Establishing conditions for weak convergence to stochastic integrals

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1 Establishig coditios for weak covergece to stochastic itegrals August 7, 26 Jiagya Peg School of Mathematical Scieces School of Maagemet ad Ecoomics Uiversity of Electroic Sciece ad Techology of Chia, Chegdu, 673, P. R. Chia Qiyig Wag School of Mathematics ad Statistics Uiversity of Sydey, NSW, 26, Australia Abstract Limit theory ivolvig stochastic itegrals plays a major role i time series ecoometrics. I earlier cotributios o weak covergece to stochastic itegrals, the literature commoly uses martigale ad semimartigale structures. Liag, et al (25 (see also Wag (25, Chapter 4.5 curretly exteded the weak covergece to stochastic itegrals by allowig for the liear process i the iovatios. While these martigale ad liear processes structures have wild relevace, they are ot sufficietly geeral to cover may ecoometric applicatios where edogeeity ad oliearity are preset. This paper provides ew coditios for weak covergece to stochastic itegrals. Our frameworks allow for log memory processes, causal processes ad ear-epoch depedece i the iovatios, which ca be applied to a wild rage of areas i ecoometrics, such as GARCH, TAR, biliear ad other oliear models. Key words ad phrases: Stochastic itegral, covergece, log memory process, ear-epoch depedece, liear process, causal process, TAR model, biliear model, GARCH model. JEL Classificatio: C22, C65 Correspodig author: School of Mathematics ad Statistics, The Uiversity of Sydey, NSW, 26, Australia. qiyig@maths.usyd.edu.au.

2 2 Itroductio I ecoometrics with ostatioary time series, it is usually ecessary to rely o the covergece to stochastic itegrals. The latter result is particularly vital to oliear coitegratig regressio. See Wag ad Phillips (29a, 29b, 26 for istace. Also see Wag (25, Chapter 5 ad the referece therei. Write Let (u j, v j j be a sequece of radom vectors o R d R ad F k = σ(u j, v j, j k. x k = d k u j, y k = k v j, where < d 2. As a bechmark, the basic result o covergece to stochastic itegrals is give as follows. See, e.g., Kurtz ad Protter (99. THEOREM.. Suppose A (v k, F k forms a martigale differece with sup k Ev 2 k < ; A2 {x, t, y, t } {G t, W t } o D R d+[, ] i the Skorohod topology. The, for ay cotiuous fuctios g(s ad f(s o R d, we have { x, t, y, t, { G t, W t, o D R 2d+2[, ] i the Skorohod topology. g(x k, g(g t dt, k= f(x k v k+ } f(g t dw t }, (. Kurtz ad Protter (99 [also see Jacod ad Shiryaev (23] actually established the result with y k beig a semimartigale istead of A. Toward a geeral result beyod the semimartigale, Liag, et al. (25 ad Wag (25, Chapter 4.5 ivestigated the extesio to liear process iovatios, amely, they provided the covergece of sample quatities f(x k w k+ to fuctioals of stochastic processes ad stochastic itegrals, where k= w k = ϕ j v k j, (.2 j= with ϕ = ϕ j ad j ϕ j <. Liag, et al. (25 ad Wag (25, Chapter j= j= 4.5 further cosidered the extesio to α-mixig iovatios.

3 3 While these results are elegat, they are ot sufficietly geeral to cover may ecoometric applicatios where edogeeity ad more geeral iovatio processes are preset. I particular, the liear structure i (.2 is well-kow restrictive, failig to iclude may practical importat models such as GARCH, threshold, oliear autoregressios, etc. The aim of this paper is to fill i the gap, providig ew geeral results o the covergece to stochastic itegrals i which there are some advatages i ecoometrical applicatios. Explicitly, our frameworks cosider the covergece of S := f(x k w k+, where the w k has the form: w k = v k + z k z k, (.3 with z k satisfyig certai regular coditios specified i ext sectio. The {w k } k i (.3 is usually ot a martigale differece, but w k = v k + z z provides a approximatio to martigale. Martigale approximatio has bee widely ivestigated i the literature. For a curret developmet, we refer to Borovskikh ad Korolyuk (997. As evideced i Sectio 3, these existig results o martigale approximatio provide importat techical support for the purpose of this paper. This paper is orgaized as follows. I Sectio 2, we establish two frameworks for the covergece of S. Theorem 2. icludes the situatio that u k is a log memory process, while Theorem 2.2 is for the u k to be a short memory process. It is show that, for a short memory u k, the additioal term z k i (.3 has a essetial impact o the limit behaviors of S, but it is ot the case whe u k is a log memory process uder mior atural coditios o the z t. Sectio 3 provides three corollaries of our frameworks o log memory processes, causal processes ad ear-epoch depedece, which capture the most popular models i ecoometrics. More detailed examples icludig liear processes, oliear trasformatios of liear processes, oliear autoregressive time series ad GARCH model are give i Sectio 4. We coclude i Sectio 5. Proofs of all theorems are postpoed to Sectio 6. Throughout the paper, we deote costats by C, C, C 2,..., which may differ at each appearace. D R d[, ] deotes the space of càdlàg fuctios from [, ] to R d. If x = (x,..., x m, we make use of the otatio x = m x j. For a sequece of icreasig σ-fields F k, we write P k Z = E(Z F k E(Z F k for ay E Z <, ad Z L p (p > if Z p = (E Z p /p <. Whe o cofusio occurs, we geerally use the idex otatio x k (y k for x,k (y,k. Other otatio is stadard. k=

