Stationary sequence

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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:

& {} \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}),

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 .

See also[edit]


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