scholarly journals Central Limit Theorem for Peaks of a Random Permutation in a Fixed Conjugacy Class of $$S_n$$

2021 ◽  
Author(s):  
Jason Fulman ◽  
Gene B. Kim ◽  
Sangchul Lee
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Conjugacy Class ◽  
Central Limit ◽  
Random Permutation

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 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.

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.

scholarly journals Combable functions, quasimorphisms, and the central limit theorem

2009 ◽  
Vol 30 (5) ◽  
pp. 1343-1369 ◽  
Author(s):  
DANNY CALEGARI ◽  
KOJI FUJIWARA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Word Length ◽  
Hyperbolic Groups ◽  
Finite State Automaton ◽  
List Type ◽  
Generating Sets ◽  
Finite State ◽  
Word Hyperbolic Groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

Sign in / Sign up

Export Citation Format

Share Document