Recent Advances in the Central Limit Theorem and Its Weak Invariance Principle for Mixing Sequences of Random Variables (A Survey)

1986 ◽  
pp. 193-223 ◽  
Author(s):  
Magda Peligrad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Recent Advances ◽  
Mixing Sequences

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.

scholarly journals STATISTICAL LIMIT LAWS FOR EQUIVARIANT OBSERVATIONS

2004 ◽  
Vol 04 (01) ◽  
pp. 1-13 ◽  
Author(s):  
IAN MELBOURNE ◽  
MATTHEW NICOL
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Probability Space ◽  
Limit Laws ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Statistical Limit

We show that statistical limit laws for ergodic stationary sequences of G-equivariant observations ϕ on a probability space Ω×G are inherited by sequences of observations ϕ(·,g0) on the probability space Ω for each fixed g0. The statistical limit laws we consider are the central limit theorem, weak invariance principle and the law of the iterated logarithm.

scholarly journals On the estimate of the remainder term in the central limit theorem for non-equally distributed random variables

1972 ◽  
Vol 12 (4) ◽  
pp. 183-194
Author(s):  
V. Paulauskas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Random Variables ◽  
The 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: В. Паулаускас. Оценка скорости сходимости в центральной предельной теореме для разнораспределенных слагаемых V. Paulauskas. Konvergavimo greičio įvertinimas centrinėje ribinėje teoremoje nevienodai pasiskirsčiusiems dėmenims

scholarly journals Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mingzhou Xu ◽  
Kun Cheng
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Uniform Convergence ◽  
Central Limit ◽  
Random Variables ◽  
Precise Asymptotics ◽  
Partial Sum ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.

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