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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2002-13
April 2002

The Moment Problem


Jørgen Hoffmann-Jørgensen


Recall that a probability measure $mu$ on the real line with finite moments of all orders is called determinate if $mu = nu$ for any probability measure $nu$ with the same moments an $mu$. There are three classical criteria for determinacy due to O. Perron, M. Riesz and T. Carleman. The Perron condition states that the Laplace transform of $mu$ is finite in an open interval around 0 and it is the most commonly criteria used in probability theory. However, the Riesz and Carleman conditions are weaker than the Perron condition but difficult to apply due to the fact that they require precise estimates of the moments. The objective of this paper is to provide equivalent forms of the Riesz and Carleman conditions which are easier to apply. In particular, I shall show that each of the two conditions are equivalent to integrability of at least one function in a specified class of functions.

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