Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

By Marcel de Jeu. 2001

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We prove in a direct fashion that a multidimensional probability measure μ is determinate if the higher-dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated Lp-spaces for 1≤p<∞. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under μ. We give practical examples of such weights, based on their classification. As in the one-dimensional case, the results on determinacy of measures supported on \Rn lead to sufficient conditions for determinacy of measures supported in a positive convex cone, that is, the higher-dimensional analogue of determinacy in the sense of Stieltjes.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08c62a693f88ee2125607241782ad00932a6eec4

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

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General Rough integration, Levy Rough paths and a Levy--Kintchine type formula (2012, background): “dly advertized by Lyons, see e.g. [8] and also [16].) In our situation, exponential integrability of K will entail that the law of any such projection enjoyes exponential integrability, and thus (see [4] and the refences therein) yields a suﬃcient condition under which the expected signature describ”