On Linear Processes with Dependent Innovations

Transcription

1 The University of Chicago Department of Statistics TECHNICAL REPORT SERIES On Linear Processes with Dependent Innovations Wei Biao Wu, Wanli Min TECHNICAL REPORT NO. 549 May 24, S. University Avenue Chicago, IL 60637

2 On linear processes with dependent innovations Wei Biao Wu, Wanli Min 1 May 24, 2004 Department of Statistics, The University of Chicago, Chicago, IL 60637, USA Abstract We investigate asymptotic properties of partial sums and sample covariances for linear processes whose innovations are dependent. Central limit theorems and invariance principles are established under fairly mild conditions. Our results go beyond earlier ones by allowing a quite wide class of innovations which includes many important non-linear time series models. Applications to linear processes with GARCH innovations and other non-linear time series models are discussed. 1 Introduction Let {ɛ t } t Z be independent and identically distributed (iid) random elements and F be a measurable function such that a t = F (..., ɛ t 1, ɛ t ) (1) is a well-defined random variable. Then {a t } t Z is a stationary and ergodic infinite-order nonlinear moving average process. Assume that a t has mean 0, finite variance and that {ψ i } i 0 is a sequence of real numbers such that i,j=0 ψ iψ j E(a 0 a i j ) <. Then the linear process X t = ψ i a t i (2) i=0 exists and E(X 2 t ) <. In this paper we consider asymptotic distributions of the properly normalized partial sum process S n = n i=1 X i and the sample covariances ˆγ h = n X tx t+h /n, h 0. We will establish an invariance principle for the former and a central limit theorem (CLT) for ˆγ h. Such results are important and useful in statistical inference of stationary processes. 1 address: wbwu,wmin@galton.uchicago.edu 1

3 In the classical time series analysis, the innovations {a t } t Z in the linear process X t are often assumed to be iid; see for example Brockwell and Davis (1991) and Box and Jenkins (1976). In this case asymptotic behaviors of the sample means and partial sum processes have been extensively studied in the literature. It would be hard to compile a complete list. Here we only mention some representatives: Davydov (1970), Gorodetskii (1977), Hall and Heyde (1980), Phillips and Solo (1992), Yokoyama (1995) and Hosking (1996). See references therein for further background. There are basically two types of results. If the coefficients ψ t are absolutely summable, then the covariances of X t are summable and we say that X t is short-range dependent (SRD). Under SRD, the normalizing constant for the sum n i=1 X i is n, which is of the same order as that in the classical CLT for iid observations. When the coefficients ψ t are not summable, then X t is long-range dependent (LRD) and the normalizing constant for n i=1 X i is typically larger than n. Fractional ARIMA model (Hosking, 1981) is an important class which may exhibit LRD. Asymptotic normality for sample covariances has also been widely discussed; see for example, Box and Jenkins (1976), Hannan (1976), Hall and Heyde (1980), Giraitis and Surgailis (1990), Brockwell and Davis (1991), Phillips and Solo (1992) and Hosking (1996). The asymptotic problem of partial sums and sample covariances becomes more difficult if dependence among a t is allowed. Recently, fractional ARIMA (FARIMA) processes with GARCH innovations have been proposed to model some econometric time series. former feature allows LRD and the latter one allows that the conditional variance can change over time, namely heteroscedasticity. Financial time series often exhibit these two features. Hence FARIMA models with GARCH innovations provide a natural vehicle for modelling processes with such features; see Baillie, Chung and Tieslau (1995), Hauser and Kunst (2001) and Lien and Tse (1999). Romano and Thombs (1996) point out that the traditional large sample inference on autocorrelations under the assumption of iid innovations is misleading if the underlying {a t } are actually dependent. Results so far obtained in this direction require that a t are m-dependent (Dianana, 1953) or martingale differences (Hannan and Heyde, 1972). Recently, Wang, Lin and Gulati (2002) consider invariance principles for iid or martingale differences a t. The However, the proof in the latter paper contains a serious error [cf Remark 1]. Chung (2002) considers long-memory processes under the assumption that a t are martingale differences with a prohibitively strong restriction [cf. (18)] that exclude the 2

4 widely-used ARCH models. In this paper, we shall apply the central limit theory and the idea of martingale approximations developed in Woodroofe (1992) and Wu and Woodroofe (2003) to obtain asymptotic distributions of the sample means and covariances of (2) under sufficiently mild conditions. Our results go beyond earlier ones by allowing a large class of important nonlinear process a t, which substantially weakens the iid or martingale differences assumptions. In particular, our conditions are satisfied if a t are the GARCH, random coefficient AR, bilinear AR and threshold AR models etc. The goal of the paper is twofold: the first one is to obtain asymptotic distributions of X n and ˆγ h which are certainly needed for statistical inferences of such processes. The second major goal is to introduce another type of dependence structure as well as to provide a unified methodology for asymptotic problems in econometrics models based on martingale approximations. With our dependence structure, martingales can be constructed which efficiently approximate the original sequences. Based on such martingale approximations, we can apply celebrated results in martingale theory such as martingale central limit theorems (MCLT) and martingale inequalities. The proposed dependence structure only involves the computation of conditional moments and it is easily verifiable. This feature is quite different from strong mixing conditions which might be too restrictive and hard to be verified. On the other hand, our condition is sufficiently mild as well. Recently Wu and Mielniczuk (2002), Hsing and Wu (2003), Wu (2003a, 2003b) apply the idea of martingale approximations and solve certain open asymptotic problems. The paper is organized as follows. Main theorems are stated in Section 2 and comparisons with earlier results are elaborated. Proofs are given in the Section 4. In Section 3 we present applications to some nonlinear processes. 2 Results Let X t be the linear process defined by (2) and recall E(a n ) = 0; let F t = (..., ɛ t 1, ɛ t ) be the shift process. Then F t is a Markov Chain. For a random variable ξ define the projections P t ξ = E(ξ F t ) E(ξ F t 1 ) and its L p (p 1) norm by ξ p = [E( ξ p )] 1/p. Let = 2 and write ξ L p if ξ p <. We say c n d n for two sequences {c n } and {d n } if lim c n /d n = 1. For n 0 write A n = i=n ψ2 i, Ψ n = n i=0 ψ i and B 2 n = n 1 i=0 Ψ2 i. 3

