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arXiv · 1812.00372

Asymptotics of a locally dependent statistic on finite reflection groups

Frank Röttger
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Abstract

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme on signed permutations, so elements of Coxeter groups of type $ B_n $, which is also known as the hyperoctahedral group. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $D_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cramér-Wold device.

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Frank Röttger. 2020-03-21. Asymptotics of a locally dependent statistic on finite reflection groups. https://doi.org/10.37236/9454

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