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}

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):

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

References

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