scholarly journals On the strong attraction of functions on dependent variables to the normal law

2020 ◽  
pp. 5-19
Author(s):  
A. G. Grin
Keyword(s):  
Symmetric Functions ◽  
Random Variables ◽  
Strong Mixing ◽  
Stationary Sequences ◽  
Mixing Condition ◽  
Strong Attraction ◽  
Dependent Variables ◽  
Normal Law ◽  
Strong Mixing Condition ◽  
General Conditions

For symmetric functions on random variables from stationary sequences satisfying the uniformly strong mixing condition, the general conditions of attraction to the normal law in terms of distributions of individual items are obtained. The main result of the paper generalizes all known to present results of this type.

A nearly independent, but non-strong mixing, triangular array

1985 ◽  
Vol 22 (03) ◽  
pp. 729-731 ◽  
Author(s):  
Donald W. K. Andrews
Keyword(s):  
Random Variables ◽  
Triangular Array ◽  
Strong Mixing ◽  
Common Condition ◽  
Mixing Condition ◽  
First Order ◽  
Triangular Arrays ◽  
The Central Limit Theorem ◽  
Independent Identically Distributed ◽  
Strong Mixing Condition

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.

On the asymptotic normality of some sums of dependent random variables

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Margarita I. Tikhomirova ◽  
Vladimir P. Chistjakov
Keyword(s):  
Asymptotic Normality ◽  
Random Sequence ◽  
Random Variables ◽  
Stationary Sequence ◽  
Dependency Graph ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Dependent Random Variables ◽  
Sum Of Functions ◽  
Strong Mixing Condition

AbstractA theorem on the asymptotic normality of the sum of dependent random variables is stated and proved. Conditions of the theorem are formulated in terms of a dependency graph which characterizes the relationships between random variables. This theorem is used to prove the asymptotic normality of the sum of functions defined on subsets of elements of the stationary sequence satisfying the strong mixing condition. As an illustration of possible applications of these theorems we give a theorem on the asymptotic normality of the number of empty cells if the random sequence of cells occupied by particles is a stationary sequence satisfying the uniform strong mixing condition.

A nearly independent, but non-strong mixing, triangular array

1985 ◽  
Vol 22 (3) ◽  
pp. 729-731 ◽  
Author(s):  
Donald W. K. Andrews
Keyword(s):  
Random Variables ◽  
Triangular Array ◽  
Strong Mixing ◽  
Common Condition ◽  
Mixing Condition ◽  
First Order ◽  
Triangular Arrays ◽  
The Central Limit Theorem ◽  
Independent Identically Distributed ◽  
Strong Mixing Condition

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.

scholarly journals On the attraction to the normal law of functions of weakly dependent variables

2020 ◽  
pp. 24-39
Author(s):  
A. G. Grin
Keyword(s):  
Symmetric Functions ◽  
Sufficient Conditions ◽  
Weak Dependence ◽  
Necessary And Sufficient Conditions ◽  
Dependent Variables ◽  
Normal Law ◽  
Necessary And Sufficient ◽  
Weakly Dependent

The paper gives a “generally accepted” condition of weak dependence and a condition on the class of symmetric functions that provide fulfilment of the necessary and sufficient conditions, obtained earlier by the author, for attracting functions of dependent quantities to the normal law.

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