4 4 2 Mai results I this sectio, we establish frameworks o covergece to stochastic itegrals. Except metioed explicitly, the otatio is the same as Sectio. THEOREM 2.. I additio to A A2, suppose that sup k E ( z k u k < ad d 2 /. The, for ay cotiuous fuctio g(s o R d ad ay fuctio f(x o R d satisfyig a local Lipschitz coditio 2, we have { x, t, y, t, { G t, W t, g(x k, g(g t dt, k= f(x k w k+ } f(g s dw s }. (2. As oticed i Liag, et al. (25, the local Lipschitz coditio is a mior requiremet ad hold for may cotiuous fuctios. If sup k E ( u k 2 + z k 2 <, it is atural to have sup k E ( z k u k < by Hölder s iequality. Theorem 2. idicates that, whe d 2 /, the additioal term z k i (.3 do ot modify the limit behaviors uder mior atural coditios o z k ad f(x. The coditio d 2 / usually holds whe the compoets of u t are log memory processes. See Sectio 3. for example. The situatio becomes very differet if d 2 / σ 2 < for a costat σ, which geerally holds for short memory processes u t. I this situatio, as see i the followig theorem, z t has a essetial impact o the limit distributios. Let Df(x = ( f x,..., f x d. The followig additioal assumptios are required for our theory developmet. A3. Df(x is cotiuous o R d ad for ay K >, Df(x Df(y C K x y β, for some β >, for max{ x, y } K, where C K is a costat depedig oly o K. A4. (i sup k E u k 2 < ad sup k E z k 2+δ < for some δ > ; 2 That is, for ay K >, there exists a costat C K such that, for all x + y < K, f(x f(y C K d x j y j.

5 5 (ii Ez k u k A = (A,..., A d, as k ; Set λ k = z k u k Ez k u k. (iii sup k 2m E ( λ k F k m = op (, as m ; or (iii sup k 2m E E ( λ k F k m = o(, as m. THEOREM 2.2. Suppose d 2 / σ 2, where σ 2 > is a costat. Suppose A A4 hold. The, for ay cotiuous fuctio g(s o R d, we have { x, t, y, t, { G t, W t, g(x k, g(g t dt, k= f(g s dw s + σ f(x k w k+ } d f } A j (G s ds. (2.2 x j Remark. Coditio A3 is similar to that i previous work. See, e.g, Liag, et al. (25 ad Wag (25. The momet coditio sup k E z k 2+δ < for some δ > i A4 (i is required to remove the effect of higher order from z k. I terms of the covergece i (2.2, sup k E z k 2 < is essetially to be ecessary. It is ot clear at the momet if the δ i A4 (i ca be reduced to zero. Remark 2. If w k satisfies (.2, we may write w k = ϕv k + z k z k, where z k = ϕ j v k j with ϕ j = ϕ m, i.e., w k ca be deoted as i the structure of (.3. See, j= m=j+ e.g., Phillips ad Solo (992. For this w k, Theorem 4.9 of Wag (25 [also see Liag, et al. 25] established a result that is similar to (2.2 by assumig (amog other coditios that, for ay i, ϕ j E ( u j+i v i F i = A, a.s., (2.3 j= where A is a costat. Sice it is required to be held for all i, (2.3 is difficult to be verified for the u k to be a oliear statioary process such as u k = F (ɛ k, ɛ k,..., eve i the situatio that (ɛ k, v k are idepedet ad idetically distributed (i.i.d. radom vectors. I compariso, A4 (ii ad (iii [or (iii ] ca be easily applied to statioary causal processes ad mixig sequeces, as see i Sectio 3. Remark 3. We have w k = v k + (z z, idicatig that provides a approximatio to the martigale w k v k, uder give coditios. However,