5 Let Ψ n = 0 for n < 0 and define σ 2 n = (Ψ n+i Ψ i ) 2. (3) i= n Before we state our main results, we now introduce the basic concept of L p weakly dependence (cf Definition 1), which unlike strong mixing conditions, only involves the estimation of the decay rate of conditional moments. On the other hand, it appears to be very mild. In Section 3 we argue that many nonlinear time series models satisfy this condition. More importantly, it provides a natural vehicle for the central limit theory for stationary processes; see Woodroofe (1992) and Lemma 3 (Lemma 1 below) in Wu (2003a). An interesting observation of Lemma 1 is that there is a close form for the asymptotic variance in the limiting normal distribution. Definition 1. The process Y n = g(f n ), where g is a measurable function, is said to be L p weakly dependent with order r (p 1 and r 0) if E( Y n p ) < and n r P 1 Y n p <. (4) If (4) holds with r = 0, then Y n is said to be L p weakly dependent. n=1 Lemma 1. (Wu (2003a)) Let Y n = g(f n ) be L 2 weakly dependent. Then n i=1 (Y i EY 1 )/ n N(0, ξ 2 ), where ξ = j=1 P 1Y j L 2 and denotes convergence in distribution. The intuition of the definition weakly dependence is that the projection of the future Y n to the space M 1 M 0 = {Z L p : Z is F 1 measurable and E(Z F 0 ) = 0} has a small magnitude. So the future depends weakly on the current states. If Y n are martingale differences, then (4) is automatically satisfied for all r 0 if Y 0 L p. Lemma 1 readily entails the following corollary for the process (2). Corollary 1. Assume that a n is L 2 weakly dependent and ψ i <. (5) i=0 Then P 1X t < and n X t/ n N(0, ξ 2 ), where ξ = P 1X t. 4

6 2.1 Invariance Principles Let W n (t), 0 t 1 be a continuous, piece-wise linear function such that W n (t) = S k / S n at t = k/n, k = 0,..., n, and W n (t) = S k / S n + (nt k)x k+1 / S n when k/n t (k + 1)/n. Let space C[0, 1] be the collection of all continuous function on [0, 1]; let the distance between two functions f, g C[0, 1] be ρ(f, g) = sup 0 t 1 f(t) g(t). Billingsley (1968) contains an extensive treatment of the convergence theory on C[0, 1]. Recall Ψ n = n i=0 ψ i for n 0, Ψ n = 0 for n < 0 and B 2 n = n 1 i=0 Ψ2 i. Theorem 1 asserts that the limiting distribution is the standard Brownian motion under (6). The norming sequence has the form S n = l (n) n for some slowly varying function l (n). If ψ n decays sufficiently slowly, then (6) is violated and we may have fractional Brownian motions (Mandelbrot and Van Ness, 1969) as limits. In Theorem 2 we assume that the coefficients have the form ψ n = l(n)/n β for n 1, where 1/2 < β < 1 and l is a slowly varying function. Then ψ n is not summable and norming sequences for S n are different. Theorem 1. Assume that for some α > 2, {a n } is L α weakly dependent with order 1, (Ψ n+i Ψ i ) 2 = o(bn) 2 (6) i=0 and B n. Then l (n) := S n / n is a slowly varying function, B n / n l (n) n 1 i=0 Ψ i /n and {W n (t), 0 t 1} converges in C[0, 1] to the standard Brownian motion W (t) on [0, 1]. Theorem 2. Let ψ n = l(n)/n β for n 1, where 1/2 < β < 1. Assume that a n is L 2 weakly dependent with order 1. Then {W n (t), 0 t 1} converges in C[0, 1] to the standard fractional Brownian motion {W H (t), 0 t 1] with Hurst index H = 3/2 β, and S n σ n η, with η = P 1a t and σ n defined in Lemma 3. Invariance principle is a powerful tool in statistical inference of econometric time series such as unit root testing problems, and it enables one to obtain limiting distributions for many statistics. This problem has a substantial history. The celebrated Donsker s theorem asserts invariance principles for iid sequence of X n. For dependent sequences, see the survey by Bradley (1986) and Peligrad (1986) for strong mixing processes, McLeish (1975, 1977) for mixingale sequences, Billingsley (1968) and Hall and Heyde (1980). 5

