Schur test: Difference between revisions
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In [[probability theory]] – specifically in the theory of [[stochastic process]]es, a '''stationary sequence''' is a [[random sequence]] whose [[joint probability distribution]] is [[Invariant (mathematics)|invariant]] over time. If a random sequence ''X''<sub> ''j''</sub> is stationary then the following holds: | |||
: <math> | |||
\begin{align} | |||
& {} \qquad 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} | |||
</math> | |||
where ''F'' is the joint [[cumulative distribution function]] of the [[random variable]]s 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 (mathematics)|mean]] (which may not be finite): | |||
: <math>E(X[n]) = \mu \quad \text{for all } n .</math> | |||
==See also== | |||
*[[Stationary process]] | |||
==References== | |||
* ''Probability and Random Processes with Application to Signal Processing: Third Edition'' by Henry Stark and John W. Woods. Prentice-Hall, 2002. | |||
[[Category:Sequences and series]] | |||
[[Category:Stochastic processes]] | |||
[[Category:Time series analysis]] | |||
{{probability-stub}} | |||
Latest revision as of 00:54, 21 June 2013
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:
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):
See also
References
- Probability and Random Processes with Application to Signal Processing: Third Edition by Henry Stark and John W. Woods. Prentice-Hall, 2002.