The functional central limit theorem for linear processes with strong near-epoch dependent innovations

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(1)J. Math. Anal. Appl. 376 (2011) 373–382. Contents lists available at ScienceDirect. Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa. The functional central limit theorem for linear processes with strong near-epoch dependent innovations ✩ Jin Qiu a,∗ , Zhengyan Lin b a b. School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang 310018, PR China Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310028, PR China. a r t i c l e. i n f o. a b s t r a c t. Article history: Received 8 January 2010 Available online 5 November 2010 Submitted by Goong Chen. This paper discusses linear processes with innovations exhibiting asymptotic weak dependence by being strong near-epoch dependent functions of mixing processes. The functional central limit theorem for the normalized partial sum process is established. The conditions given essentially improve on existing results in the literature in terms of the “size” requirement for the amount of dependence. It is also shown that two important econometric models, ARMA and GARCH models, are strong near-epoch dependent sequences. © 2010 Elsevier Inc. All rights reserved.. Keywords: Linear process Strong near-epoch dependence Functional central limit theorem ARMA model GARCH model. 1. Introduction In econometric models, such as integrated and cointegrated processes, innovations generally possess dependence structure. Econometricians usually adopt martingale differences and mixing processes to describe the dependence property of the innovations. Obviously, martingale differences are sequences of a rather special kind. One-step-ahead unpredictability is not a feature we can always expect to encounter in observed time series. Various mixing concepts are the most wellknown concepts of weak dependence. Econometric literature usually consider uniform (ϕ -) mixing and strong (α -) mixing random variables. However, the mixing concept has a serious drawback from the viewpoint of applications in time series modeling, in that a function of a mixing sequence (even an independent sequence) that depends on an infinite number of lags and/or heads of the sequence is not generally mixing. Further, martingale differences and mixing have limitation in practical modeling, because both of them concentrate on characterizing the dependence structure of the random variables themselves. Ibragimov [9] developed an idea that reflecting weak dependence, which had been formalized in different ways by many other researchers and was later called near-epoch dependence. Let { V n , n 1} be a sequence of random variables, put Fst = σ ( V s , . . . , V t ), Ets X = E( X |Fst ). For p > 0, put X p = (E| X | p )1/ p . Definition 1.1. Let p > 0. { X n , n 1} will be called L p -near-epoch dependent (L p -NED) on { V n , n 1} if there exist sequences {dn } and {ν (m)} of nonnegative constants, where ν (m) → 0 as m → ∞, such that for n 1 and m 0,. Xn − En+m Xn ν (m)dn . n−m p. ✩ Project supported by the grant of National Social Science Foundation of China (07CTJ001), National Research Project for Statistics (2009LY056), National Natural Science Foundation of China (10901136, 71072113). Corresponding author. E-mail address: [email protected] (J. Qiu).. *. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.10.078. ©. 2010 Elsevier Inc. All rights reserved..

