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:

${\displaystyle {\begin{aligned}&{}\quad F_{X_{n},X_{n+1},\dots ,X_{n+N-1}}(x_{n},x_{n+1},\dots ,x_{n+N-1})\\&=F_{X_{n+k},X_{n+k+1},\dots ,X_{n+k+N-1}}(x_{n},x_{n+1},\dots ,x_{n+N-1}),\end{aligned}}}$

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):

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