Let X 0 , . . . , X N−1 be a sample of size N from a stationary process with an unknown mean and an SDF S(·) defined over [−1/2, 1/2] (i.e., the sampling interval ∆t is taken to be 1). At lag τ = 0...

Let X₀, . . . , X_N−1
be a sample of size N from a stationary process with an unknown mean and an SDF S(·) defined over [−1/2, 1/2] (i.e., the sampling interval ∆t is taken to be 1). At lag τ = 0 both the unbiased and biased estimators of the ACVS reduce to

(a) Show that

(b) Show that

where F(·) is Fejer’s kernel (see Equation (174c)).

(c) Plot 1 − F(f)/N versus f for, say, N = 64. Based upon this plot, for what kind of S(·) would there be a large discrepancy between