Why would this statement be true? "Every iid process is strictly stationary, but it may not be weakly stationary."

Question

After learning the basic definitions of stationarity and iid, there was a question on my exercise:

"Every iid process is strictly stationary, but it may not be weakly stationary."

On the definition of weak stationarity, it doesn't usually highlight E[X(t)^2]< ∞ condition, but is this the only difference? If so, what does this condition actually imply? I'm guessing it's relevant to the second-order moments or variance, but this part is still confusing for me.

Thank you in advance for your response!

Answer 1

This is a FAQ. Strict stationarity only means that the joint distribution does not change with time, it does not guarantee the existence of finite moments.

On the other hand, weakly stationary processes require that the mean and autocorrelation do not vary with time and the variance is finite. For a strictly stationary process to be weakly stationary, it must have finite mean and covariance. An example is i.i.d. Cauchy distribution, which is strictly stationary but not weakly stationary since both its mean and variance are undefined.