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From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

In: Probability, Combinatorics and Control

Author

Listed:
  • Victor Korolev
  • Andrey Gorshenin
  • Alexander Zeifman

Abstract

This chapter contains a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. The bases for this explanation are limit theorems for random sums and statistics constructed from samples with random sizes. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of random sums of independent random vectors with finite covariance matrices to multivariate elliptically contoured stable and Linnik distributions. Also, conditions are presented for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate Student distributions. The joint asymptotic behavior of sample quantiles is also considered.

Suggested Citation

  • Victor Korolev & Andrey Gorshenin & Alexander Zeifman, 2020. "From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes," Chapters, in: Andrey Kostogryzov & Victor Korolev (ed.), Probability, Combinatorics and Control, IntechOpen.
    • Handle: RePEc:ito:pchaps:199769
    DOI: 10.5772/intechopen.89659
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    File URL: https://www.intechopen.com/chapters/69677
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    More about this item

    Keywords

    random sum; random sample size; multivariate normal mixtures; heavy-tailed distributions; multivariate stable distribution; multivariate Linnik distribution; Mittag-Leffler distribution; multivariate Student distribution; sample quantiles; AMS 2000 Subject Classification: 60F05; 60G50; 60G55; 62E20; 62G30;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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