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Criteria for the Unique Determination of Probability Distributions by Moments
Author(s) -
Pakes Anthony G.,
Hung WenLiang,
Wu JongWuu
Publication year - 2001
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00158
Subject(s) - mathematics , infinity , mathematical proof , moment (physics) , zero (linguistics) , probability density function , sequence (biology) , power law , function (biology) , moment generating function , power (physics) , mathematical analysis , pure mathematics , combinatorics , statistics , quantum mechanics , geometry , physics , linguistics , philosophy , evolutionary biology , biology , genetics
A positive probability law has a density function of the general form Q ( x )exp(− x 1/λ L ( x )), where Q is subject to growth restrictions, and L is slowly varying at infinity. This law is determined by its moment sequence when λ< 2, and not determined when λ> 2. It is still determined when λ= 2 and L ( x ) does not tend to zero too quickly. This paper explores the consequences for the induced power and doubled laws, and for mixtures. The proofs couple the Carleman and Krein criteria with elementary comparison arguments.