On the rate of convergence in the central limit theorem for hierarchical Laplacians

2019 ◽  
Vol 23 ◽  
pp. 68-81
Author(s):  
Alexander Bendikov ◽  
Wojciech Cygan
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Ultrametric Space ◽  
Standard Normal Distribution ◽  
Arithmetic Means ◽  
The Central Limit Theorem ◽  
Standard Normal ◽  
Variation Distance

Let (X,d) be a proper ultrametric space. Given a measuremonXand a functionB↦C(B) defined on the set of all non-singleton ballsBwe consider the hierarchical LaplacianL=LC. Choosing a sequence {ε(B)} of i.i.d. random variables we define the perturbed functionC(B,ω) and the perturbed hierarchical LaplacianLω=LC(ω). We study the arithmetic means λ̅(ω) of theLω-eigenvalues. Under certain assumptions the normalized arithmetic means (λ̅−Eλ̅) ∕ σ(λ̅) converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

scholarly journals The central limit theorem for $\sum f(\theta^nx)g(\theta^{n^2}x)$

2000 ◽  
Vol 20 (5) ◽  
pp. 1335-1353 ◽  
Author(s):  
KATUSI FUKUYAMA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
The Central Limit Theorem ◽  
Distribution Of Values

In this paper, it is proved that the distribution of values of $N^{-1/2}\sum_{n=1}^N f_1(\theta^{p_1(n)}x)\dots f_K(\theta^{p_K(n)}x)$ converges to normal distribution. Here $p_k(n)$ are polynomials.

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