Problem Set 3 Solutions
نویسنده
چکیده
(b) If a process X(t) is strongly stationary, all of its moments have to be independent of t. The third moment of X(t), E(X(t)) = (EA)(cos(ωt) + sin(ωt)), is not a constant with respect to t unless EA = 0. Clearly we can find many examples of distributions for A so that the third moment is not zero while E(A) = 0. For example take A to be a discrete random variable with P (A = −1) = 1/3, P (A = 1/2) = 2/3. Then E(A) = −1 × 1/3 + 1/2 × 2/3 = 0 and E(A) = −1× 1/3 + 1/8× 2/3 = −1/4. We could just as well consider the fourth order moment:
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