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On the preservation of determinacy under convolution

Author: Christian Berg
Journal: Proc. Amer. Math. Soc. 93 (1985), 351-357
MSC: Primary 60E07; Secondary 44A35, 44A60
MathSciNet review: 770553
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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.

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Keywords: Determinate probability, Nevanlinna extremal measure, infinitely divisible measure, Carleman's condition
Article copyright: © Copyright 1985 American Mathematical Society