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Recent developments on the moment problem

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  • Gwo Dong Lin

    (Academia Sinica)

Abstract

We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on the checkable moment-(in)determinacy criteria including Cramér’s condition, Carleman’s condition, Hardy’s condition, Krein’s condition and the growth rate of moments, which help us solve the problem more easily. Both Hamburger and Stieltjes cases are investigated. The former is concerned with distributions on the whole real line, while the latter deals only with distributions on the right half-line. Some new results or new simple (direct) proofs of previous criteria are provided. Finally, we review the most recent moment problem for products of independent random variables with different distributions, which occur naturally in stochastic modelling of complex random phenomena.

Suggested Citation

  • Gwo Dong Lin, 2017. "Recent developments on the moment problem," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-17, December.
    • Handle: RePEc:spr:jstada:v:4:y:2017:i:1:d:10.1186_s40488-017-0059-2
    DOI: 10.1186/s40488-017-0059-2
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    References listed on IDEAS

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    1. Christian Kleiber, 2014. "The Generalized Lognormal Distribution and the Stieltjes Moment Problem," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1167-1177, December.
    2. Gwo Dong Lin, 1997. "On the moment problems," Statistics & Probability Letters, Elsevier, vol. 35(1), pages 85-90, August.
    3. Ostrovska, Sofiya & Stoyanov, Jordan, 2005. "Stieltjes classes for M-indeterminate powers of inverse Gaussian distributions," Statistics & Probability Letters, Elsevier, vol. 71(2), pages 165-171, February.
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    5. Stoyanov, Jordan & Tolmatz, Leonid, 2004. "New Stieltjes classes involving generalized gamma distributions," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 213-219, August.
    6. Martin A. Lariviere, 2006. "A Note on Probability Distributions with Increasing Generalized Failure Rates," Operations Research, INFORMS, vol. 54(3), pages 602-604, June.
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    Cited by:

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