Volume 6, Number 5
In 1944 M.G. Krein proposed a condition throwing light on the moment problem for absolutely continuous probability distributions. This condition, implying non-uniqueness, is expressed in terms of a normalized logarithmic integral of the density and has different forms in the Hamburger moment problem (for distributions on the whole real line) and in the Stieltjes moment problem (for distributions on the positive real line). Other forms of the Krein condition, together with new conditions (smoothing and growth condition on the density) suggested by G.D. Lin and based on a work by H. Dym and H.P. McKean, led to a unique solution to the moment problem. We present new results, give new proofs of previously known results and discuss related topics.
 Akhiezer, N.I. (1965) The Classical Moment Problem. New York: Hafner.
 Berg, C. (1988) The cube of the normal distribution is indeterminate. Ann Probab., 16, 910-913.
Mathematical Reviews (MathSciNet): MR929086
 Berg, C. (1995) Indeterminate moment problems and the theory of entire functions. J. Comput. Appl. Math., 65, 27-55.
 Berg, C. (1998a) On some indeterminate moment problems for measures on a geometric progression. J. Comput. Appl. Math., 99, 67-75.
 Berg, C. (1998b) From discrete to absolutely continuous solutions of indeterminate moment problems. Preprint No. 11, September 1998, University of Copenhagen.
 Berg, C. and Christensen, J.P.R. (1981) Density questions in the classical theory of moments. Ann. Inst. Fourier, 31, 99-114.
Mathematical Reviews (MathSciNet): MR638619
 Crow, E.L. and Shimizu, K. (1988) Lognormal Distributions: Theory and Applications. New York: Marcel Dekker.
Mathematical Reviews (MathSciNet): MR939191
 Devroye, L. (1986) Non-Uniform Random Variate Generation. New York: Springer-Verlag.
Mathematical Reviews (MathSciNet): MR836973
 Dym, H. and McKean, H.P. (1976) Gaussian Processes, Function Theory and the Inverse Spectral Problem. New York: Academic Press.
Mathematical Reviews (MathSciNet): MR448523
 Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. New York: Wiley.
Mathematical Reviews (MathSciNet): MR270403
 Gradshteyn, I.S. and Ryzhik, I.M. (1980) Tables of Integrals, Series and Products. New York: Academic Press.
 Heyde, C.C. (1963) On a property of the lognormal distribution. J. Roy. Statist. Soc. Ser. B, 29, 392- 393.
Mathematical Reviews (MathSciNet): MR171336
 Koosis, P. (1988) The Logarithmic Integral, Vol. 1, Cambridge: Cambridge University Press.
Mathematical Reviews (MathSciNet): MR961844
 Krein, M.G. (1944) On one extrapolation problem of A.N. Kolmogorov. Dokl. Akad. Nauk SSSR, 46(8), 339-342 (in Russian).
 Krein, M.G. and Nudelman, A.A. (1977). The Markov Moment Problem and Extremal Problems. Providence, RI: American Mathematical Society. Landau, H.J. (ed.) (1987) Moments in Mathematics, Proc. Sympos. Appl. Math. 37. Providence, RI: American Mathematical Society.
Mathematical Reviews (MathSciNet): MR458081
 Leipnik, R. (1981) The lognormal distribution and strong nonuniqueness of the moment problem. Theory Probab. Appl., 26, 850-852.
Mathematical Reviews (MathSciNet): MR636784
 Lin, G.D. (1997) On the moment problem. Statist. Probab. Lett., 35, 85-90.
 Lin, G.D. and Huang, J.S. (1997) The cube of the logistic distribution is indeterminate. Austral. J. Statist., 39, 247-252.
 Lukacs, E. (1970) Characteristic Functions, 2nd edn. London: Griffin.
Mathematical Reviews (MathSciNet): MR346874
 Pakes, A.G. and Khattree, R. (1992) Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist., 34, 307-322.
 Pedersen, H.L. (1998) On Krein's theorem for indeterminacy of the classical moment problem. J. Approx. Theory, 95, 90-100.
 Prohorov, Yu.V. and Rozanov, Yu.A. (1969) Probability Theory. Berlin: Springer-Verlag.
Mathematical Reviews (MathSciNet): MR251754
 Sapatinas, Th. (1995) Identifiability of mixtures of power-series distributions and related characterizations. Ann. Inst. Statist. Math., 47, 447-459.
 Seshadri, V. (1993) The Inverse Gaussian Distribution. Oxford: Clarendon Press.
 Shohat, J.A. and Tamarkin, J.D. (1943) The Problem of Moments. Providence, RI: American Mathematical Society.
Mathematical Reviews (MathSciNet): MR8438
 Simon, B. (1998) The classical moment problem as a self-adjoint finite difference operator. Adv. Math., 137, 82-203.
 Slud, E.V. (1993) The moment problem for polynomial forms of normal random variables. Ann. Probab., 21, 2200-2214.
 Stieltjes, T.J. (1894) Recherches sur les fractions continués. Ann. Fac. Sci. Toulouse, 8, 1-122.
 Stoyanov, J. (1997) Counterexamples in Probability, 2nd edn. Chichester: Wiley.
Mathematical Reviews (MathSciNet): MR930671
 Stoyanov, J. (1999) Inverse Gaussian distribution and the moment problem. J. Appl. Statist. Sci., 9, 61- 71.
 Targhetta, M.L. (1990) On a family of indeterminate distributions. J. Math. Anal. Appl., 147, 477- 479.