7 In classical theory of invariance principles for linear processes, it is often assumed that innovations {a n } are iid or martingale differences; see Davydov (1970), Gorodetskii (1977), Hall and Heyde (HH, pp. 146, 1980) and the recent work by Wang, Lin and Gulati (WLG, 2002) among others. We believe that even for the special case that {a n } are martingale differences, Theorem 1 imposes the weakest sufficient conditions on the coefficients {ψ n }. Remark 1. WLG (2002) attempted to generalize previous results on invariance principle and wanted to establish {S kn (t)/σ n, 0 t 1} {W (t), 0 t 1} in D[0, 1], where X n is defined by (2) with iid a t s, D[0, 1] is the collection of all right continuous functions with left limits on [0, 1] and k n (t) = sup{m : B 2 m tb 2 n} (cf Theorem 2.1 in the latter paper). However, there is a serious error in their argument. A key step in their paper is to use their equation (36), namely the distributional identity k n(t) k=1 k n(t) a k Ψ kn (t) k = D to establish the invariance principle via k=1 a k Ψ k 1, 0 t 1, (7) {S kn (t)/b n, 0 t 1} {W (t), 0 t 1} (8) k n(t) {Bn 1 k=1 a k Ψ k 1, 0 t 1} {W (t), 0 t 1}. (9) The claim (7) holds only for a single t and it fails to be valid jointly for 0 t 1. To see this, choose t 1 and t 2 such that k n (t 1 ) = 1 and k n (t 2 ) = 2. In this case it is easily seen that (7) fails since the random vectors (a 1 ψ 0, a 1 (ψ 0 + ψ 1 ) + a 2 ψ 0 ) and (a 1 ψ 0, a 1 ψ 0 + a 2 (ψ 0 + ψ 1 )) generally have different distributions (even though the marginal distributions are the same). The invariance principle certainly requires the joint behavior over 0 t 1. Remark 2. It would be interesting to compare our result with previous ones including WLG even though the latter paper contains an error. Our Theorem 1 differs from HH (pp. 146, 1980) and WLG (2002) in several important aspects. First, we allow a fairly general class of a n which includes many nonlinear time series models. If a n are iid or martingale differences, then a n are automatically L α weakly 6

8 dependent with any order provided a 0 L α. Our moment conditions is slightly stronger since α > 2 is required. In HH (1980) and WLG (2002), a n are assumed to iid or martingale differences and the existence of E(a 2 n) is required. Second, in WLG (2002), the conditions on ψ k 1 B n max Ψ j 0 and 1 j n n j=0 A 1/2 j = o(b n ) (10) are imposed, which are stronger than (6). To see this, assume that {a n } are stationary martingale differences with a n = 1. Then E(S n F 0 ) = i=0 (Ψ n+i Ψ i ) 2, and for j 0, E(X j F 0 ) = A 1/2 j. Observe that E(S n F 0 ) n j=1 E(X j F 0 ). Thus (10) implies (6). In HH (1980), two sufficient conditions for the invariance principle are presented. WLG s (2002) assumption (10) weakens one sufficient condition (cf Inequality (5.38) of HH s (1980)) j=1 when one-sided linear processes is considered. (5.37) in HH (1980) A 1/2 j < (11) However, the other sufficient condition ( ψ l ) 2 < (12) n=1 l=n cannot be derived from WLG s (10). For example, let ψ n = ( 1) n n 2/3, n 1 and ψ 0 = 1. Then l=n ψ l = O(n 2/3 ) and (12) holds. However, WLG s (10) does not hold since A n = m=n ψ2 n 3n 1/3 and n j=0 A1/2 j 1.2 3n 5/6. Interestingly, (12) does imply our condition (6) by noting that Ψ n+i Ψ i = l=i+1 ψ l l=i+1+n ψ l. Thus (6) unifies (10), (11) and (12). Third, WLG (2002) posed the open problem whether B n can be replaced by σ n. Theorem 1 provides an affirmative answer to this problem. Let {a n } be martingale differences with a n = 1. Since E(S n F 0 ) and S n E(S n F 0 ) are orthogonal, S n 2 = E(S n F 0 ) 2 + S n E(S n F 0 ) 2, which implies that σ 2 n = i=0 (Ψ n+i Ψ i ) 2 + B 2 n B 2 n by (6). Moreover, our Theorem 1 asserts that σ n necessarily has the form l (n) n and reveals the inner relations B n / n l (n) n 1 i=0 Ψ i /n. 7

9 Finally, the form of our result {W n (t), 0 t 1} {W (t), 0 t 1} in C[0, 1] is a typical one for invariance principles. It is slightly different from the one in WLG s (8). Our form seems more convenient for application and it actually implies the latter. To see this, let Ŵn(t) and Ŵ (t) be defined on a (possibly richer) probability space such that Ŵ D n = W n, Ŵ = D W and sup 0 t 1 Ŵn(t) Ŵ (t) P 0. Here η D 1 = η 2 denotes that η 1 and η 2 have the same distribution and P means convergence in probability. Since Bn/n 2 is a slowly varying function, by Lemma 4 with γ = 1, lim max Bk 2 1 k n k Bn 2 n = 0, (13) and since Ŵ has a version with continuous path, max 1 k n Ŵ (B2 k /B2 n) Ŵ (k/n) P 0. Hence max 1 k n Ŵn(k/n) Ŵ (B2 k /B2 P n) 0, which implies the invariance principle of WLG (2002). Example 1. Let ψ k = l(k)/k, k 1, where l is a slowly varying function such that k=1 ψ k =. By Lemma 3, Ψ n is slowly varying, Ψ n n k=0 ψ k, σn 2 nψ 2 n and lim (1/σn) 2 j=n (Ψ j Ψ j n ) 2 = 0. Hence (6) is satisfied. 2.2 Sample Covariances Covariances play a fundamental role in the study of stationary processes. In this section we present asymptotic results for sample covariances. Theorem 3. Assume that {a n } is L 4 weakly dependent, and ψ i A i+1 < (14) i=0 P 1 (a i a j ) <. (15) i,j=0 Then P 1X 2 t < ; namely X 2 t is L 2 weakly dependent. Theorem 3 is useful in proving Corollary 2 concerning asymptotic normality of sample covariances. 8

