If {Xt} and {Yt} are uncorrelated stationary processes, i.e. if Xs and Yt are uncorrelated for every s and t, illustrate that {Xt+Yt } is stationary and calculate its auto-covariance function in terms of auto-covariance functions of {Xt} and {Yt}.

Let et~IID N(0,1) and state Xt=etet-1. Illustrate that {Xt} is white noise sequence. This gives the example of dependent white noise sequence. Two bonus points for rigorously proving Xt and Xt+1 are not independent.

Let Xt=1+et-0.5et-1, t=1,2,3,..., where et~IID N(0,2), variance of et is 2.

a. Calculate the mean of Xt.
b. Calculate the variance of Xt.
c. Determine the correlation between Xt and et-1.