I asked this question on [stats.stackexchange.com](https://stats.stackexchange.com/questions/1441/example-of-a-stochastic-process-that-is-1st-and-2nd-order-stationary-but-not-str) a little while back but didn't get an answer. It was suggested that I post it here at the time. There appears to be some migratory problems going on over there. Hopefully, this question is seen as appropriate.


Let $X(t)$ be a stochastic process. We say that $X(t)$ is *Nth-order stationary* if for every set of ''times'' $t\_1, t\_2, \dots, t\_N$ we have that the joint cumulative density functions
$$F\_{X(t\_1),X(t\_2),\dots,X(t\_N)} = F\_{X(t\_1 + \tau),X(t\_2 + \tau),\dots,X(t\_N + \tau)}$$
for all ''time shifts'' $\tau$. This is quite a strong condition, it says that the joint statistics don't change at all as time shifts. 


For example, a 1st order stationary process is such that $F\_{X(t\_1)} = F\_{X(t\_2)}$ for all $t\_1$ and $t\_2$. That is, the $X(t)$ are all identically distributed. It is quite easy to see that a 1st order stationary process need not be 2nd order stationary. Simply assign a correlation structure to say $X(t)$, $X(t+1)$, $X(t+2)$ that *does not* correspond to a (symmetric) Toeplitz matrix. That is, in vector form, the covariance matrix of $[ X(t), X(t+1), X(t+3)]$ could be given as
$$\left[\begin{array}{cc} 
\sigma^2 & a & b \newline 
a & \sigma^2 & c \newline
b & c& \sigma^2
\end{array}\right]$$
for $a,b,c$ distinct. This is now not 2nd order stationary because $E[X(t)X(t+1)] = a$ and, time shifting by 1 we have $E[X(t+1)X(t+2)] = c \neq a$.


In a similar way (presumably) a process that is 1st and 2nd order stationary need not be 3rd order stationary and this leads to my question:



> 
> Does somebody have a nice example of a stochastic process that is both 1st and 2nd order stationary, but not 3rd order stationary?
> 
> 
> 


**Motivation:** Some of the material I have been expected to teach recently has included stochastic processes and I feel this is a gap in my knowledge. Although no student has actually asked this question yet, I think it is natural enough for it to be asked at some point and I would like to have a neat and simple example.



A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary
How about this: 


Consider $X\_1,X\_2$ i.i.d. with $\mathbb{P}(X\_i = 1) = \mathbb{P}(X\_i = 0) = 0.5$. Now define $X\_3 = X\_1 + X\_2 (\text{mod }2)$. Notice that $X\_3$ is independent from $X\_1$ and from $X\_2$ individually. However the three variables $X\_1,X\_2,X\_3$ are not jointly independent.


Now consider a sequence $X\_1,X\_2,X\_3,\ldots$ such that each triplet $(X\_{3n+1},X\_{3n+2},X\_{3n+3})$ has the same distribution as $(X\_1,X\_2,X\_3)$ and all these triplets are independent.


With this one has that all $X\_n$ have the same distribution as $X\_1$, so that the process is 1 stationary. Also, all pairs $X\_i,X\_j$ with $i\neq j$ are independent and hence have the same distribution (this gives 2 stationarity). However the triplet $(X\_2,X\_3,X\_4)$ has a different distribution from $(X\_1,X\_2,X\_3)$ (in fact, $X\_2,X\_3$ and $X\_4$ are jointly independent).


