Note that +Et4-1+….+Et +ct2-1+….+E Since(Eta, Eta-1., Et1+1) is independent of (Et4, Et-1, . Eta+1) it follow that Yi.- Yta and Yto -yt 1 are independe Wr(r4)-Wr(r3) 2(et4+et4-1+…+et2+1)/ is independent of (et2+E2-1+….+t1+1) P For given0≤a<b≤1,Wr(b)-Wr(a)→N(0,b-a)asT→ Proof: by definition (b)-Wr(a) Et t=ITa+l =T-1/(b-md)2x(rb-r)∑s t=[Ta1*+1 The last term(Tb"-Tal*-1/22-p -t=ITa] +1 Et -N(O, 1) by the CLt, and T-12(T-Ta)12=(T-Tam)2→(b-a)/2asT→∞. Hence Wr(b)-Wr(a)-N(0, b-a)Proof: Note that Yt4 − Yt3 = εt4 + εt4−1 + ... + εt3+1, Yt2 − Yt1 = εt2 + εt2−1 + ... + εt1+1. Since (εt2 ,εt2−1,...,εt1+1) is independent of (εt4 ,εt4−1,...,εt3+1) it follow that Yt4 − Yt3 and Yt2 − Yt1 are independent. Consequently, WT (r4) − WT (r3) = T −1/2 (εt4 + εt4−1 + ... + εt3+1)/σ is independent of WT (r2) − WT (r1) = T −1/2 (εt2 + εt2−1 + ... + εt1+1)/σ. Proposition: For given 0 ≤ a < b ≤ 1, WT (b) − WT (a) L−→ N(0,b − a) as T → ∞. Proof: By definition WT (b) − WT (a) = T −1/2 [T b] X∗ t=[T a] ∗+1 εt = T −1/2 ([Tb] ∗ − [Ta] ∗ ) 1/2 × ([Tb] ∗ − [Ta] ∗ ) −1/2 [T b] X∗ t=[T a] ∗+1 εt . The last term ([Tb] ∗ − [Ta] ∗ ) −1/2 P[T b] ∗ t=[T a] ∗+1 εt L−→ N(0, 1) by the CLT, and T −1/2 ([Tb] ∗ − [Ta] ∗ ) 1/2 = (([Tb] ∗ − [Ta] ∗ )/T) 1/2 → (b − a) 1/2 as T → ∞. Hence WT (b) − WT (a) L−→ N(0,b − a). 5