10 Proposition 1. A sufficient condition for (14) is tψ 2 t <. (16) Corollary 2. Let the conditions of Theorem 3 be satisfied. For a fixed integer h 0 let the column random vector X t,h = (X t h,..., X t ) T, where T Γ(h) = E(X 0 X h,h ). Then 1 n where Σ h = E(ξ h ξ T h ) and ξ h = t= P 1(X t X t+h,h ) L 2. stands for transpose and n [X t X t+h,h Γ(h)] N(0, Σ h ) (17) In Section 3.1, we will illustrate how to apply Theorem 3 and Corollary 2 to the GARCH models. Notice that (14) and (16) allow some non-summable sequences ψ k. Consider the Fractional ARIMA(0, d, 0) model (1 B) d X n = a n, where B is the back-shift operator (BX n = X n 1 ) and 1/2 < d < 1/2. Then X n = (1 B) d a n = i=0 ψ ia n i and as j, ψ j = Γ(j + d)/[γ(j)γ(d)] j d 1 /Γ(d). If 1/4 > d > 0, then (16) holds. Asymptotic distribution for sample correlations can be easily obtained from Corollary 2. Remark 3. In Theorem 6.7 of HH (pp 188, 1980), a CLT for sample correlations is derived under the condition (16), {a t } t Z are martingale differences and E(a 2 t F t 1 ) = a positive constant. (18) Relation (18) generalizes the case in which {a n } are iid. Chung (2002) recently derives various limit theorems for martingale differences {a t } t Z satisfying (18). Unfortunately the latter assumption appears too restrictive and it excludes many important models. One of the most interesting case is the ARCH model. To see this, let a t = ɛ t θ θ 2 2a 2 t 1 be the ARCH(1) model, where {ɛ t } t Z are iid with mean 0 and variance 1 and θ 1 and θ 2 are parameters. Then E(a 2 t F t 1 ) = θ θ 2 2a 2 t 1, which cannot be almost surely constant unless θ 2 = 0. Thus limit theorems by Chung (2002) and HH (1980) can not be directly applied to linear processes with ARCH innovations. Our results avoids this severe limitation. Remark 4. If a n are martingale differences, then the L p weakly dependence condition trivially holds if a n L p. Since P 1 a i a j = 0 for i > j 1, condition (15) is reduced to P 1 a 2 i <. i=1 9

11 On the other hand, if (18) holds, then for i 2, E(a 2 i F 1 ) = E[E(a 2 i F i 1 ) F 1 ] is almost surely a constant and hence P 1 a 2 i = 0 almost surely. 3 Applications The class of non-linear time series models that (1) represents is clearly huge. For example, it includes threshold autoregressive model (TAR, Tong, 1990), bilinear model (Meyn and Tweedie, 1994), generalized autoregressive conditional heteroscedastic model (GARCH, Bollerslev, 1986), random coefficient autoregressive model (RCA) (Nicholls and Quinn, 1982) etc. Here we argue that those widely used non-linear time series models are L p weakly dependent, and moreover, P 1 a n p decays to 0 exponentially fast. Namely there exists C > 0 and ρ (0, 1) such that P 1 a n p Cρ n for all n 0. In this section C > 0 and ρ (0, 1) stand for constants which may vary from line to line. Let {ɛ t} t Z be an iid copy of {ɛ t } t Z and a n = F (..., ɛ 1, ɛ 0, ɛ 1,..., ɛ n ) be a coupled version of a n. We say that {a t } is geometrically moment contracting (GMC) if there exists α > 0 such that for all n 0, E( a n a n α ) Cρ n. (19) The GMC condition (19) implies that the process {a t } forgets the past F 0 at an exponential rate by measuring the distance between a n and its coupled version a n. Recently Hsing and Wu (2003) obtains asymptotic theory for U-statistics of processing satisfying (19). Proposition 2. (i) If (19) holds with some α 1, then E(a n F 0 ) α Cρ n and hence P 1 a n α Cρ n. (ii) Assume that {a n } satisfies (19) for some α 0 > 0 and a n L q for some q > 1. Then (19) holds for all α (0, q). Proposition 3. Assume that {a t } L 4 and (19) holds with α = 4. Then there exist C > 0 and ρ (0, 1) such that for all t, k 0, P 1 (a t a t+k ) Cρ t+k. (20) Hence {a t } satisfies (15). Propositions 2 and 3 make Theorems 1, 2, and Corollary 2 applicable by providing easily verifiable and mild conditions. An important special class of (1) is the so-called 10

