Auto-covariance function. PLEASE HELP!!!?

Question

Let {Xt} be a stationary process with auto-covariance function

gamma(h). Show that for any choice of constants Cj, j= 0,1,2... the process Yt= CoX(t) + C1X(t-1) + C2X(t-2). Find its auto-covariance function.

Any help would be much appreciated. thanks!

Answer 1

Your question seems to be incomplete.

"Show that for any choice of constants Cj, j= 0,1,2... the process Yt= CoX(t) + C1X(t-1) + C2X(t-2)"

We are to show what, exactly?  That Y is stationary? That ....?

We know that E{X(t)X(t+h)} = gamma(h).

Then

E{Y(t)Y(t+h)}

is the sum of nine terms. They are

E{C0^2*X(t)X(t+h) } = C0^2*gamma(h)

E{C0*C1*X(t)*X(t+h-1)} = C0*C1*gamma(h-1)

E{C0*C2*X(t)*X(t+h-2)} = C0*C2*gamma(h-2)

E{C1*C0*X(t-1)*X(t+h)} = C0*C1*gamma(h+1)

E{C1^2*X(t-1)*X(t+h-1)} = C1^2*gamma(h)

E{C1*C2*X(t-1)*X(t+h-2)} = C1*C2*gamma(h-1)

E{C2*C0*X(t-2)*X(t+h)} = C0*C2*gamma(h+2)

E{C2*C1*X(t-2)*X(t+h-1)} = C1*C2*gamma(h+1)

E{C2^2*X(t-2)*X(t+h-2)} = C2^2*gamma(h)

That sum can be simplified to

C0*C2*gamma(h-2) +

(C0*C1+ C1*C2)*gamma(h-1) +

(C0^2+C1^2+C2^2)*gamma(h) +

(C0*C1+C1*C2)*gamma(h+1) +

C0*C2*gamma(h+2)