Method for constructing Stieltjes classes for M-indeterminate probability distributions

Abstract Let F be a distribution function with density f , finite moments of any positive integer order and such that the problem of moments for F has a nonunique solution ( F is M-indeterminate). We are looking for explicit Stieltjes classes S  = { f e  =  f [1 +  eh ], e  ∈ [−1, 1]} of distributions (here in terms of their densities) all sharing the same moments as those of F . We suggest a general method for constructing such classes. Then we apply our method to describe the M-indeterminacy of power transformations of popular distributions such as lognormal, inverse Gaussian and logistic. The Stieltjes classes presented here are new. We also find numerical values of the index of dissimilarity for some particular cases.

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

[2]  B. Simon.  The Classical Moment Problem as a Self-Adjoint Finite Difference Operator , 1998, math-ph/9906008.

[3]  Philippe Biane,et al.  Poissonian Exponential Functionals, q-Series, q-Integrals, and the Moment Problem for log-Normal Distributions , 2004 .

[4]  E. Crow,et al.  Lognormal Distributions: Theory and Applications , 1987 .

[5]  Aldo Tagliani,et al.  Numerical aspects of finite Hausdorff moment problem by maximum entropy approach , 2001, Appl. Math. Comput..

[6]  C. C. Heyde,et al.  On a Property of the Lognormal Distribution , 1963 .

[7]  Allan Gut,et al.  The Moment Problem , 2002, Encyclopedia of Special Functions: The Askey-Bateman Project.

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

[9]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[10]  Jordan Stoyanov,et al.  New Stieltjes classes involving generalized gamma distributions , 2004 .

[11]  S. Schwartz,et al.  Properties and evolution of aerosols with size distributions having identical moments , 1998 .

[12]  T. Stieltjes.  Recherches sur les fractions continues , 1995 .

[13]  Leonid Tolmatz.  Addenda: On the distribution of the square integral of the Brownian bridge , 2002 .

[14]  Jordan Stoyanov,et al.  Counterexamples in Probability , 1988 .

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

[16]  Christian Berg,et al.  Density questions in the classical theory of moments , 1981 .

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

[18]  Jordan Stoyanov,et al.  Stieltjes classes for moment-indeterminate probability distributions , 2004, Journal of Applied Probability.

[19]  Gwo Dong Lin.  On the moment problems [Statist. Probab. Lett. 35 (1997) 85–90]☆ , 2000 .

[20]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[21]  Gianluca Fusai,et al.  AN ACCURATE VALUATION OF ASIAN OPTIONS USING MOMENTS , 2002 .

[22]  Gwo Dong Lin,et al.  THE CUBE OF A LOGISTIC DISTRIBUTION IS INDETERMINATE , 1997 .