Stationary sequence

{{Short description|Random sequence whose joint probability distribution is invariant over time}}

{{one source |date=March 2024}}

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):

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