scholarly journals Asymptotics of a Locally Dependent Statistic on Finite Reflection Groups

10.37236/9454 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Frank Röttger
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Coxeter Groups ◽  
Random Permutation ◽  
Wasserstein Distance ◽  
Reflection Groups ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Dependency Structures

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.

Estimating the Error of a Permutational Central Limit Theorem

1996 ◽  
Vol 10 (4) ◽  
pp. 533-541 ◽  
Author(s):  
Chern-Ching Chao ◽  
Lincheng Zhao ◽  
Wen-Qi Liang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Error Bound ◽  
Central Limit ◽  
Normal Approximation ◽  
Random Permutation ◽  
Stein’S Method ◽  
Real Numbers ◽  
Sorting Problem ◽  
Two Measures

Motivated by two measures of presortedness, number of runs and oscillation of a permutation, related to the sorting problem, we derive an error bound for normal approximation to the distribution of Here, αij's are given real numbers and π is a uniformly distributed random permutation of {l,…, n}. The derivation is based on Stein's method.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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