Exercise 6.9.1. Show that the following functions are nonnegative definite. (a) (k) = 1

Exercise 6.9.1. Show that the following functions are nonnegative definite.

(a)

σ (k) =

⎧⎪⎨

⎪⎩

1 ifk = 0 14 27 if k = ±1 4

27 if k = ±2 0 other k .

(b)

σ (k) =

⎧⎪⎨

⎪⎩

1 ifk = 0

ρ if k = ±1

ρ2 if k = ±2 0 other k .

The function σ (k) is nonnegative definite if, for any n, the n×n matrix Σ ≡ [σ (i−

j)] is nonnegative definite.

Exercise 6.9.2. Let α and β be random variables with E(α) = E(β) =

0,Var(α) = Var(β) =σ 2, and Cov(α,β) = 0. Show that yt =α cos(2πνt)+β sin(2πνt)

is a second-order stationary process.

Exercise 6.9.3. Suppose et is second-order stationary. Which of the following processes are second-order stationary?

(a) yt = exp[et ].

(b) yt = yt−1+et .

(c) yt = xtet , where xt is another second-order stationary process independent of et .