A real-valued function g(t), defined on the integers, is non-negative definite if and only if Xn i=1 Xn j=1 aig(ti − tj )aj ≥ 0 for all positive integers n and for all vectors a = (a1, a2, . . . , an)0 and t =

(t1, t2, . . . , tn)0

. For the matrix G = {g(ti − tj ); i, j = 1, 2, . . . , n}, this implies that a0 Ga ≥ 0 for all vectors

a. It is called positive definite if we can replace

‘≥’ with ‘>’ for all a 6= 0, the zero vector.

(a) Prove that γ(h), the autocovariance function of a stationary process, is a non-negative definite function.

(b) Verify that the sample autocovariance γb(h) is a non-negative definite function.

Section 1.7