6 6 w k is ot a semi-martigale as cosidered i Kurtz ad Protter (99, sice we do ot require the coditio sup E z k z k <. As a cosequece, Theorems provide a essetial extesio for the covergece to stochastic itegrals, rather tha a simple corollary of the previous works. 3 Three useful corollaries This sectio ivestigates the applicatios of Theorems 2. ad 2.2. Sectio 3. cosiders the situatio that u k is a log memory process ad w k is a statioary causal process. Sectio 3.2 cotributes to the covergece for both u k ad w k beig statioary causal processes. Fially, i Sectio 3.3, we ivestigate the impact of ear-epoch depedece i covergece to stochastic itegrals. The detailed verificatio of assumptios for more practical models such as GARCH ad oliear autoregressive time series will be preseted i Sectio Log memory process Let (ɛ i, η i i Z be i.i.d. radom vectors with zero meas ad Eɛ 2 = Eη 2 =. Defie a log memory liear process u k by u k = ψ j ɛ k j, where ψ j j µ h(j, /2 < µ < ad h(k is a fuctio that is slowly varyig at. Let F be a measurable fuctio such that w k = F (..., η k, η k, k Z, is a well-defied statioary radom variable with Ew = ad Ew 2 <. The w k is kow as a statioary causal process that has bee extesively discussed i Wu (25, 27 ad Wu ad Mi (25. k Defie x k = d u j ad y k = covergece of k= k w j, where d 2 = var( u j. To ivestigate the f(x k w k+, we first itroduce the followig otatio. Write F k = σ(ɛ i, η i, i k ad assume i P w i 2 <. The latter coditio implies

7 7 that E(vk 2 + z2 k <, where v k = P k w i+k, z k = i= E(w i+k F k. See Lemma 7 of Wu ad Mi (25, amely, (35 there. All processes w k, v k ad z k are statioary satisfyig the decompositio: w k = v k + z k z k. (3. ( We ext let ρ = Eɛ v = ρ Eɛ w i, Ω = i= ρ Ev 2, (B t, B 2t be a bivariate Browia motio with covariace matrix Ω t ad B t be a stadard Browia motio idepedet of (B t, B 2t. We further defie a fractioal Browia motio B H (t depedig o (B t, B t by B H (t = A(d [(t s d ( s d ] db s + t (t s d db s, where ( [ ] 2ds /2. A(d = 2d + + ( + s d s d After these otatio, a simple applicatio of Theorem 2. yields the followig result i the situatio that u k is a log memory process ad w k is a statioary causal process. THEOREM 3.. Suppose i P w i 2 < ad, for some ɛ >, i +ɛ E w i wi 2 <, (3.2 where w k = F (..., η, η, η,..., η k, η k ad {η k } k Z is a i.i.d. copy of {η k } k Z ad idepedet of (ɛ k, η k k Z. The, for ay cotiuous fuctio g(s ad ay fuctio f(x satisfyig a local Lipschitz coditio, we have { x, t, y, t, g(x k, { B 3/2 µ (t, B 2t, k= g [ B 3/2 µ (t ] dt, f(x k w k+ } f [ B 3/2 µ (t ] db 2t }. (3.3 We remark that coditio i P w i 2 < is close to be ecessary. As show i the proof of Theorem 3. (see Sectio 6, coditio (3.2 ca be replaced by [ 2 E P k (w i+k wi+k], as k, i=

8 8 which is required to remove the correlatio betwee ɛ j ad v j for j so that a bivariate process (B H (t, B 2t depedig o (B t, B t, B 2t ca be defied o D R 2[, ]. Without this coditio or equivalet, the limit distributio i (3.3 may have a differet structure. Coditio (3.2 is quite weak, which is satisfied by most of the commoly used models. Examples icludig oliear trasformatios of liear processes, oliear autoregressive time series ad GARCH model will be give i Sectio Causal processes As i Sectio 3., suppose that (ɛ i, η i i Z are i.i.d. radom vectors with zero meas ad Eɛ 2 = Eη 2 =. I this sectio, we let u k = F (..., ɛ k, ɛ k ; w k = F 2 (..., η k, η k, k Z, where F ad F 2 are measurable fuctios such that both u k ad w k are well-defied statioary radom variables with Eu = Ew = ad Eu 2 + Ew 2 <, amely, both u k ad w k are statioary causal processes. This sectio ivestigates the covergece of To this ed, let F k = σ(ɛ i, η i, i k, k= f(x k w k+, where x k = k u j. z k = v k = E(u i+k F k, z 2k = E(w i+k F k P k u i+k, v 2k = i= i= P k w i+k. The followig assumptio is used i this sectio. A5 (i i P u i 2 < ; (ii i P w i 2+δ <, for some δ > ; Set λk = u k z 2k Eu k z 2k. (iii sup k 2m E ( λk F k m = op (, as m ; or (iii sup k 2m E E ( λk F k m = o(, as m. As oticed i Sectio 3., all u k, w k, z ik ad v ik, i =, 2, are statioary, havig the decompositios: u k = v k + z,k z k, w k = v 2k + z 2,k z 2k. (3.4