12 iterated random functions. Let G(, ) be a bivariate measurable function with Lipschitz constant L ε = sup G(x, ε) G(x, ε) / x x and Z x x n be defined recursively by Z n = G(Z n 1, ε n ). (21) Diaconis and Freedman (1999) show that {Z n } has a unique stationary distribution if E(log L ε ) < 0, E(L α ε ) < and E[ z 0 G(z 0, ε) α ] < (22) hold for some α > 0 and z 0. The same set of conditions actually also imply the GMC (19) (cf Lemma 3 in Wu and Woodroofe, 2000). Example 2. Let the TAR(1) a n = φ 1 max(a n 1, 0) + φ 2 max( a n 1, 0) + ɛ n, the condition (22) is satisfied when L ɛ = max( φ 1, φ 2 ) < 1 and E( ɛ 0 α ) < for some α > 0. For the bilinear model a n = (α 1 + β 1 ɛ n )a n 1 + ɛ n, where α 1 and β 1 are real parameters and E( ɛ 0 α ) < for some α > 0, we have a close form of the Lipschitz constant L ɛ = α 1 +β 1 ɛ and (22) holds if E(L α ɛ ) < 1. Similarly for the RCA model a n = (φ 1 + η n )a n 1 + ɛ n, where η n are iid, then the Lipschitz constant L ɛ = φ 1 + η n. 3.1 GARCH Models Let ɛ t, t Z be iid random variables with mean 0 and variance 1; let a t = h t ɛ t and h t = α 0 + α 1 a 2 t α q a 2 t q + β 1 h t β p h t p (23) be the generalized autoregressive conditional heteroscedastic model GARCH(p, q), where α 0 > 0, α j 0 for 1 j q and β i 0 for 1 i p. Chen and An (1998) show that if q α j + j=1 p β i < 1, (24) i=1 then a t is strictly stationary. Notice that a t form martingale differences. Hence P 1 a t = 0 for t 2, P 1 a 1 = a 1 and (4) holds for any r 0 if a 1 L p. The existence of moments for GARCH models has been widely studied; see He and Teräsvirta (1999), Ling (1999), and Ling and McAleer (2002) and references therein. 11

13 Let Y t = (a 2 t,..., a 2 t q+1, h t,..., h t p+1 ) T and b t = (α 0 ɛ 2 t, 0,..., 0, α 0, 0,..., 0) T. It is well known that GARCH models admits the following representation (see for example Theorem in Taniguchi and Kakizawa (2000)): α 1 ɛ 2 t... α q 1 ɛ 2 t α q ɛ 2 t β 1 ɛ 2 t... β p 1 ɛ 2 t β p ɛ 2 t Y t = M t Y t 1 + b t, where M t = α 1... α q 1 α q β 1... β p 1 β p For a square matrix M let ρ(m) be its largest eigenvalue. Let be the usual Kronecker product; let Y be the Euclidean length of the vector Y. Assume E(ɛ 4 t ) <. Ling (1999) shows that if ρ[e(mt 2 )] < 1, then a t has a stationary distribution and E(a 4 t ) <. Ling and McAleer (2002) argue that the condition ρ[e(mt 2 )] < 1 is also necessary for the finiteness of the fourth moment. Our Proposition 4 states that the same condition actually implies (19) as well. Proposition 4. Assume that ɛ t (25) are iid with mean 0 and variance 1, E(ɛ 4 t ) < and ρ[e(m 2 t )] < 1. Then E( a n a n 4 ) Cρ n for some C < and ρ (0, 1). Namely (19) holds. Under the conditions of Proposition 4, it is clear that Proposition 3 and hence Theorem 3 are applicable since Pa i a j and Pa i decays to zero exponentially fast. To derive asymptotic distributions related to stationary processes, traditional approaches normally require strong mixing conditions. However, it is a challenging problem to obtain strong mixing conditions of GARCH models; see Carrasco and Chen (2002) for an recent attempt. Our approach, however, provides a framework that completely avoids strong mixing conditions. 12

14 4 Proofs We assume E(a n ) = 0 throughout. Recall A n = i=n ψ2 i, Ψ n n 1 i=0 Ψ2 i and (3) for σn. 2 = n i=0 ψ i and B 2 n = Lemma 2. Let {D i, i N} be a martingale difference sequence in L p for some p 2. Then i=1 D i p C p [ i=1 D i 2 p] 1/2, where C p = 18p 3/2 /(p 1) 1/2. Proof of Lemma 2. It is a straightforward consequence of Burkholder s inequality (cf Theorem in Chow and Teicher (1978)) and Minkowski s inequality ( ) p/2 [ ] p/2 D i p p CpE p Di 2 Cp p Di 2 p/2 i=1 i=1 since p/2 becomes a norm when p/2 1. Part (i) of Lemma 3 is also used in WLG (2002). For the sake of completeness, we provide a proof here. Part (ii) is well-known and it is an easy consequence of Karamata s theorem. Lemma 3. (i) Let ψ k = l(k)/k, k 1 and assume that k=1 ψ k =. Then Ψ n is slowly varying, l(n)/ψ n 0, Ψ n n k=0 ψ k, σn 2 nψ 2 n and lim (1/σn) 2 j=n (Ψ j Ψ j n ) 2 = 0. (ii) Let ψ k = l(k)/k β, k 1 and 1/2 < β < 1. Then Ψ n n 1 β l(n)/(1 β) and σ n n 3/2 β l(n)c β, where c 2 β = 0 [x1 β max(x 1, 0) 1 β ] 2 dx/(1 β) 2. Proof of Lemma 3. (i) Since l is slowly varying, there exists N 0 N such that either l(n) > 0 for all n N 0 or l(n) < 0 for all n N 0. Without loss of generality we assume the former. So k=1 ψ k = implies that lim Ψ n / n k=0 ψ k = 1. For any 0 < δ < 1 and G > 1, 0 Ψ n Ψ δn 1 n k=δn ψ k δn 1 k=δn/g ψ k = i=1 l(n) n k=δn 1/k l(δn) δn 1 k=δn/g log δ 1 = 1/k log G which approaches 0 as G. Thus by definition Ψ n is a slowly varying function. The same argument also implies that lim l(n)/ψ n = 0. By Karamata s theorem, σn 2 n 1 j=0 Ψ2 n nψ 2 n since Ψ n is slowly varying. For any fixed δ > 0, 1 nψ 2 n j=(1+δ)n (Ψ j Ψ j n ) 2 = 13 1 nψ 2 n j=(1+δ)n O(nψ j ) 2 = 0

