Let {xt ; t = 0, 1, 2, . . .} be iid(0, 2 ). (a) For h 1 and k 1, show

Let {xt

; t = 0, ±1, ±2, . . .} be iid(0, σ2

).

(a) For h ≥ 1 and k ≥ 1, show that xt xt+h and xs xs+k are uncorrelated for all s , t.

(b) For fixed h ≥ 1, show that the h × 1 vector

σ

−2 n

−1/2Õn t=1

(xt xt+1, . . ., xt xt+h)

0 d→ (z1, . . ., zh)

0 where z1, . . ., zh are iid N(0, 1) random variables. [Hint: Use the Cramér-Wold device.

(c) Show, for each h ≥ 1, n −1/2 "Õn t=1 xt xt+h −
Õn−h t=1 (xt − x¯)(xt+h − x¯)
#
p→ 0 as n → ∞
where x¯ = n −1 Ín t=1 xt .

(d) Noting that n −1 Ín t=1 x 2 t p→ σ
2 by the WLLN, conclude that n 1/2 [ρˆ(1), . . ., ρˆ(h)]
0 d→ (z1, . . ., zh)
0 where ρˆ(h) is the sample ACF of the data x1, . . ., xn.