9 9 Furthermore A5 (i [(ii, respectively] implies that E ( v 2 + z 2 < [E ( v2 2+δ + z 2 2+δ <, respectively]. As a cosequece, it follows that E u k z 2k < ad A := Eu z 2 = E(u w i <. ( Ev 2 We further let Ω = Ev v 2 Ev v 2 Ev2 2 ad (B t, B 2t be a bivariate Browia motio with covariace matrix Ω t. We have the followig result by makig a applicatio of Theorem 2.2. THEOREM 3.2. Suppose that A3 (with d = ad A5 hold. The, for ay cotiuous fuctio g(s, we have where y k = { x, t, y, t, { B t, B 2t, k w j. g(x k, g(b s ds, k= f(x k w k+ } f(b s db 2s + A f [B s ]ds }, (3.5 Theorem 3.2 provides a quite geeral result for both u t ad w t are causal processes. I a related research, usig a quite complicated techique origiated from Jacod ad Shiryaev (23, Li ad Wag (25 cosidered the specified situatio that u t = w t. I compariso, by usig Theorem 2.2, our proof is quite simple, as see i Sectio 6. Furthermore our coditio A5 is easy to verify. A illustratio is give i the followig corollary, ivestigatig the case that u k is a short memory liear process ad w k is a geeral statioary causal process. COROLLARY 3.. Suppose that u t = ϕ j ɛ t j, where i ϕ i <. Result (3.5 j= holds true, if, i additio to A3 (with d =, k w k w k 2+δ <, for some δ >, (3.6 where w k = F 2(..., η, η, η,..., η k ad {η k } k Z is a i.i.d. copy of {η k } k Z ad idepedet of (ɛ k, η k k Z. Coditio (3.6 is required to establish A5 (ii. Whe u t = ϕ j ɛ t j with i ϕ i <, A5 (iii ca be established uder less restrictive coditio: j= k w k w k 2 < as

10 see i the proof of Corollary 3. give i Sectio 6. Some examples for w k satisfyig (3.6, icludig oliear trasformatios of liear processes, oliear autoregressive time series ad GARCH model are discussed i Sectio Near-epoch depedece Let {A k } k be a sequece of radom vectors whose coordiates are measurable fuctios of aother radom vector process {η k } k Z. Defie F t s = σ(η s,..., η t for s t ad deote by F t for F t. As i Davidso (994, {A k } k is said to be ear-epoch depedece o {η k } k Z i L P -orm for p > if A t E ( A t F t+m t m p d t ν(m, where d t is a sequece of positive costats, ad ν(m as m. For short, {A k } k is said to be L P -NED of size µ if d t A t p ad ν(m = O(m µ ɛ for some ɛ >. k k For k, let x k = u j ad y k = w j, where (u k, w k k defied o R d+ is a statioary process. This sectio ivestigates the covergece of the followig coditios: A6 (i η k = (η k,..., η km, k Z, is α-mixig of size 6 3 ; (ii (u k k is L 2 -NED of size ad u k is adapted to F k ; (iii (w k k is L 2+δ -NED of size for some δ > ; (iv E(u, w = ad E ( u 4 + w 4 <. f(x k w k+ i k= Due to the statioarity of (u k, w k k, it follows easily from A6 that where M k = (u k, w k ad Ω := lim ( E(M im Ω ρ j = ρ, (3.7 Ω 2 i, Ω = Eu u + 2 Eu u i, Ω 2 = Ew Ew w i, ρ = Eu w + (Eu w i + Eu iw. 3 For a defiitios of α-mixig, we refer to Davidso (994.

11 For a proof of (3.7, see Sectio 6. I terms of (3.7 ad A6, Corollary 29.9 of Davidso (994, Page 494 yields that, as, ( x,[t], y,[t] (Bt, B 2t, (3.8 where (B t, B 2t is a d+-dimesioal Browia motio with covariace matrix Ω t. Now, by usig Theorem 2.2, we have the followig theorem. THEOREM 3.3. Suppose A3 ad A6 hold. For ay cotiuous fuctio g(s o R d, we have { x, t, y, t, { B t, B 2t, where A = E(u w i. g(x k, g(b s ds, k= f(x k w k+ } f(b s db 2s + A Df[B s ]ds }, (3.9 Theorem 3.3, uder less momet coditios, provides a extesio of Theorem 3. i Liag, et al. (25 [see also Theorem 4. of Wag (25] from α mixig sequece to ear-epoch depedece. We metioed that NED approach also allows for our results to be used i may practical importat models such as biliear, GARCH, threshold autoregressive models, etc. For the details, we refer to Davidso (22. 4 Examples: verificatios of (3.2 ad (3.6 As i Sectio 3. ad 3.2, defie a statioary causal process by w k = F (..., η k, η k, k Z, where η i, i Z, are i.i.d. radom variables with mea zero ad Eη 2 measurable fuctio such that Ew = ad Ew 2 <. = ad F is a I this sectio, we verify (3.2 ad (3.6 for some practical importat examples, icludig liear processes, oliear trasformatios of liear processes, oliear autoregressive time series ad GARCH model. These examples partially come from Wu (25 ad Wu ad Mi (25. For the coveiece of readig, except metioed explicitly, we use the otatio as i Sectio 3, i particular, we recall the otatio that {η k } k Z is a i.i.d. copy of {η k } k Z ad idepedet of (ɛ k, η k k Z, ad w k = F (..., η, η, η,..., η k, η k ad w k = F (..., η, η, η,..., η k, η k.

