# Example of a 2nd order stationary, but not strictly stationary process

Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?

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I would like to add a stochastic processes tag, but don't have the reputation, perhaps somebody could edit this for me? – Robby McKilliam Aug 9 '10 at 6:51
retag done ! – robin girard Aug 9 '10 at 6:59

## 1 Answer

Take any process $(X_t)_t$ with independent components that has a constant first and second moment and put a varying third moment.

It is second order stationnary because $E[ X_t X_{t+h} ]=0$ and it is not strictly stationnary because $P( X_t \geq x_t, X_{t+1} \geq x_{t+1})$ depends upon $t$

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Perhaps I am a little confused, or we are using different definitions?  Am I correct in thinking that a process is 2nd order stationary if the joint marginal cdfs $F_{X(t),X(t+τ)}$ are equal for all $\tau$?  Similarly for a process to be 1st-order stationary the marginal cdfs $F_{X(t)}$ need to be the same for every $t$. So all the moments of the $X(t)$ must be equal.   2nd-order stationary implies 1st-order stationary, correct? – Robby McKilliam Aug 9 '10 at 7:44
Extending this, a process is $N$th order stationary if for every $t_1, t_2, \dots, t_N$ the marginal cdfs $F_{X(t_1 + \tau), X(t_2 + \tau)\dots,X(t_N+\tau)}$ are the same for all $\tau$.  Strictly stationary is Nth order stationary for all N. – Robby McKilliam Aug 9 '10 at 7:45
– robin girard Aug 9 '10 at 8:05
– Robby McKilliam Aug 9 '10 at 8:24
@robby OK... I didn't know this "second order stationnarity" I think you should not say second order stationnarity in the question and give the definintion instead. For more clarity you should ask another question. do you have a paper that refers to this second order stationnarity ? – robin girard Aug 9 '10 at 8:32
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