$ on the real line such that $\nu *\nu$ is indeterminate. The relation to previous work of Heyde and to the condition of Carleman is discussed.