Asymptotic correlations of metrics on the symmetric groups

2011/10/04 by Yunjiang Jiang, Jiang, Yunjiang
Computer Science · Mathematics · #Advanced Algebra and Geometry #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1110.0807

openalex publication_date 2011/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the asymptotic joint distributions among several families of well-known metrics on S_n, the symmetric group. These include the bi-invariant metrics such as the Cayley and Hamming distance, and the left-invariant metrics such as Spearman's footrule, Kendall's tau, and the Ulam distance. We also introduce a natural limit of the Spearman family, ρ_∞, and study its asymptotic distribution and relation with other metrics. This is a continuation of earlier work on the asymptotic independence of bi-invariant metrics on both S_n and general linear groups over a finite field. The technique is based on some simple observation about the record map and Hammersley's device. In several cases, we give near-optimal estimate of the error term for asymptotic independence. This simplifies significantly the proof of a central limit theorem by Bai, Chao, and Liang regarding the oscillation of a permutation.

Citations

• The Proof and Measurement of Association between Two Things

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