(2) 374. J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. We will call { X n , n 1} “an L p -NED sequence of size −λ” if ν (m) in Definition 1.1 is of size −λ, that is ν (m) = O (m−λ−ε ) for some ε > 0. The scale constants dn allow for possible non-stationarity. { V n } is called the companion sequence. Nearepoch dependence is not an alternative to a mixing assumption; it is a property of the mapping from { V n } to { X n }, not of the random variables themselves. It is a nonparametric restriction on the memory of a process and has the advantage of holding in cases where mixing fails. From the viewpoint of applications, near-epoch dependence captures nicely the characteristic of a stable dynamic econometric model in which a dependent variable X n depends mainly on the recent histories of a collection of explanatory variables or shock processes V n . NED becomes useful when combined with a mixing condition on V n , and in particular, independence of this series. The concept of NED was first introduced to econometricians by Gallant and White [7]. A wide scope of econometric time series models, such as ARMA models, linear processes, a range of popular nonlinear models, etc., possess the property of L p -NED on an independent process (cf., e.g., Davidson [2,3]). The most general central limit theorem (CLT) and functional central limit theorem (FCLT) for NED sequences were provided by De Jong [5] and De Jong and Davidson [6] respectively. To weaken the sizes of both NED and mixing dependence that are required in CLT and FCLT for NED sequences, Lin [11] imposed a weak restriction on NED sequence by introducing a new class of dependent random variables, which is a subclass k+n of NED sequence, but also “approximately” mixing. We call such sequence a strong NED sequence. Put S k (n) = t =k+1 X t . Definition 1.2. Let p > 0. { X n , n 1} will be called a strong L p -NED sequence on { V n , n 1} if there exist sequences {dn } and {ν (m)} of nonnegative constants, where ν (m) → 0 as m → ∞, such that for all k > 0, n 1 and m 0,. n 1/2 2 S k (n) − Ek+n+m S k (n) ν (m) dk+ j . k+1−m p j =1. For strong NED sequences, the terminology size, which has been defined for NED, is also applicable. We will call. { Xn , n 1} “a strong L p -NED sequence of size −λ” if ν (m) in Definition 1.2 is of size −λ. Lin [11] gave two examples of strong NED sequences, linear processes and a sort of popular nonlinear models defined by nonlinear difference equations, which are given as examples of NED sequences in Davidson [2]. Indeed, we can verify many other popular econometric models satisfy strong NED property. In Section 3, we will show two familiar cases, ARMA models and GARCH models, are strong NED sequences under general conditions. Lin [11] established a maximal inequality on p-th moment, p > 2, under quite weak dependence sizes. By applying this inequality, Lin and Qiu [12,16] derived a CLT and FCLT for strong L p -NED under the same moment condition but weaker size requirement for dependence than that in Theorem 2 of De Jong [5] and Theorem 3.1 of De Jong and Davidson [6]. We state the results in the following proposition. Let { K n (ξ ), n 1} be a sequence of integer-valued, right continuous, nondecreasing functions of ξ , with K n (0) = 0, K n (1) = n for all n 1; K n (ξ ) is nondecreasing in n for all ξ ∈ [0, 1]; and K n (ξ2 ) − K n (ξ1 ) → ∞ as n → ∞ if ξ2 > ξ1 . Proposition 1.1. Let { X n , n 1} be a sequence of random variables with E X n = 0 and E| X n | p M < ∞ for some p > 2, n = 1, 2, . . . . Suppose that { X n , n 1} is strong L p -NED on a ϕ -mixing sequence { V n , n 1} with. . . ϕ (n) = O (log n)− p(1+δ/2) ,. (1.1). {dn } and {ν (m)} satisfying lim sup sup n→∞ k0. . n j =1. dk2+ j /n = B < ∞,. ν (m) = O (log m)−(1+δ/2). (1.2). . (1.3). and. σn2 := E S n2 → ∞ as n → ∞, where S n =. n. j =1. (1.4). X j . Then we have. S n /σn → N (0, 1). in distribution.. In addition, assume that. η(ξ ) := lim E Xn (ξ )2 n→∞. exists for all ξ ∈ [0, 1], where X n (ξ ) = σn−1. . . (1.5). K n (ξ ) j =1. . X j , and that. . lim sup lim sup K n min(ξ + δ, 1) − K n (ξ ) /n = 0.. δ→0 ξ ∈[0,1] n→∞. (1.6).