15 since l(n)/ψ n 0 and both l(n) and Ψ n are slowly varying. Thus 1 nψ 2 n j=n (Ψ j Ψ j n ) 2 = 1 nψ 2 n 1 nψ 2 n (1+δ)n j=n (1+δ)n j=n (Ψ j Ψ j n ) 2 2(Ψ 2 j + Ψ 2 j n) 4δ which completes the proof of (i) since δ > 0 is arbitrarily chosen. Lemma 4. Let γ > 0 and l(n) be a slowly varying function. Then In particular, lim max 1 k n (k/n) γ l(k)/l(n) = 1. Proof of Lemma 4. lim max 1 k n (k/n)γ l(k)/l(n) 1 = 0. (26) Let l(m) have the representation l(m) = c m e m 1 η(u)/udu (Bingham, Goldie and Teugels, 1987) with c m c > 0. Choose K 0 N be sufficiently large such that i γ/4 l(i) i γ/4 holds for all i K 0. Then max K 0 k n and for any δ (0, 1), max (k/n) γ l(k)/l(n) 1 = 0, (27) 1 k K 0 ( k n )γ l(k) l(n) max (k nδ k n n )γ l(k) l(n) 1 = 1 By (27), (28) and (29), for sufficiently large n, max K 0 k n ( k n )γ k γ/4 n γ/4 = 0, (28) max l(k) 1 = 0. (29) nδ k n l(n) max 1 k n (k/n)γ l(k)/l(n) 1 max (k/n) γ l(k)/l(n) 1 n k δn δ γ + max (k/n) γ c k /c n e n k η(u)/udu n k δn δ γ + max (k/n) γ e max n u δn η(u) n k 1/udu δ γ + δ γ/2. n k δn Thus the lemma follows since max n u δn η(u) γ/2 for sufficiently large n and δ is arbitrarily chosen. 14

16 Lemma 5. Let {η k, k Z} be a stationary and ergodic process with mean 0 and define n = j=0 (1/σ2 n)(ψ j Ψ j n ) 2 η j, where {ψ k, k 0} satisfies conditions in (ii) of Theorem 2. Then lim E n = 0. Proof of Lemma 5. Let T n = n 1 k=0 η k and δ k = sup n k E T n /n. By the ergodic theorem, δ k 0. We first consider the case 1 < β < 1. For any fixed M > 1 write 2 n 1 n = [ j=0 (1+M)n + j=n + j=(1+m)n+1] 1 (Ψ σn 2 j Ψ j n ) 2 η j =: n,1 + Ω n,m + Ξ n,m. (30) In the sequel we will show that lim E n,1 = 0 and lim E Ω n,m = 0. Using the Abelian summation technique, n,1 = Ψ 2 n 1T n /σ 2 n + n 1 j=1 (1/σ2 n)(ψ 2 j Ψ 2 j 1)T j. Thus Ψ 2 E n,1 n 1E T n σn 2 + n 1 Ψ 2 j Ψ 2 j 1 E T j j=1 σ 2 n (31) Since E T n = o(n) and nψ 2 n 1/σ 2 n = O(1) by Lemma 3, the first term in the proceeding display vanishes. For the second one, let K 1 be a fixed integer. Then K 1 Ψ 2 j Ψ 2 j 1 E T j E n,1 σ 2 j=1 n + δ K n 1 j Ψ 2 j Ψ 2 j 1. (32) Observe that as j, Ψ 2 j Ψ 2 j 1 = ψ j 2Ψ j 1 + ψ j 2j 1 2β l 2 (j)/(1 β). Hence n 1 j Ψ 2 j Ψ 2 j 1 j=1 σ 2 n = n 1 j=1 O[j 2 2β l 2 (j)] σ 2 n j=k σ 2 n O[n 3 2β l 2 (n)] = σn 2 = O(1) by Karamata s theorem. So (32) implies that E n,1 = 0 since K is arbitrarily chosen and δ K 0 as K. The claim lim E Ω n,m = 0 can be similarly proved. Actually, by the same arguments in (31) and (32), it suffices to show the analogy of (33): (33) 1 σ 2 n nm j=1 j (Ψ n+j 1 Ψ j 1 ) 2 (Ψ n+j Ψ j ) 2 <. (34) Simple algebra gives (Ψ n+j 1 Ψ j 1 ) 2 (Ψ n+j Ψ j ) 2 2 ψ n+j ψ j Ψ n+j 1 Ψ j 1 + ψ n+j ψ j 2. (35) 15

