Stationary sequence

Short description: Random sequence whose joint probability distribution is invariant over time

In probability theory – specifically in the theory of stochastic processes, a stationary sequence is a random sequence whose joint probability distribution is invariant over time. If a random sequence X _j is stationary then the following holds:

[math]\displaystyle{ \begin{align} & {} \quad F_{X_n,X_{n+1},\dots,X_{n+N-1}}(x_n, x_{n+1},\dots,x_{n+N-1}) \\ & = F_{X_{n+k},X_{n+k+1},\dots,X_{n+k+N-1}}(x_n, x_{n+1},\dots,x_{n+N-1}), \end{align} }[/math]

where F is the joint cumulative distribution function of the random variables in the subscript.

If a sequence is stationary then it is wide-sense stationary.

If a sequence is stationary then it has a constant mean (which may not be finite):

[math]\displaystyle{ E(X[n]) = \mu \quad \text{for all } n . }[/math]

See also

• Stationary process

References

• Probability and Random Processes with Application to Signal Processing: Third Edition by Henry Stark and John W. Woods. Prentice-Hall, 2002.