Is Weak Stationarity of Y_k Achievable in AR(1) Process?

[fireb]

Consider AR(1) process \(X_t=bX_{t-1}+e_t\)
where \(e_t\) with mean of 0 and variance of \(\sigma^2\)
and |b| <1
Let \( a_k \) be a recursive sequence with \( a_1 \) =1 and \( a_{k+1} = a_k + P_k +1\) for \( k = 1, 2 ,...,\) where \(P_k \) is Poisson iid r.v with mean = 1
also, assume \(P_t\) and \(X_t\) are independent.
Is \(Y_k\)= \(X_{a_k}\) for k =1,2,... weakly stationary?There arent any similar problems in the my textbook and i have no clue how to begin
Im not looking for a straight answer, just something to point me in the right direction.
Thanks in advance

 

[dave777]

Consider AR(1) process \(X_t=bX_{t-1}+e_t\)
where \(e_t\) with mean of 0 and variance of \(\sigma^2\)
and |b| <1
Let \( a_k \) be a recursive sequence with \( a_1 \) =1 and \( a_{k+1} = a_k + P_k +1\) for \( k = 1, 2 ,...,\) where \(P_k \) is Poisson iid r.v with mean = 1
also, assume \(P_t\) and \(X_t\) are independent.
Is \(Y_k\)= \(X_{a_k}\) for k =1,2,... weakly stationary?There arent any similar problems in the my textbook and i have no clue how to begin
Im not looking for a straight answer, just something to point me in the right direction.
Thanks in advance

Hi

I have some experience with AR but don't claim to be an expert or provide you with a correct solution. I can't post links yet...

According to a basic definition of weakly stationary (google wikipedia and weakly stationary) we need to see whether the mean and variance of the AR model changes over time. If it does, then the process is not weakly stationary. From your definition above the variance and mean of \(e_t\) is not modified and does not change over time. Whilst I cannot state or be confident that this is the answer, my hint would be that this points to weakly stationary.

However according to section 4.1 (google weakly stationary definition and see the result from ohio state) you need to show the expectation of \(X_t\) is finite and does not depend on t (which I don't think it does but this is where the problem may lie)

Hope this helps, again this is not the correct solution just some hints.
Cheers