12 2 We metio that, due to the statioarity of w k ad i.i.d. properties of η k, E P w p E w w p ( C p E w w p + E w + w+ p, (4. for ay p, where C p is a costat depedig oly o p. As a cosequece, both (3.2 ad (3.6 hold if we ca prove E w w 2+δ C 4 3δ, (4.2 for some δ > ad all sufficietly large. 4. Liear process ad its oliear trasformatio Cosider a liear process w k defied by w k = θ j η k j with Eη =. Routie calculatio show that w k w k = θ k(η η ad w k wk = θ j+k (η j η j. Hece, if j θ j <, j= j= j 2+δ θj 2 < ad E η 2+δ < for some δ >, the (3.2 ad (3.6 hold true. Ideed (3.6 follows from k w k w k 2+δ k θ k η η 2+δ < ; ad (3.2 from i +δ w i wi 2 2 = i +δ E[ θ j (η i j ηi j] 2 i +δ j=i θ 2 j E[(η η ] 2 C j=i j 2+δ θj 2 <. The result above ca be easily exteded to a oliear trasformatio of w k. To see the claim, let h k = G(w k EG(w k, where G is a Lipschitz cotiuous fuctio, i.e., there exists a costat C < such that G(x G(y C x y, for all x, y R. (4.3 It is readily see that (3.2 ad (3.6 still hold true with the w k beig replaced by h k by usig the followig facts: h k h k C w k w k ad h k h k C w k wk.

13 3 4.2 Noliear autoregressive time series Let w be geerated recursively by where R is a measurable fuctio of its compoets. Let w = R(w, η, Z, (4.4 R(x, η R(x, η L η = sup x x x x be the Lipschitz coefficiet. Suppose that, for some q > 2 ad x, E(log L η < ad E(L q η + x R(x, η q <. (4.5 Lemma 2 (i of Wu ad Mi (25 proved that there exist C = C(q > ad r q (, such that, for all N, E w w q Cr q. (4.6 Sice (4.6 implies (4.2, the w defied by (4.4 satisfies (3.2 ad (3.6. We metio that the w defied by (4.4 is a oliear autoregressive time series ad the coditio (4.5 ca be easily verified by may popular oliear models such as threshold autoregressive (TAR, biliear autoregressive, ARCH ad expoetial autoregressive (EAR models. The followig illustratios come from Examples 3-4 i Wu ad Mi (25. TAR model: w = φ max(w, + φ 2 max( w, + η. implies that if L η satisfied. Simple calculatio = max( φ, φ 2 < ad E( η q < for some q >, the (4.5 is Biliear model: w = (α + β η w + η, where α ad β are real parameters ad E( η q < for some q >. Note that L η = α + β η. (4.5 holds if oly E(L q η <. 4.3 GARCH model Let {w t } t be a GARCH(l, m model defied by w t = h t η t ad h t = α + m l α i wt i 2 + β j h t j, (4.7 where η t i.i.d. with Eη = ad Eη 2 =, α >, α j for j m, β i m for i l, ad h = O p (. It is well-kow that, if α i + l β j <, the w t

14 4 is a statioary process havig the followig represetatio (see, e.g., Theorem i Taiguchi ad Kakizawa (2: Y t = M t Y t + b t with M t = (θηt 2, e,..., e m, θ, e m+,..., e l+m T, where Y t = (wt 2,..., wt m+, 2 h t,..., h t l+ T ad b t = (α ηt 2,,...,, α,,..., T ad θ = (α,..., α m, β,..., β l T ; e i = (,...,,,,..., T is the uit colum vector with ith elemet beig, i l + m. Suppose that E η 4 < ad ρ[e(m 2 t ] <, where ρ(m is the largest eigevalue of the square matrix M ad is the usual Kroecker product. Propositio 3 i Wu ad Mi (25 implies for some C < ad r (,, E( w w 4 Cr. (4.8 Sice (4.8 implies (4.2, the w defied by (4.7 satisfies (3.2 ad ( Coclusio O weak covergece to stochastic itegrals, we have show that the commoly used martigale ad semimartigale structures ca be exteded to iclude the log memory processes, the causal processes ad the ear-epoch depedece i the iovatios. Our frameworks ca be applied to GARCH, TAR, biliear ad other oliear models. I ecoometrics with o-statioary time series, it is usually ecessary to rely o the covergece to stochastic itegrals. The authors hope these results derived i this paper prove useful i the related areas, particularly, i oliear coitegratig regressio where edogeeity ad oliearity play major roles. 6 Proofs This sectio provides the proofs of all theorems. Except metioed explicitly, the otatio used i this sectio is the same as i previous sectios.

