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{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}

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

References


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