(Solution) - Let ei t 1 0 1 be -(2025 Original AI-Free Solution)

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Let {ei: t = - 1, 0, 1, ...} be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by
x, = et - (l/2)e1-1 + (l/2)e1-2, t = 1,2,....
(i) Find E(xt) and Var(xt). Do either of these depend on t?
(ii) Show that Corr(xt" xt+1) = -1/2 and Corr(xt, x1+2) = 1/3. (It is easiest to use the formula in Problem 11.1.)
(iii) What is Corr(x1, xl+h) for h > 2?
(iv) Is {xt} an asymptotically uncorrelated process?