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**lpd** Can someone help me with the follow textbook question from Time Series: Theory and Methods by Brockwell.

$\displaystyle Suppose\, that\, m_t = c_0 + c_1 + c_2t^2, \,t=0,\, \pm1,\, \pm2, ...$

a) $\displaystyle Show \, that\, m_t = \sum^2_{i=-2} a_im_{t+1} = \sum^3_{i=-3} b_im_{t+1}, \,t=0,\, \pm1,\, \pm2, ...$

$\displaystyle \, where \, a_2=a_{-2}=\frac{3}{35}, \, a_1=a_{-1}=\frac{7}{35}, a_0=\frac{17}{35}, \, b_3=b_{-3}=\frac{2}{21},\, b_2=b_{-2}=\frac{3}{21},\, b_1=b_{-1}=\frac{6}{21}, b_0=\frac{7}{21}$

I can do this with ease.

But i am stuck with this one :

b. $\displaystyle Suppose\, that\, X_t =m_t + Z_t \,where\, (Z_t, \, t=0,\, \pm1,\, \pm2, ...),$ $\displaystyle \, is\, a\, sequence\, of\, independent\, normal\, random\, variables,\, each\, with\, mean\, 0\, and\, variance\, \sigma^2.$

$\displaystyle \, Let\, U_t = \sum^2_{i=-2}{a_iX_{t+i}}\, and\, V_t=\sum^3_{i=-3}{a_iX_{t+i}}.$

$\displaystyle i Find\, the\, means\, and\, variances\, of\, U_t\, and\, V_t\,$

$\displaystyle ii. Find\, the\, Correlations\, between\, U_t\,and\,U_{t+1}\,and\,between\,V_t\,and\,V_{t+1}$

Thank=yoU!!

Thankyou!!