Recall that the auto-covariance function (h) of a stationary process is a function that for all h satisfies (i) (0) 0, (ii) (h) (0), (iii) (h) = (h) (even function) and (iv) is non-negative definite. In fact we can prove that "any real-valued function defined on the integers is the auto-covariance function of a stationary time series if and only if it is even and non-negative definite". Also recall that a real-valued function k(i) defined on the integers is non-negative definite if for all positive integers n and values a1, ..., an R, we have

nn

Problem 1 . ( 10 mark ) Recall that the auto-covariance function y ( h ) of a stationary process is a function that for all h satisfies ( 1 ) ( ) 2 0 , ( 11 ) ( )