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<p>The present work improves the authors’ result [A. Karbonskis and E. Manstavičius, Variance of a strongly additive function defined on random permutations, <i>Lith. Math. J.</i>, 64(3):302–314, 2024] on the variance of a strongly additive function defined on the symmetric group endowed with the uniform probability measure. Previously, we established an upper estimate containing the optimal asymptotic constant; however, the remainder term bound, when compared to that obtained by Kubilius [J. Kubilius, Improved estimate of the second central moment for additive arithmetic functions, <i>Lith. Math. J.</i>, 25(3):250–254,1985] in number theory, remained unsatisfactory. Inspired by his result, we now reach an analogous level by applying a novel methodology. In particular, we take advantage of the power method in the eigenvalue localization algorithm.</p>

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Variance of statistics defined on random permutations

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • Arvydas Karbonskis,
  • Eugenijus Manstavičius

摘要

The present work improves the authors’ result [A. Karbonskis and E. Manstavičius, Variance of a strongly additive function defined on random permutations, Lith. Math. J., 64(3):302–314, 2024] on the variance of a strongly additive function defined on the symmetric group endowed with the uniform probability measure. Previously, we established an upper estimate containing the optimal asymptotic constant; however, the remainder term bound, when compared to that obtained by Kubilius [J. Kubilius, Improved estimate of the second central moment for additive arithmetic functions, Lith. Math. J., 25(3):250–254,1985] in number theory, remained unsatisfactory. Inspired by his result, we now reach an analogous level by applying a novel methodology. In particular, we take advantage of the power method in the eigenvalue localization algorithm.