## The Annals of Probability

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

Marcel de Jeu

#### Abstract

We prove in a direct fashion that a multidimensional probability measure $\mu$ 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 $L_p$-spaces for $1\leq p<\infty$. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under $\mu$. 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.

#### Article information

Source
Ann. Probab., Volume 31, Number 3 (2003), 1205-1227.

Dates
First available in Project Euclid: 12 June 2003

https://projecteuclid.org/euclid.aop/1055425776

Digital Object Identifier
doi:10.1214/aop/1055425776

Mathematical Reviews number (MathSciNet)
MR1988469

Zentralblatt MATH identifier
1050.44003

#### Citation

de Jeu, Marcel. Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Probab. 31 (2003), no. 3, 1205--1227. doi:10.1214/aop/1055425776. https://projecteuclid.org/euclid.aop/1055425776

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