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

Let {xt; t = 0, ±1, ±2, . . .} be iid(0, σ2).

(a) For h ≥ 1 and k ≥ 1, show that xtxt+h and xsxs+k are uncorrelated for all s 6= t.

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

σ−2n−1/2Xn t=1

(xtxt+1, . . . , xtxt+h)

0 d

→ (z1, . . . , zh)

0 where z1, . . . , zh are iid N(0, 1) random variables. [Note: the sequence

{xtxt+h; t = 1, 2, . . .} is h-dependent and white noise (0, σ4). Also, recall the Cram´er-Wold device.]

(c) Show, for each h ≥ 1, n−1/2

"

Xn t=1 xtxt+h −

n X−h t=1

(xt − x¯)(xt+h − x¯)

#

p

→ 0 as n → ∞

where ¯x = n−1 Pn t=1 xt.

(d) Noting that n−1 Pn t=1 x2 t

p

→ σ2, conclude that n1/2 [ρb(1), . . . , ρb(h)]0 d

→ (z1, . . . , zh)

0 where ρb(h) is the sample ACF of the data x1, . . . , xn.