Response to the following question:
Question 1. Suppose that {Y_t} is stationary with autocovariance function ?_k.
a) Show that W_t=?Y_t=Y_t-Y_(t-1) is stationary by finding the mean and autocovariance function for {W_t} .
b) Show that U_t=?^2 Y_t=?[Y_t-Y_(t-1) ]= Y_t-2Y_(t-1)+Y_(t-2) is stationary. (you need not find the mean and autocovariance function for {U_t}.)
Question 2. Suppose Cov(X_t,X_(t-k))=?_k is free of t but that E(X_t )=3t.
a) Is {X_t} stationary?
b) Let Y_t=7-3t+X_t. Is {Y_t} stationary?
Question 3. Two processes {Z_t} and {Y_t} are said to be independent if for any time points t_1, t_2, ............t_m and
s_1,s_2, ..............s_nthe random variables {Z_(t_1 ),Z_(t_2 ),...............,Z_(t_m )} are independent of the random variables {Y_(s_1 ),Y_(s_2 ),...............,S_(s_n )}.
a) Show that if {Z_t} and {Y_t} are independent stationary processes, then W_t=Z_t+Y_t is stationary.