Rates of Weak Convergence for the Multidimensional Central Limit Theorem

1970 ◽  
Vol 15 (1) ◽  
pp. 68-86 ◽  
Author(s):  
R. N. Bhattacharya
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Multidimensional Central Limit Theorem

scholarly journals On the estimation of the remainder term in the multidimensional central limit theorem

1971 ◽  
Vol 11 (4) ◽  
pp. 905-910
Author(s):  
H. Jasiūnas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Multidimensional Central Limit Theorem

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Г. Ясюнас. Об оценке остаточного члена в многомерной центральной предельной теореме H. Jasiūnas. Liekamoje nario įvertinimas daugiamatėje centrinėje ribinėje teoremoje

An almost everywhere central limit theorem

1988 ◽  
Vol 104 (3) ◽  
pp. 561-574 ◽  
Author(s):  
Gunnar A. Brosamler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Finite Measure ◽  
Standard Normal Distribution ◽  
Almost Everywhere ◽  
Real Functions ◽  
Standard Normal ◽  
Image Position

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean thatfor all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS I

2000 ◽  
Vol 16 (5) ◽  
pp. 621-642 ◽  
Author(s):  
Robert M. de Jong ◽  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Unit Root ◽  
Functional Central Limit Theorem ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Unit Root Testing ◽  
Functional Central Limit

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

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