Summary
A sufficient condition for statistical completeness of location families generated by a probability density in euclidean space is given. As an application, completeness of families generated by a symmetric stable law is proved. Our criterion, complementing a classical result of Wiener and recent work of Isenbeck and Rüschendorf, is in terms of regularity of the generating density and zerofreeness of its characteristic function. Its proof rests on a local version of the convolution theorem for Fourier transforms of tempered distributions. A more general version of the criterion is applicable to apparently different problems, as is illustrated by giving a simultaneous proof of a theorem on translated moments by P. Hall and a uniqueness result of M. Riesz in potential theory.
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Mattner, L. Completeness of location families, translated moments, and uniqueness of charges. Probab. Th. Rel. Fields 92, 137–149 (1992). https://doi.org/10.1007/BF01194918
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DOI: https://doi.org/10.1007/BF01194918