15 5 Proof of Theorem 2.. We may write f(x k w k+ = f(x k (v k+ + z k z k+ = f(x k v k+ + [ f ( x k f ( x,k ] z k + o p ( = f(x k v k+ + R + o P (, say. (6. Write Ω K = {x i : max i x i K}. Sice f satisfies the local Lipschitz coditio, it is readily see from sup k E z k u k < that, as, E R I(Ω K C K d E z k u k C K (/d 2 /2. This implies that R = o P ( due to P (Ω K, as K. Theorem 2. follows from Theorem.. Proof of Theorem 2.2. We may write f(x k w k+ = f(x k (v k+ + z k z k+ = f(x k v k+ + [ f ( x k f ( x,k ] z k + o p ( = f(x k v k+ + (x k x,k Df(x,k z k + R ( + o p ( = f(x k v k+ + d where the remaider terms are E(z k u k Df(x,k + R ( + R 2 ( + o p (,(6.2 R ( = z k [f(x k f(x,k (x k x,k Df(x,k ] R 2 ( = [z k u k E(z k u k ]Df(x,k. d By virtue of Theorem., to prove (2.2, it suffices to show that R i ( = o P (, i =, 2. (6.3

16 6 To prove (6.3, write Ω K = {x i : max i x i K}. Note that A3 implies that, for ay K > ad max{ x, y } K, Df(x C K ad f(x f(y (x y Df(x C K x y +β, where β = mi{δ/(2 + δ, β} for δ > give i A4(i. The, E R ( I ( Ω K C K E ( x k x,k +β z k C K (+β /2 E( u k +β z k = O( β /2, (6.4 where we have used the fact that, due to A4(i, sup k E( u k +β z k sup k ( E uk 2 (+β /2 ( sup E zk 2+δ /(2+δ <. k This implies that R ( = O P ( β /2 due to P (Ω K as K. It remais to show R 2 ( = o P (. To this ed, let m = m ad m log. By recallig λ k = z k u k E(z k u k, we have R 2 ( = σ + σ 2m λ k Df(x,k + σ k=2m k=2m λ k Df(x,k m λ k [ Df(x,k Df(x,k m ] = R 2 ( + R 22 ( + R 23 (. As i the proof of (6.4, it is readily see from A3 that E R 2 ( I ( Ω K CK m sup E λ k C K log, k E R 23 ( I ( Ω K CK E ( x,k x,k m β λ k C K β /2 k j=k m E( u j β λ k C K β /2 log, where β = mi{δ/(2 + δ, β}. Hece R 2 ( + R 23 ( = o P ( due to P (Ω K as K. To estimate R 22 (, write IR ( = σ IR 2 ( = σ k=2m k=2m [ λk E(λ k F k m ] x k, E(λ k F k m x k,

17 7 where x k = Df(x,k m I(max j k m x j K. Due to A4 (iii ad A3, IR 2 ( C K E(λ k F k m sup E(λ k F k m = o P (. k 2m Similarly, if A4 (iii ad A3 hold, the E IR 2 ( C K E E(λ k F k m sup E E(λ k F k m = o(, k 2m which yields IR 2 ( = o P (. O the other had, we have IR ( = m IR j (, j= where IR j ( = σ k=2m [ E(λk F k j E(λ k F k j ] x k. Let λ k (j = [ E(λ k F k j E(λ k F k j ] x k. Note that, for each j, IR j ( = σ k=2m λ k (j is a martigale with sup k E λ k (j +δ C sup k E λ k +δ < for some δ >. The classical result o strog law for martigale (see, e.g., Hall ad Heyde (98, Theorem 2.2, Page 4 yields IR j ( = o a.s (log 2, for each j m log, implyig IR ( = m IR j ( = o P (. We ow have R 22 ( = o P ( due to P (Ω K as K, ad the fact that, o Ω k, R 22 ( = σ k=2m j= λ k x k = IR ( + IR 2 ( = o P (. Combiig these results, we prove R 2 ( = o P ( ad also complete the proof of (2.2. Proof of Theorem 3.. Except metioed explicitly, otatio used i this sectio is the same as i Sectio 3.. First ote that d 2 = var( u j c µ 3 2µ h 2 (, with c µ = ( µ(3 2µ x µ (x + µ dx,

