(strict sense) stationarity of AR processes?

HI all,

I have a question about the stationary of AR processes in the strict
sense?

Take AR(1),

X_n=a*X_(n-1) + e_n,  |a|<1.

where e_n's are white noise process(uncorrelated with common mean 0 and
common sigma^2), but not neccessarily IID process, i.e., each e_n may
not need to be independently and identically distributed.

(This is a definition from Brockwell's Time Series book).

My question is: is X_n stationary in the strict sense?

My guess is that it is not strictly stationary. So I just have to
provide a counter-example:

Suppose the process starts from X0 which is a random variable.

Then X1=a * X0 + e1, will not have same distribution as X0,

(this looks to me like a triviality, but I am not sure if there will be
some hidden traps...)

But what if the process starts from minus infinity?

Staring from minus infinity, will X0 and a * X0 + e1 have the same
distribution? 

Please shed some light on it! Thanks a lot!

comtech wrote:

> HI all,
>
> I have a question about the stationary of AR processes in the strict
> sense?
>
> Take AR(1),
>
> X_n=a*X_(n-1) + e_n,  |a|<1.
>
> where e_n's are white noise process(uncorrelated with common mean 0 and
> common sigma^2), but not neccessarily IID process, i.e., each e_n may
> not need to be independently and identically distributed.
>
> (This is a definition from Brockwell's Time Series book).
>
> My question is: is X_n stationary in the strict sense?
>
> My guess is that it is not strictly stationary. So I just have to
> provide a counter-example:

I before you provide a counter-example, you should write down the
defintion of stationary. I think before you reach the end of writing,
you'll know if the process you define is stationary or not.

Regards,
Andor