Solution of Assignment 2
Problem 2.3
a. The process Xt = Zt + 0.3Zt1 0.4Zt2 where Zt W N (0, 1) is stationary. Note
that
X (t) = E [Xt ] = 0;
and
X (h) = Cov (Xt+h ; Xt ) = Cov (Zt+h + 0.3Zt+h1 0.4Zt+h2 ; Zt + 0.3Zt1 0.4Zt2 )
1.25 if h = 0
0.18 if |h| =
Solution of Assignment 3
Problem 2.15
By the properties on the best linear prediction (Brockwell and Davis (2002), page 68), we have
for n > p
Pn Xn+1 = Pn (1 Xn + . + p Xnp+1 ) + Pn (Zn+1 )
= Pn (1 Xn ) + . + Pn (p Xnp+1 ) + Pn (Zn+1 )
= 1 Pn (Xn ) + . +
Solution of Assignment 4
Problem 4.6
By Problem 4.5, let cfw_Yt and cfw_Wt be two uncorrelated stationary processes with spectral
distribution functions FY (.) and FW (.). Then, the spectral distribution function of cfw_Yt + Wt
is given by FW (.) + FY