Stationary sequence

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:

{\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}}

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])=\mu \quad {\text{for all }}n.

References

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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