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Covariance stationary

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A sequence of random variables is covariance stationary if:

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.

If a sequence is stationary, all couple of terms of the sequence that are separated by the same number of terms also have the same covariance.

Table of Contents


Covariance stationary sequences are also called:

Often, we also use the term time series instead of sequence or process.


This is a formal definition.

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

In words:

  1. all the terms of the sequence have mean mu;

  2. the covariance [eq3] depends only on the relative position $j$ and not on the absolute position n.

Note that[eq4]which implies that a weakly stationary process has constant variance.

Definition for random vectors

The definition above applies without modifications to sequences of random vectors.

In the case of a sequence of vectors, [eq5] is a vector of expected values and [eq6] is a matrix of covariances between the entries of the two vectors X_n and $X_{n-j}$ (a cross-covariance matrix).


Let us make some examples.

Example 1 - White noise

The simplest example is the so called white noise process, a sequence [eq7] that satisfies the following three conditions for any n and $j>0$:[eq8]where sigma^2 is a positive constant.

Example 2 - Autoregressive process

Let [eq9] be the white noise process of the previous example.

A first-order autoregressive process is a sequence [eq10] whose terms satisfy[eq11]where $
ho >0$ is a constant and the recursion starts from a random variable $X_{0}$ uncorrelated with the terms of [eq12].

The expected values of the terms of the sequence are[eq13]

For the process [eq10] to be weakly stationary, the first condition that needs to be satisfied is [eq15]which is satisfied only if [eq16] or [eq17]. The latter possibility wil be ruled out below.

The variances are[eq18]

The variances remain finite as n grows only if $
ho <1$. Furthermore, the condition[eq19]is satisfied only if[eq20]which can be shown, for example, by solving[eq21]

As proved in the lecture on autocorrelation, the covariance between any two terms of the sequence is[eq22]which satisfies the condition stated in the definition of weak stationarity (the covariance depends only on $j$).

Thus, the sequence [eq10] is covariance stationary only if[eq24]

Weakly vs strictly stationary

A stronger concept of stationarity is that of strict stationarity.

A sequence [eq10] is said to be strictly stationary if and only if [eq26]and [eq27]have the same joint distribution for any n, k and $q$.

How covariance stationarity is used

The concept of covariance stationarity is often used in probability, statistics and time-series analysis.

Here are some examples:

More details

Other concepts related to covariance stationarity can be found in the lecture on sequences of random variables.

Keep reading the glossary

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How to cite

Please cite as:

Taboga, Marco (2021). "Covariance stationary", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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