I have questions about weakly stationary in time series analysis.

I cannot get autocovariance for the weakly stationary time series and I am confused with #5. I am not sure how I show that the time series is weakly stationary.

Could you show me them?

Thanks,

4. (10 points) Let {X,, t c Z) be a weakly stationary time series. Define Xt, if t is even Ytix, + 1, if tis odd. i. Express the autocovariance function yr(s, t) of {Y., t c Z} in terms of yx(s, t) and conclude if it depends only on the lagh = s -t or not. ii. Conclude if {Y., t c Z} is weakly stationary or not. 5. (15 points) A time series (X,, t c Z) is said to be strictly stationary if the joint distribution of (Xt ,Xt,) ..)Xt) is same as the joint distribution of (X, +,Xt,+ ...,Xin+:) for any n E N and TEZ. Let { ,t E Z) be Gaussian white noise with unit variance, i.e. &, ~i. i. d. N(0,1). Define Et, if t is even X. = 1 2 (E? - 1), if tis odd. Show that the time series {X,,t c Z) is weakly stationary but not strictly stationary