Problem 4 Consider the time series process {X} with mean E[X,] = 0 and covariance function y(i, j) = E[XX,]. Define the nth innovation as Un = Xn - Xn, where if n = 1, Xn = P (Xml X n - 1 , ..., X1) if n = 2,3, .... 4.i In the innovations algorithm, show that for each n 2 2, the innovation Un = X, - Xn is uncor- related with X1, ..., Xn-1. Conclude that Un = Xn - Xn is uncorrelated with the innovations X1 - X1, ..., Xn-1 - Xn-1. 4.ii Derive the update step for 0 in the innovations algorithm, i.e., derive the expression On. n - k = Vx1 ( y ( n + 1, k + 1) - > k. k -jenn-juj ), Ock 0mn-juj. 1=0 Hint: Start with the expression Xnti = >Onj(Xn+1-j - Xn+1-j) and multiply each side by innovation Uk+1 = (Xk+1 - Xx+1)<.>