18 8 i.e., d 2 /. See, e.g., Wag, Li ad Gullati (23. By recallig (3. ad usig Theorem 2., Theorem 3. will follow if we may verify A2, i.e., o D R 2[, ], ( [t] d u j, [t] ( w j B3/2 µ (t, B 2t. (6.5 We ext prove (6.5. Sice {(ɛ k, v k, F k } k forms a statioary martigale differece with covariace matrix Ω, a applicatio of the classical martigale limit theorem [see, e.g., Theorem 3.9 of Wag (25] yields that [t] ( ɛ j, [t] v j ( Bt, B 2t, (6.6 o D R 2[, ]. Recall that, for k, w k = F (..., η, η, η,..., η k, η k, where {ηk } k Z is a i.i.d. copy of {η k } k Z ad idepedet of (ɛ k, η k k Z. Let vk = P k wi+k. Note that ɛ i is idepedet of (ɛ i, vi for i. If we have the coditio: i= it follows from (6.6 that max k k (v j vj = o P (, (6.7 [t] ( ɛ j, [t] ɛ j, [t] ( v j Bt, B t, B 2t, (6.8 o D R 3[, ], where B t is a stadard Browia motio idepedet of ( B t, B 2t. Note that max k Result (6.8 implies that k w j k v j max k z k / = o P (. [t] ( ɛ j, [t] ɛ j, [t] ( w j Bt, B t, B 2t, o D R 3[, ]. As a cosequece, (6.5 follows from the cotiuous mappig theorem ad similar argumets to those i Wag, Li ad Gullati (23.

19 9 It remais to show that (3.2 implies (6.7. I fact, by otig {v k v k, F k} k forms a martigale differece, it is readily see from martigale maximum iequality that, for ay ɛ >, ( P max k 2 ɛ 2 k (v j vj ɛ 2 ɛ 2 E(v j vj 2 [ 2. E P k (w i+k wi+k] (6.9 i= By Hölder s iequality ad (3.2, we have [ 2 E P k (w i+k wi+k] i= [ ] 2 (i + k ɛ (i + k +ɛ E P k (w i+k wi+k i= C i= i +ɛ E(w i wi 2, i=k as k. Takig this estimate ito (6.9, we yield (6.7 ad also complete the proof of Theorem 3.. Proof of Theorem 3.2. As i the proof of Theorem 3., by recallig (3.4 ad usig Theorem 2.2, we oly eed to verify A2, i.e., o D R 2[, ], [t] ( u k, [t] w k (Bt, B 2t. (6. I fact, by otig that { (v k, v 2k, F k forms a statioary martigale differece with }k E ( v 2 + v2 2 <, the classical martigale limit theorem [see, e.g., Theorem 3.9 of Wag (25] yields that [t] [t] ( v k, v 2k (Bt, B 2t, o D R 2[, ], where (B t, B 2t t is a 2-dimesioal Gaussia process with zero meas, statioary ad idepedet icremets, ad covariace matrix: As a cosequece, we have Ω t = lim [t] [ ( v cov k v 2k (vk, ] v 2k = Ω t. ( x,[t], y,[t] = ( [t] [t] v k, v 2k + R,t (B t, B 2t,

20 2 due to the fact that, by recallig E( z 2 + z 2 2 <, sup R,t max ( z k + z 2k / = o P (. t k This yields (6., ad also completes the proof of Theorem 3.2. Proof of Corollary 3.. We oly eed to verify A5. First of all, simple calculatio shows that P k u i+k = ϕ i ɛ k. As a cosequece, i P u i 2 <, that is, A5 (i holds. Due to (4., A5 (ii is implied by (3.6. It remais to show that A5 (iii holds true if t w t w t 2 <, as the latter is a cosequece of (3.6. I fact, by lettig j = t= if j < k, we may write i=k E ( λk F k m = = = k m j= i= i= P j (u k z 2,k = ( max{m,i} ϕ i j=m max{m,i} ϕ i j=m + i= ϕ i j=m j=max{m,i}+ P (ɛ j i z 2,j + P k j (ɛ k i z 2,k P (ɛ j i z 2,j i= ϕ i j=max{m,i}+ P (ɛ j i w t+j := A m + A 2m. (6. t= It is readily see from E z 2k 2 = E z 2 2 < that E A m 2 i ϕ i (Eɛ 2 /2 (Ez2 2 /2, i=m as m. As for A 2m, by otig P (ɛ j i w t+j = E [ ] ɛ j i (w t+j w t+j F wheever j > i, we have E A 2m ϕ i E ɛ j i (w t+j w t+j i= C C j=m+ j=m+ t=+j t= w t w t 2 t w t w t 2, t=m as m. Takig these estimates ito (6., we obtai E [ sup E ( λk ] F k m E Am + E A 2m, k 2m

