8.3 In the same setting as Exercise 8.2, suppose it is desired to predict a new signal S0 given observations Z = [Z1, ..., Zn]

T. Here, it is assumed that

[S0, ST]

T are jointly multivariate normal with E(S0) = ????0, var(S0) = ????00 and cov(S0, Si

) = ????0i

, i = 1, ..., n. The best linear predictor takes the form Ŝ

0 = ????0 + cov(S0, Z)

T var(Z)

−1(Z − E(Z))

with prediction variance

????00 − cov(S0, Z)

T var(Z)

−1 cov(Z, S0).

Show that cov(S0, Z) and var(Z) have elements cov(S0, Zi

) = ????0i gi

, cov(zi

, zj

) = {gi gj

{

exp (

????ij)

− 1

}

, i ≠ j, g2 i

{

exp (

????ii)

− 1

}

+ gi

, i = j, for i ≠ j = 1, ..., n and where gi = exp(????i + 1 2

????ii).