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Multivariate distributions and the moment problem

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  • Kleiber, Christian
  • Stoyanov, Jordan

Abstract

For any multivariate distribution with finite moments we can ask, as in the univariate case, whether or not the distribution is uniquely determined by its moments. In this paper, we summarize, unify and extend some results that are widely scattered in the mathematical and statistical literature. We present some new results showing how to use univariate criteria together with other arguments to characterize the moment (in)determinacy of multivariate distributions. Among our examples are some classical multivariate distributions including the class of elliptically contoured distributions. Kotz-type distributions receive particular attention. We also describe some Stieltjes classes comprising distinct multivariate distributions that all possess the same set of moments. Some challenging open questions in this area are briefly outlined.

Suggested Citation

  • Kleiber, Christian & Stoyanov, Jordan, 2013. "Multivariate distributions and the moment problem," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 7-18.
    • Handle: RePEc:eee:jmvana:v:113:y:2013:i:c:p:7-18
    DOI: 10.1016/j.jmva.2011.06.001
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    References listed on IDEAS

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    1. Mnatsakanov, Robert M., 2011. "Moment-recovered approximations of multivariate distributions: The Laplace transform inversion," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 1-7, January.
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    Cited by:

    1. Kuoch, Kevin & Redig, Frank, 2016. "Ergodic theory of the symmetric inclusion process," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3480-3498.
    2. Christophe Gaillac & Eric Gautier, 2021. "Nonparametric classes for identification in random coefficients models when regressors have limited variation," Working Papers hal-03231392, HAL.
    3. Yi-Hsuan Lin, 2020. "Random Non-Expected Utility: Non-Uniqueness," Papers 2009.04173, arXiv.org.
    4. Werner Kirsch & Gabor Toth, 2020. "Two Groups in a Curie–Weiss Model with Heterogeneous Coupling," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2001-2026, December.
    5. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    6. Octav Olteanu, 2013. "Moment Problems on Bounded and Unbounded Domains," International Journal of Analysis, Hindawi, vol. 2013, pages 1-7, January.
    7. Octav Olteanu, 2020. "From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications," Mathematics, MDPI, vol. 8(8), pages 1-16, August.
    8. Octav Olteanu, 2013. "New Results on Markov Moment Problem," International Journal of Analysis, Hindawi, vol. 2013, pages 1-17, February.
    9. Sanjay Mehrotra & David Papp, 2013. "A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization," Papers 1306.3437, arXiv.org, revised Aug 2014.
    10. Octav Olteanu, 2020. "Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications," Mathematics, MDPI, vol. 8(10), pages 1-12, September.
    11. Nail Kashaev, 2018. "Identification and estimation of multinomial choice models with latent special covariates," Papers 1811.05555, arXiv.org, revised Mar 2022.

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