The Invariance Principle for Stationary Random Fields with a Strong Mixing Condition

1983 ◽  
Vol 27 (2) ◽  
pp. 380-385 ◽  
Author(s):  
V. V. Gorodetskii
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Stationary Random Fields ◽  
Strong Mixing Condition

On the weak invariance principle for non-adapted stationary random fields under projective criteria

2021 ◽  
Author(s):  
Han-Mai Lin
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Long Memory ◽  
Sufficient Conditions ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Projective Criteria ◽  
Stationary Random Fields ◽  
Do So

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

Parameter estimation for finite-parameter stationary random fields

1986 ◽  
Vol 23 (A) ◽  
pp. 311-318
Author(s):  
M. Rosenblatt
Keyword(s):  
Parameter Estimation ◽  
Asymptotic Distribution ◽  
Random Fields ◽  
Strong Mixing ◽  
Parameter Estimates ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Stationary Random Fields

The concept of strong mixing is used to obtain a generalization of results on the asymptotic distribution of finite-parameter estimates of linear processes and extend them for stationary sequences and random fields.

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.

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