Remarks on converse Carleman and Krein criteria for the classical moment problem

Abstract The key theme is converse forms of criteria for deciding determinateness in the classical moment problem. A method of proof due to Koosis is streamlined and generalized giving a convexity condition under which moments satisfying implies that c a positive constant. A contrapositive version is proved under a rapid variation condition on f (x), generalizing a result of Lin. These results are used to obtain converses of the Stieltjes versions of the Carleman and Krein criteria. Hamburger versions are obtained which relax the symmetry assumption of Koosis and Lin, respectively. A sufficient condition for Stieltjes determinateness of a discrete law is given in terms of its mass function. These criteria are illustrated through several examples.

[1]  Paul Koosis,et al.  The Logarithmic Integral I: Frontmatter , 1988 .

[2]  D. Applebaum Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[3]  A. Pakes Length Biasing and Laws Equivalent to the Log-Normal , 1996 .

[4]  Jong-Wuu Wu,et al.  Criteria for the Unique Determination of Probability Distributions by Moments , 2001 .

[5]  Jordan Stoyanov,et al.  Counterexamples in Probability , 1989 .

[6]  Gwo Dong Lin,et al.  On the moment problems , 1997 .

[7]  Eric V. Slud,et al.  The Moment Problem for Polynomial Forms in Normal Random Variables , 1993 .

[8]  P. R. Nelson The algebra of random variables , 1979 .

[9]  Z. Nehari Bounded analytic functions , 1950 .

[10]  T. Gamelin,et al.  The Logarithmic Integral , 2001 .

[11]  Jordan Stoyanov,et al.  Krein condition in probabilistic moment problems , 2000 .

[12]  SOME REMARKS ON THE MOMENT PROBLEM (II) , 1963 .

[13]  A. Pakes Characterization by invariance under length-biasing and random scaling , 1997 .

[14]  Christian Berg,et al.  Indeterminate moment problems and the theory of entire functions , 1995 .

[15]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[16]  A maximum entropy approach to the classical moment problem , 1992 .

[17]  Paul Koosis,et al.  The Logarithmic Integral , 1986 .

[18]  Henrik L. Pedersen On Krein's Theorem for Indeterminacy of the Classical Moment Problem , 1998 .

[19]  V. Zolotarev One-dimensional stable distributions , 1986 .

[20]  A note on the Carleman condition for determinacy of moment problems , 1987 .