17 Since ψ n+j ψ j = O[j β l(j)] and Ψ n+j 1 Ψ j 1 = O[n 1 β l(n)] for 1 j nm, (34) follows from nm j=1 jj β l(j) = O[n 2 2β l 2 (n)] = O(σ 2 n/n) and nm j σ 2 j=1 n ψ n+j ψ j Ψ n+j 1 Ψ j 1 = nm j=1 O[j 1 β l(j)n 1 β l(n)] σ 2 n = O[n3 2β l 2 (n)] σ 2 n by Karamata s theorem. Let M 1. Since l is slowly varying, there exists a constant c > 0 such that for all sufficiently large n, Ψ j Ψ j n c nj β l(j) holds for all j j n = (1 + M)n + 1. Therefore by (30) and Karamata s theorem, E n σ 2 j=(1+m)n+1 n E n,1 + 1 [c nj β l(j)] 2 E η 1 = E Ω n,m + E Ξ n,m for some c 1 <. Hence lim E n = 0 by letting M. jn 1 2β l 2 (j n ) c 2 (2β 1)σ E η n 2 1 = c 1 M 1 2β For the case β = 1, by (ii) of Lemma 3 and (30), it follows as in the the proof of the case 1/2 < β < 1 from n 1 j Ψ 2 j Ψ 2 j 1 j=1 σ 2 n = n 1 j=1 O(j ψ j Ψ j ) σ 2 n = n 1 j=1 O( l(j)ψ j ) σ 2 n since l(j)ψ j is also a slowly varying function and nl(n)ψ n /σ 2 n = 0 by Lemma 3. = 0 Lemma 6. Assume that {a n } is L p (p 2) weakly dependent with order 1. Then for S n = n i=1 X i, there exists a constant C, independent of n, such that for all n N, S n p Cσ n. (36) Proof of Lemma 6. Observe that E(a t F 0 ) = 0 k= P ka t. Then 0 E(a t F 0 ) p P k a t p = P 1 a j p = t P 1 a t p <. (37) t=0 t=0 k= t=0 j=t+1 Hence b k := t=k E(a t F k ) L p and the Poisson equation b k = a k +E(b k+1 F k ) holds. Let d k = b k E(b k F k 1 ), which by definition are stationary and ergodic martingale differences. Let Xt = j=0 ψ jd t j, X # t = X t Xt, Sn = n i=1 X t and S n # = n i=1 X# t. Then Sn = (Ψ j Ψ j n )d n j and S n # = ψ j [E(b 1 j F j ) E(b n+1 j F n j )]. (38) j=0 j=0 16

18 The essence of our approach is to approximate S n by S n, which admits martingale structures. By Lemma 2, S n p C p σ n d 0 p. Inequality (37) implies that d 0 p 2 b 0 p <. To establish (36), it then remains to verify that j=0 ψ je(b 1 j F j ) L p, which entails S n # p = O(1). To this end, for k 0 let V k = 0 i= ψ ip i k b 1+i. Then j=0 ψ je(b 1 j F j ) = k=0 V k. Since P i k b 1+i forms martingale differences in i, by Lemma 2, Therefore, V k 2 p C 2 p 0 i= ψ i P i k b 1+i 2 p = Cp( 2 ψj 2 ) P 1 b 1+k 2 p. j=0 ψ j E(b 1 j F j ) p j=0 k=0 V k p C p A 1/2 0 P 1 b 1+k p k=0 which is finite in view of P 1 b 1+k p P 1 E(a t F k+1 ) p = k=0 k=0 t=k+1 P 1 a t p = k=0 t=k+1 t P 1 a t p <. Here we have applied P 1 E(a t F k+1 ) = P 1 a t for t k Thus S # n p = O(1). Proof of Corollary 1. It follows from Lemma 1 in view of P 1 X t ψ i P 1 a t i = i=0 ψ i P 1 a t i = i=0 t=i+1 ψ i P 1 a k <. i=0 k=1 since P 1 a k = 0 for k 0. Proof of Theorem 1. By the weak convergence theory of random functions, it suffices to establish (i) finite dimensional convergence and (ii) tightness of the random function sequence W n (t). To show the finite dimensional convergence, let b k = t=k E(a t F k ) L 2 and d k = b k E(b k F k 1 ), which are in L 2 from the proof of Lemma 6. Recall (38) for S # n and S n. Then (6) implies that E(S n F 0 ) = o( S n ) and S n B n d 1. By Theorem 1 in 17

19 Wu and Woodroofe (2003), l 1 (n) = S n / n is a slowly varying function, and moreover for H ni = Ψ n d i, where Ψ n = n 1 j=0 Ψ j/n, we have max 1 k n S k Then S n 2 k i=1 H ni 2 = n H n1 2 and since S # n = O(1), k H ni = o( Sn ). (39) i=1 l (n) := S n n S n n = l 1 (n) Ψ n. Therefore, the finite dimensional convergence trivially follows from (39) since for 0 < t < 1, S nt n Ψn = nt i=1 H ni n Ψn + o P (1) = nt i=1 d i n + o P (1) N(0, t d 1 2 ). For the tightness, by Theorem 12.3 in Billingsley (1968), we need to show that there exists a constant C < and τ > 1 such that for all 1 k n, E[ W n (k/n) α ] C(k/n) τ. (40) We claim that (40) holds for τ = (2 + α)/4 > 1. By Lemma 6, (40) is reduced to max (n 1 k n k ) 2+α 4α S k S n = max (n 1 k n k ) 2+α 4α kl (k) nl (n) <. By Lemma 4, the limit in the proceeding display is actually 1. Proof of Theorem 2. As in the proof of Theorem 1, we need to verify the finite dimensional convergence and the tightness. Since {a n } are L 2 weakly dependent with order 1, from the proof of Lemma 6, we can define b k = t=k E(a t F k ) L 2 and d k = b k E(b k F k 1 ). Recall (38) for S n # and Sn. Note that S n # = O(1), it suffices to show that Sn/σ n N(0, d 1 2 ). To this end, we shall apply the martingale central limit theorem. Let ζ n,i = (Ψ i Ψ i n )/σ n. Then n X t /σ n = i=0 ζ n,id n i and i=0 ζ2 n,i = 1 for each n. The Lindeberg condition is obvious in view of sup ζ n,i sup ζ n,i + sup ζ n,i = O(nsup ψ j /σ n ) + O(n 1 β l(n)/σ n ) = O(1/ n). i 0 i>2n 0 i 2n j n 18