(3) J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. 375. Then. X n (ξ ) ⇒ X (ξ ), where X (ξ ) is a Gaussian process having almost surely (a.s.) continuous sample paths and independent increments. Based on these results, we can derive the limit distribution of many important processes with strong NED innovations. Linear processes arise very frequently in econometric and physical science modeling. Consider a linear process defined by. Zt =. ∞ . θ j Xt − j ,. (1.7). j =0. where { X t , −∞ < t < +∞} is a zero mean L p -bounded, p > 2, and strong near-epoch dependent sequence on another sequence { V t , −∞ < t < +∞}, which might be assumed to be mixing, {θ j , j 0} is a sequence of real numbers that satisfying ∞ . |θ j | < ∞.. (1.8). j =0. From (1.8), the covariances of X t are summable and we say that X t is short-range dependent. X t is generically called to be long-range dependent if its covariances are not absolutely summable. We talk about short-range dependent processes in this paper. The FCLT and CLT for linear processes are extremely useful in characterizing the asymptotic distribution of various test statistics arising from the inference for econometric models. There is an extensive literature concerning this topic. Here we only cite some papers talking about processes with dependent innovations. For example, Kim and Baek [10] imposed a stationary linear positively quadrant dependent (LPQD) assumption, Wang et al. [18] considered the martingale difference case, Lu [13] supposed the innovations to be a negative associated (NA) sequence, Moon [14] assumed stationary ρ -mixing condition. The above literature all focused on the innovations with a relatively specific dependence structure respectively. In terms of generically adapted and stationary sequences, Wu and Min [19] established results under L p weak dependence conditions, which is satisfied by a wide class of dependent innovations. Peligrad and Utev [15] discussed the CLT for linear processes with dependent innovations including mixingale type of assumptions, which is a weaker form of the strongly mixing coefficient. For non-adapted circumstance, Dedecker et al. [4] studied the CLT and FCLT for sums of non-adapted stationary sequences under conditions involving the conditional expectation of the variables with respect to a given σ -algebra. The conditions they assumed are satisfied by a large variety of examples including linear processes with dependent innovations. Davidson [3] derived sufficient conditions for the FCLT to hold in linear processes with near-epoch dependent innovations, which may assumed to be non-adapted and non-stationary sequences, on mixing sequences. In this paper, we establish the FCLT for linear processes defined by (1.7) with innovations exhibiting asymptotic weak dependence by being strong NED functions of mixing processes. We aim at improving on the requirements on the amount of dependence. Main results and their proofs are given in Section 2. In Section 3, we show that two important econometric models, ARMA models and GARCH models, are strong NED sequences under general conditions. The proofs of the lemmas and theorems in Section 3 are gathered in Appendix A. 2. Main results and proofs We are interested in the limit behavior of the partial sum processes of Z t in (1.7). Let. Z n (ξ ) = where. [nξ ] 1 . τn. Zt ,. (2.1). t =1. n. τn2 = υθ2 E(. t =1. X t )2 = υθ2 σn2 ,. υθ =. ∞. j =0 θ j. > 0.. Theorem 2.1. Let { V n , −∞ < n < +∞} be a ϕ -mixing sequence with mixing coefficient ϕ (n) satisfying (1.1) and let { X n , −∞ < n < +∞} be a zero mean L p -bounded and strong L p -NED sequence on { V n }, p > 2, with {dn } and {ν (m)} satisfying (1.2) and (1.3). Assume that (1.4) holds and for all k 0,. lim. n→∞. E S k2 (n) E S n2. = 1,. (2.2). and that the sequence {θ j } satisfies (1.8), then we have. Z n (ξ ) ⇒ W (ξ ), where W (ξ ) is a standard Wiener process.. (2.3).

