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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
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up vote 4 down vote accepted

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
    
see statistik.tuwien.ac.at/public/dutt/vorles/geost_03/node49.html for order 2 stationnarity. Anyway, I have tryed to clarify my answer, hope it is better now... –  robin girard Aug 9 '10 at 8:05
    
I believe that article describes stationary in the weak sense and this is not what I meant. I should clarify this in the question. See en.wikipedia.org/wiki/Stationary_process for a description of the various types of stationarity. –  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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