A sequence of random variables is covariance stationary if:
all the terms of the sequence have the same mean;
the covariance between any two terms of the sequence depends only on the relative position of the two terms and not on their absolute position.
By relative position of two terms we mean how far apart they are located from each other in the sequence.
Instead, the absolute position refers to where they are located in the sequence.
Covariance stationary sequences are also called:
weakly stationary sequences;
covariance stationary processes;
weakly stationary processes.
This is a formal definition.
Definition A sequence of random variables is covariance stationary if and only if
In words:
all the terms of the sequence have mean ;
the covariance depends only on the relative position and not on the absolute position .
Note thatwhich implies that a weakly stationary process has constant variance.
The definition above applies without modifications to sequences of random vectors.
In the case of a sequence of vectors, is a vector of expected values and is a matrix of covariances between the entries of the two vectors and (a cross-covariance matrix).
Let us make some examples.
The simplest example is the so called white noise process, a sequence that satisfies the following three conditions for any and :where is a positive constant.
Let be the white noise process of the previous example.
A first-order autoregressive process is a sequence whose terms satisfywhere is a constant and the recursion starts from a random variable uncorrelated with the terms of .
The expected values of the terms of the sequence are
For the process to be weakly stationary, the first condition that needs to be satisfied is which is satisfied only if .
The variances are
The variances remain finite as grows only if . Furthermore, the conditionis satisfied only ifwhich can be shown, for example, by solving
As proved in the lecture on autocorrelation, the covariance between any two terms of the sequence iswhich satisfies the condition stated in the definition of weak stationarity (the covariance depends only on ).
Thus, the sequence is covariance stationary only if
A stronger concept of stationarity is that of strict stationarity.
A sequence is said to be strictly stationary if and only if and have the same joint distribution for any , and .
Other concepts related to covariance stationarity can be found in the lecture on sequences of random variables.
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Please cite as:
Taboga, Marco (2021). "Covariance stationary", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/covariance-stationary.
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