(4) 376. J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. Remark 2.1. From Theorem 2.1, we can conclude that any strong L p -NED sequence satisfying the conditions (1.1)–(1.4) of Proposition 1.1 can be substituted by a linear process generated by itself, and the invariance principle is preserved so long as the conditions (1.8) and (2.2) are satisfied. Remark 2.2. Note that the scale constants dn in the definition of both NED and strong NED allow for possible nonstationarity. Therefore, in our Theorem 2.1, we do not require the sequence to be adapted and stationary. This is a significant feature that differs from the situations discussed in Kim and Baek [10], Wang et al. [18], Lu [13], Wu and Min [19], Peligrad and Utev [15], Dedecker et al. [4], Moon [14], etc. Remark 2.3. Davidson [3] considered the linear process defined by (1.7) with { X t } satisfies the following assumptions: 1. X t is L 2 -NED of size −1/2 on a process { V s } with respect to constants dt X t r , where V s is either −r /(r − 2) for r > 2 or ϕ -mixing of size −r /(2r − 2) for r 2. 2. supt E| X t − E X t |r < ∞, and if r = 2 then {( X t − E X t )2 } is uniformly integrable. 3. σn2 /n → σ 2 > 0, as n → ∞.. ∞. α -mixing of size. ∞ n+k. 2 Then if |θ j | is regularly varying at infinity, and 0 < | j =0 θ j | < ∞ and k=0 ( j =k+1 θ j ) = o (n), then the FCLT for Z t is obtained. Comparing our Theorem 2.1 with the result above, firstly, we find that under the same moment condition on X t , the “size” requirements on both NED and mixing are essentially weakened, that is the polynomial rate is reduced to logarithmic rate. Our condition on the norm of NED is slightly stronger since p > 2 is required. Secondly, Davidson [3] required dt X t r , r > 2, which is stronger than (1.2). Thirdly, assumption 3, σn2 /n → σ 2 > 0 as n → ∞, is not imposed as a precondition in our paper. In fact, it can be shown in a strong NED case. Finally, the conditions imposed on {θ j } in Davidson [3] is weaker than absolute summability, which is required in most of the literature considering linear processes being short-range dependent.. To prove Theorem 2.1, we need the following lemmas. Throughout the sequel, C will represent a positive constant although its value may change from one appearance to the next. Denote b j = d j ∨ X j 2 , j 1. Lemma 2.1. (See Lin [11].) Let { V n , n 1} be a ϕ -mixing sequence with mixing coefficient ϕ (n) satisfying (1.1) and let { X n , n 1} be a zero mean L p -bounded and strong L p -NED sequence on { V n }, p > 2, with {dn } and {ν (m)} satisfying (1.2) and (1.3). Then there exists a finite constant C depending only on {ϕ (·)} and {ν (·)} such that for all integers k 0 and n 1,. E. . p max S k (i ) C ( Dn) p /2 ,. 1i n. (2.4). where D = B ∨ supn1 X n 2p . Lemma 2.2. (See Qiu and Lin [17].) Let { V n , n 1} be a ϕ -mixing sequence with mixing coefficient ϕ (n) satisfying (1.1) with p = 2 and let { X n , n 1} be a zero mean L 2 -bounded and strong L 2 -NED sequence on { V n }, with {ν (m)} satisfying (1.3). If there exist a constant β 1 and an integer N 0 > 0 such that. max b2j β min b2j j>N0. j>N0. (2.5). and for all integers k 0,. E S k2 (n). min j > N 0 b2j. → ∞,. (2.6). then there exist a positive integer N 0∗ > 2N 0 and a finite constant C ∗ > 0 depending only on {ϕ (·)}, {ν (·)} and β such that for all integers k 0 and n > N 0∗ ,. E S k2 (n) C ∗ n min b2j . j>N0. (2.7). Lemma 2.3. (See Qiu and Lin [17].) If, in addition to the assumptions of Lemma 2.2, (2.2) is satisfied, then we have. lim. n→∞. σn2 n. = σ 2 > 0.. (2.8).

