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On the central limit theorem for the two-sided descent statistics in Coxeter groups. (English) Zbl 1434.60041

Summary: In [Math. Comput. 89, No. 321, 437–464 (2020; Zbl 1480.20093)], T. Kahle and C. Stump raised the following problem: identify sequences of finite Coxeter groups \(W_n\) for which the two-sided descent statistics on a uniform random element of \(W_n\) is asymptotically normal. Recently, Brück and Röttger provided an almost-complete answer, assuming some regularity condition on the sequence \(W_n\). In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of C. L. Mallows [Ann. Math. Stat. 43, 508–515 (1972; Zbl 0238.60017)].

MSC:

60C05 Combinatorial probability
60F05 Central limit and other weak theorems
05E16 Combinatorial aspects of groups and algebras
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