20 Let η n,i = E(d 2 n i F i 1 ) E(d 2 n i). Then the convergence of conditional variance easily follows from Lemma 6, which asserts that lim E i=0 ζ2 n,iη n,i = 0. For the tightness, we shall show that (40) holds with α = 2 and τ = H + 1/2. By (ii) of Lemma 3, l (n) = σ n /n H is slowly varying. Then (40) is equivalent to max (n 1 k n k )H 1/2 [ l (k) l (n) ]2 <, which is an easy consequence of Lemma 4. Proof of Theorem 3. Let z t,1 = t 1 i=0 ψ ia t i and z t,0 = i=t ψ ia t i. Then X t = z t,1 + z t,0. Since z t,0 is measurable with respect to F 0, P 1 z 2 t,0 = 0, and by the triangle inequality, P 1 X 2 t P 1 z 2 t,1 + 2 z t,0 P 1 z t,1 P 1 z 2 t,1 + 2 z t,0 4 P 1 z t,1 4. We will show P 1z 2 t,1 < and z t,0 4. P 1 z t,1 4 <. For the former, 1 2 P 1z 2 t,1 = ψ i ψ i P 1 a t i a t i ψ i ψ i P 1 a t i a t i 0 i i<t ψ i ψ i+j P 1 a k a k+j i =0 i=i t=i+1 k=1 j=0 i=0 k=1 j=0 A 0 P 1 a k a k+j, where the last inequality is due to i=0 ψ iψ j i=0 ψ2 i. Hence (15) entails the former inequality. By (14), the latter one holds if we can show that z t,0 4 C A t for some constant C > 0 in view of z t,0 4 P 1 z t,1 4 C t 1 A t ψ j P 1 a t j 4 = C At ψ j P 1 a t j 4 C j=0 ψ j A j+1 ( j=0 t=j+1 j=0 t=j+1 P 1 a t j 4 ) <. To prove z t,0 4 C A t, define U k = j=t ψ je(a t j F t j k ), k 0. Then U 0 = z t,0 and U k U k+1 = j=t ψ jp t j k a t j = t i= ψ ip t+i k a t+i. Observe that the summands ψ i P t+i k a t+i of U k U k+1 form martingale differences in i. By Lemma 2, { t U k U k+1 4 C 4 i= } 1/2 ψ i P t+i k a t+i 2 4 = C 4 A 1/2 t P 0 a k 4, 19

21 which implies that z t,0 4 = (U k U k+1 ) 4 k=0 k=0 U k U k+1 4 = O(A 1/2 t ) since {a k } are L 4 weakly dependent, namely k=1 P 0a k 4 <. Proof of Proposition 1. By the Cauchy inequality, the proposition follows from [ ψ t ] 2 [ ] [ ] [ ] [ ] i 1 A t+1 tψ 2 t A t+1 t 1/2 tψ 2 t ψi 2 t 1/2 in view of i 1 t 1/2 2 i for i 2. Proof of Corollary 2. We first consider h = 1. Let ψ 0 = ψ 0, ψ k = ψ k + ψ k 1 for k 1 and A k = i=k ψ i 2. Since ψ k are square summable, it is easily seen that (14) implies i=0 ψ i A i+1 <, which by Theorem 3 entails P 1(X t + X t+1 ) 2 < in view of X t +X t+1 = k=0 ψ k a t+1 k. Similarly, P 1(X t X t+1 ) 2 <. Hence by the elementary identity X t X t+1 = [(X t + X t+1 ) 2 (X t X t+1 ) 2 ]/4, we have P 1(X t X t+1 ) < by the triangle inequality. Therefore, P 1(X t X t+h,h ) < easily follows and (17) holds by the Cramer-Wold device. Proof of Proposition 3. We shall show that P 1 a t a t+k Cρ k and P 1 a t a t+k Cρ t hold respectively, which imply that P 1 a t a t+k C min(ρ k, ρ t ) Cρ (k+t)/2. In this proof constants C > 0 and ρ (0, 1) may change from line to line. For the former, for t, k 0, by Cauchy s inequality, E(a t a t+k F 0 ) = E(E(a t a t+k F t ) F 0 ) E(a t a t+k F t ) = E(a 0 a k F 0 ) = E(a 0 (a k a k) F 0 ) a 0 4 a k a k 4 Cρ k. Thus P 1 a t a t+k E(a t a t+k F 1 ) + E(a t a t+k F 0 ) Cρ k. For the latter, observe that γ k = E(a t a t+k ) = E(a ta t+k F 0), E(a t a t+k F 0 ) γ k = E(a t a t+k a ta t+k) F 0 a t a t+k a ta t+k i=2 a t (a t+k a t+k) + (a t a t)a t+k a t 4 a t+k a t+k 4 + a t a t 4 a t+k 4 Cρ t+k + Cρ t Cρ t, 20

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