对称群上度量的渐近相关
Asymptotic correlations of metrics on the symmetric groups
摘要：
我们考虑$S_n$（对称群）上几个著名度量族之间的渐近联合分布。这些度量包括双不变度量，如Cayley和Hamming距离，以及左不变度量，如Spearman的foottrule，Kendall的tau和Ulam距离。我们还引入了Spearman族的一个自然极限$ρ_\infty$，并研究了它的渐近分布以及与其它度量的关系.这是一个延续的早期工作的双不变度量的渐近独立性的$S_n$和一般线性群在有限域。这项技术是基于对记录图和哈默斯利装置的一些简单观察。在几种情况下，我们给出了渐近独立的误差项的近似最优估计。这大大简化了白、赵和梁关于置换振荡的中心极限定理的证明。
英文摘要：
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, $ρ_\infty$, 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.
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