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Stationary sequence

A sequence of random variables is said to be stationary if the joint distribution of a group of successive terms of the sequence is independent of the position of the group.

Synonyms

Stationary sequences are also called strictly stationary.

Definition

The following is a formal definition.

Definition A sequence of random variables [eq1] is said to be stationary if and only if the two random vectors [eq2]and [eq3]have the same joint distribution for any n, k and $q$.

This implies that if a sequence is stationary, then all terms of the sequence have the same distribution, that is, X_n and $X_{n+k}$ have the same distribution for any n and k (this is obtained by setting $q=1$ in the definition above). Of course, since they have the same distribution, all terms of the sequence also have the same expected value and the same variance, provided these moments exist and are finite.

The concept extends in a straightforward manner to sequences of random vectors (the definition remains exactly the same when [eq4] is a sequence of random vectors).

Other concepts

There are also other concepts of stationarity. See, for example, the glossary entry entitled Covariance stationary.

More details

The concept of stationary sequence is explained in more detail in the lecture entitled Sequences of random variables.

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