A short proof of Motoo's combinatorial central limit theorem using Stein's method

1988 ◽  
Vol 78 (2) ◽  
pp. 249-252 ◽  
Author(s):  
W. Schneller
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Short Proof ◽  
Stein’S Method ◽  
Stein's Method

Stein’s method of normal approximation for dynamical systems

2019 ◽  
Vol 20 (04) ◽  
pp. 2050021 ◽  
Author(s):  
Olli Hella ◽  
Juho Leppänen ◽  
Mikko Stenlund
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Multiplicative Constant ◽  
Correlation Decay ◽  
The Central Limit Theorem

We present an adaptation of Stein’s method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumptions.

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 Convergence Rates for Generalized Descents

10.37236/723 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
John Pike
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Explicit Formula ◽  
Central Limit ◽  
Convergence Rates ◽  
Stein’S Method ◽  
Permutation Statistics ◽  
Mean And Variance ◽  
The Mean

d-descents are permutation statistics that generalize the notions of descents and inversions. It is known that the distribution of d-descents of permutations of length n satisfies a central limit theorem as n goes to infinity. We provide an explicit formula for the mean and variance of these statistics and obtain bounds on the rate of convergence using Stein's method.

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