Stationary sequence

Random sequence whose joint probability distribution is invariant over time

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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:

F X n , X n + 1 , … , X n + N − 1 ( x n , x n + 1 , … , x n + N − 1 ) = F X n + k , X n + k + 1 , … , X n + k + N − 1 ( x n , x n + 1 , … , x n + N − 1 ) , {\displaystyle {\begin{aligned}&{}\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{aligned}}}

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):

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

## See also

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* Stationary process

## References

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

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