Jo w(r) t=1 (d)T-∑Y-1lo2(1)2-1 (e)T-i[w(1)-o w(r)drI (r∑=1nW() (g)T3∑tY2 ∫r{W(r)dr a joint weak convergence for the sample moments given above to their respective limits is easily established and will be used below Proof: (a) is a straightforward results of Donsker's Theorem with r= 1 (b) First rewrite T->Y2, in term of Wr(rt-1)=T-/Y-10=T-1/2>us/o Wr(r)is constant for(t-l)/T <r</T, we have 7 'vrIvri-1)2. Because where rt-1=(t-1)/T, so that T-2Y21=02T->(r ∑Wr(n-12=∑/Wr()3tr t=1 Wr(r)dr The continuous mapping theorem applies to h(Wr)=o wr(r)2dr. It follows that hi(Wr)→h(), so that t-2∑1Y21→→o2/bw(n)ab, as claimed (c). The proof of item(c)is analogous to that of(b). First rewrite T-3/2>tYt n term of Wr(rt-1=T-12Yi-1/o=T-1/2e us/ o, where rt-1=(t-1)/T so that T-3/2yT ta wr(rt-1). Because Wr(r) is constant for(c) T − 3 2 P T t=1 Yt−1 L−→ σ R 1 0 W(r)dr, (d) T −1 P T t Yt−1ut L−→ 1 2 σ 2 [W(1)2 − 1], (e) T − 3 2 P T t=1 tut L−→ σ[W(1) − R 1 0 W(r)dr], (f) T − 5 2 P T t=1 tYt−1 L−→ σ R 1 0 rW(r)dr, (g) T −3 P T t=1 tY 2 t−1 L−→ σ 2 R 1 0 r[W(r)]2dr. A joint weak convergence for the sample moments given above to their respective limits is easily established and will be used below. Proof: (a) is a straightforward results of Donsker’s Theorem with r = 1. (b) First rewrite T −2 P T t=1 Y 2 t−1 in term of WT (rt−1) ≡ T −1/2Yt−1/σ = T −1/2 t P−1 s=1 us/σ, where rt−1 = (t − 1)/T, so that T −2 P T t=1 Y 2 t−1 = σ 2T −1 P T t=1 WT (rt−1) 2 . Because WT (r) is constant for (t − 1)/T ≤ r < t/T, we have T −1X T t=1 WT (rt−1) 2 = X T t=1 Z t/T (t−1)/T WT (r) 2 dr = Z 1 0 WT (r) 2 dr. The continuous mapping theorem applies to h(WT ) = R 1 0 WT (r) 2dr. It follows that h(WT ) =⇒ h(W), so that T −2 PT t=1 Y 2 t−1 =⇒ σ 2 R 1 0 W(r) 2dr, as claimed. (c). The proof of item (c) is analogous to that of (b). First rewrite T −3/2 PT t=1 Yt−1 in term of WT (rt−1) ≡ T −1/2Yt−1/σ = T −1/2 Pt−1 s=1 us/σ, where rt−1 = (t − 1)/T, so that T −3/2 PT t=1 Yt−1 = σT −1 PT t=1 WT (rt−1). Because WT (r) is constant for 10