Asymptotics of a Locally Dependent Statistic on Finite Reflection Groups

Abstract

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was listed as an interesting direction by Chatterjee and Diaconis (2017). 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 to signed permutations, i.e. elements of Coxeter groups of type $ \mathtt{B}_n $, which are also known as the hyperoctahedral groups. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $\mathtt{D}_n$ 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.

A Central Limit Theorem for the Two-Sided Descent Statistic on Coxeter Groups

“…As was shown by Chatterjee-Diaconis [7] and Röttger [16], the sequences T An , T Bn and T Dn satisfy the CLT. This allows us to apply Lemma 21 if the sequence (W n ) n satisfies the following property:…”

“…The subsequence (M f n ) n∈L is again well-behaved and as noted in Remark 27, we can assume that every irreducible component of M f n is of type A, B or D. Thus, it follows from [7], [16] and Lemma 21 that the sequence (T M f n ) n∈L satisfies the CLT. The asymptotic normality of (T n ) n∈L now follows from Lemma 22 and Lemma 23: Either V(T W f n ) is of the same order as V(T n ); because every component of W f n is f -small, this implies that after possible passing to a further subsequence, T W f n satisfies the CLT.…”

“…Special cases of Theorem 1 were known before: For the case where W n = Sym(n + 1), the irreducible Coxeter group of type A n , the result is due to Vatutin [19] and was later, with different methods, reproven by Chatterjee-Diaconis [7] and Özdemir [13]. Following the approach of Chatterjee and Diaconis, Röttger [16] generalised this to the cases where W n is an irreducible Coxeter group of type B n or D n . Technical difficulties of these proofs lie in the dependencies between des(w) and des(w −1 ), which require probabilistic methods as for example interaction graphs, see [6], to establish the CLT.…”

Extreme Values of Permutation Statistics

“…Chatterjee & Diaconis [7,8] used Stein's method on interaction graphs to prove a CLT for the sum of descents and inverse descents. Röttger & Brück [6,21] extended this to other types of Coxeter groups. Work of Conger & Viswanath [9] contains CLTs for permutations on multisets, and He [14] studied a CLT for the two-sided descent statistic on a Mallows-distributed permutation.…”

A central limit theorem for descents of a Mallows permutation and its inverse

“…The central limit theorem for des(w)+des(w −1 ) for the uniform measure on finite Coxeter groups was conjectured by Kahle and Stump [33]. It was shown to hold in a series of works [8,20,41,44]. The proof relies on the classification of finite Coxeter groups to reduce the problem to analyzing certain infinite families.…”