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Joint Extremes of Inversions and Descents of Random Permutations

Author

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  • Philip Dörr

    (Ruhr University Bochum)

  • Johannes Heiny

    (Stockholm University)

Abstract

We provide asymptotic theory for the joint distribution of $$X_\textrm{inv}$$ X inv and $$X_\textrm{des}$$ X des , the numbers of inversions and descents of random permutations. Recently, [14] proved that $$X_\textrm{inv}$$ X inv , respectively, $$X_\textrm{des}$$ X des , is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that $$(X_\textrm{inv}, X_\textrm{des})$$ ( X inv , X des ) is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $$(X_\textrm{inv}, X_\textrm{des})$$ ( X inv , X des ) and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.

Suggested Citation

  • Philip Dörr & Johannes Heiny, 2025. "Joint Extremes of Inversions and Descents of Random Permutations," Journal of Theoretical Probability, Springer, vol. 38(2), pages 1-37, June.
  • Handle: RePEc:spr:jotpro:v:38:y:2025:i:2:d:10.1007_s10959-025-01407-y
    DOI: 10.1007/s10959-025-01407-y
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    References listed on IDEAS

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    1. Arslan, İlker & Işlak, Ümit & Pehlivan, Cihan, 2018. "On unfair permutations," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 31-40.
    2. Mark Conger & D. Viswanath, 2007. "Normal Approximations for Descents and Inversions of Permutations of Multisets," Journal of Theoretical Probability, Springer, vol. 20(2), pages 309-325, June.
    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
    4. Sourav Chatterjee & Persi Diaconis, 2017. "A central limit theorem for a new statistic on permutations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 561-573, December.
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