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Covariance stationary

A sequence of random variables is covariance stationary if all the terms of the sequence have the same mean, and if the covariance between any two terms of the sequence depends only on the relative positions of the two terms, that is, on how far apart they are located from each other, and not on their absolute position, that is, on where they are located in the sequence.

Synonyms

Covariance stationary sequences are also called weakly stationary.

Definition

This is a formal definition.

Definition A sequence of random variables [eq1] is covariance stationary if and only if[eq2]

In other words, all the terms of the sequence have mean mu, and the covariance [eq3] depends only on $j$ and not on n.

The definition can be generalized to sequences of random vectors, in which case [eq4] is a vector of expected values and [eq5] is a matrix of covariances between the entries of the two vectors X_n and $X_{n-j}$.

More details

More details about covariance stationary sequences can be found in the lecture entitled Sequences of random variables.

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