21 2 implyig A5 (iii. Proof of Theorem 3.3. First ote that, uder A6, it follows from Theorem 7.5 of Davidso (994 that w k, k Z, is a statioary L 2+δ -mixigale of size with costat w 4, E(w k F k m 2+δ C w 4 m γ, (6.2 w k E(w k F k+m 2+δ C w 4 m γ, (6.3 hold for all k, m ad some γ >. Furthermore, by Theorem 6.6 of Davidso (994, we may write w k = v k + z k z k, where, as i Sectio 3.2, v k = P k w i+k, z k = i= E(w i+k F k. It is readily see that both v k ad z k are statioary ad (v k, F k k forms a martigale differece with Ev 2 2Ew 2 + 4Ez 2 <, sice, by (6.2, the followig result holds (implyig Ez 2 < : z k,j 2+δ E(w i F 2+δ C w 4 i=j+ i=j+ i γ <, (6.4 for ay j, where z k,j = have i=j+ E(w i+k F k. By (6.2 ad (6.3, for ay k, we also E(w w k E ( w w w k + E [ w E(w k F k/2 ] w 2 { w w 2 + E(w k F k/2 2 } C w 2 w 4 k γ, (6.5 where w = E(w F k/2. The result (6.5 will be used later. Sice w k has structure (.3 with the v k satisfyig A, (3.8 implies A2 ad A6 (iii ad (6.4 with j = imply A4 (i, by usig Theorem 2.2, Theorem 3.3 will follow if we prove (3.7 ad sup E E ( λ k F k m, (6.6 k 2m

22 22 where λ k = z k u k Ez k u k, as m. By recallig the statioarity of (u k, w k k, to prove (3.7, it suffices to show that Ω, Ω 2 ad ρ are fiite. I fact (6.5 implies that Ω 2 Ew 2 + C j γ <. Similarly, we may prove that (u k k is a statioary L 2 -mixigale of size with costat u 4. As a cosequece, the same argumet yields Ω < ad ρ <. I order to prove (6.6, let zk = z k z k,αm = αm E(w i+k F k, λ k, = z ku k Ez ku k, λ k,2 = z k,αm u k Ez k,αm u k, where α m ad z k,αm is give as i (6.4. Due to (6.4, we have E E ( λ k,2 F k m E λk,2 2 z k,αm 2 u 2, (6.7 as m, uiformly for ay k 2m ad ay iteger sequece α m. By recallig that u k is adapted to F k ad F k m F k, we may write E E ( α m λ k, F k m E E ( A k F k m, where A k = u k w i+k Eu k w i+k. Sice both u k ad w k are L 2 -NED of size, Corollary 7. of Davidso (994 implies that A k is L -NED of size. As a cosequece, as i the proof of (6.2, there exist a sequece of v m such that v m ad Hece, uiformly for k 2m, E E ( A k F k m C vm. E E ( λ k, F k m C αm v m, as m, by takig α m to be such a iteger sequece that α m ad α m v m. This, together with (6.7, yields sup E E ( λ k F k m C (αm v m + 2 z k,αm 2 u 2, k 2m as m, as required. The proof of Theorem 3.3 is ow complete. Ackowledgemets: This work is completed whe the first author Jiagya Peg visited the Uiversity of Sydey uder the fiacial support of Chia Scholarship Coucil (CSC from Chiese govermet. Peg thaks the Uiversity of Sydey for providig friedly research eviromet. Peg also ackowledges research support from the

23 23 Natioal Natural Sciece Foudatio of Chia (project o: 7525, Applied Basic Project of Sichua Provice (grat umber: 26JY257 ad Chia Postdoctoral Sciece Foudatio (grat umber: 25M Wag ackowledges research support from Australia Research Coucil. REFERENCES Borovski, Y. ad Korolyuk, V. (997. Martigale approximatio. VSP, Utrecht. Davidso, J. (994. Stochastic Limit Theory: A Itroductio for Ecoometricias, Oxford Uiversity Press. Davidso J. (22. Establishig coditios for the fuctioal cetral limit theorem i oliear ad semiparametric time series processes. Joural of Ecoometrics, 6, Hall, P. ad Heyde, C. C. (98. Martigale limit theory ad its applicatio. Academic Press. Jacod, J. ad A. N. Shiryaev (987/23. Limit Theorems for Stochastic Processes. New York: Spriger Verlag. Kurtz, T. G. ad Protter, P. (99. Weak limit theorems for stochastic itegrals ad stochastic differetial equatios. Aals of Probability, 9, Liag, H. Y., Phillips, P. C. B., Wag, H. C. ad Wag Q. (25. Weak covergece to stochastic itegrals for ecoometric applicatios. Ecoometric Theory,, -27. Li, Z. ad Wag, H. (25. O covergece to stochastic itegrals. Joural of theoretical probability, -2. Phillips, P. C. B. ad V. Solo (992. Asymptotics for Liear Processes, A. Statist. 2, 97-. Taiguchi, M ad Kakizawa, Y. (2. Asymptotic Theory of Statistical Iferece for Time Series, New York: Spriger. Wag, Q., Li, Y. X. ad Gulati, C. M. (23. Asymptotics for geeral fractioally itegrated processes with applicatios to uit root tests. Ecoometric Theory, 9, Wag, Q. ad Phillips, P. C. B. (29a. Asymptotic theory for local time desity estimatio ad oparametric coitegratig regressio. Ecoometric Theory, 25,

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