(5) J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. 377. Remark 2.4. Noting the effect of d j as an upper bound in Definition 1.2, we can redefine d∗j 2 = d2j + 1 and b∗j = d∗j ∨ X j 2 to ensure that lim inf j →∞ b∗j > 0. By substituting d∗j and b∗j for d j and b j respectively, the results about strong L p -NED. sequence, such as Proposition 1.1 and Lemmas 2.1–2.3, still hold with B ∗ = B + 1 and D ∗ = B ∗ ∨ supn1 X n 2p . So we always assume that lim inf j →∞ d j > 0. Proof of Theorem 2.1. Since for each integer m 1, we have m . Zt =. m ∞ . t =1. θ j Xt − j =. t =1 j =0. t −1 m t =1. . θ j Xt − j +. ∞ m t =1. j =0. θ j Xt − j. j =t. m−l ∞ m m = θ j Xl + θ j Xt − j l =1. =. . j =0. m . ∞ . t =1. j =0. t =1. θ j Xt +. . j =t. m . ∞ . t =1. j =t. θ j Xt − j −. m t =1. ∞ . θ j Xt. j =m−t +1. =: I 1 (m) + I 2 (m) + I 3 (m). Therefore, it follows that for any ξ ∈ [0, 1], we have −1. Z n (ξ ) = τn. [nξ ] . . . . . . . Z t = τn−1 I 1 [nξ ] + τn−1 I 2 [nξ ] + τn−1 I 3 [nξ ] .. (2.9). t =1. By the definition of. . τn , we get. . τn−1 I 1 [nξ ] = σn−1. [nξ ] . Xt .. t =1. By Remark 2.4, we may assume that lim inf j →∞ d j > 0. Combining it with the assumption (1.2), there exists an integer N 0 > 0 such that. 0 < min b2j < max b2j < ∞, j>N0. j>N0. which implies (2.5). Moreover, by the assumptions (1.4) and (2.2), we derive (2.6). Therefore, by applying Lemma 2.3, we obtain (2.8), which implies that. . lim E. n→∞. −1. σn. 2. [nξ ] . Xt. = ξ.. t =1. So the conditions of Proposition 1.1 are satisfied. Hence we conclude that. . . τn−1 I 1 [nξ ] ⇒ W (ξ ), where W is a standard Wiener process. Thus by Slutsky’s lemma, Theorem 2.1 will be proved if we can show that for any ε>0.

(6). lim sup P n→∞. and n→∞. . . (2.10). 0ξ 1.

(7) lim sup P. . sup I 2 [nξ ] ετn = 0. . sup I 3 [nξ ] ετn = 0.. (2.11). 0ξ 1. First, using Minkowski’s inequality and Lemma 2.1, we have. m ∞. ∞ j ∧m. p p. . . E max. θ j X t − j = E max. θ j Xt − j. 1mn. 1mn. t =1. j =t. j =1 t =1. j ∧m. p ∞. j ∧m. p 1 / p p . . . . E |θ j | max. Xt − j. |θ j | E max. Xt − j. 1mn. 1mn. t =1 t =1 j =1 j =1 ∞ p 1/2 1/2 |θ j |C D ( j ∧ n) , ∞ . j =1.

(8) 378. J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. where D = B ∨ supn1 X n 22 , B = lim supn→∞ supk0. n. 2 j =1 dk+ j /n.. Thus by Markov’s inequality, we obtain that. m ∞ p ∞ 1/2 p .

(9). . . j ∧n. − p − p − p P sup I 2 [nξ ] ετn ε τn E max. θ j Xt − j C ε |θ j |. 1mn. n 0ξ 1 t =1 j =t j =1 n p 1/2 ∞ j = C ε− p |θ j | + |θ j | → 0 as n → ∞, n. j =1. j =n+1. where the fact that the first term in the last line converges to zero follows from Kronecker’s lemma and the condition (1.8). Hence (2.10) is proved. Next, we consider (2.11). For each integer m 1, we have. I 3 (m) = −. m . . ∞ . t =1. θ j Xt = −. j =m−t +1. m . θj. j =1. . m . Xt −. t =m− j +1. m . θj. ∗ (m) = − = θ j I[ j s]. Letting I 31. m. Xt. t =1. j =m+1. =: I 31 (m) + I 32 (m). For any fixed integer s 1, define θ ∗j. . ∞ . j =1. θ∗. (2.12). m. t =m− j +1. j. X t , then. m m. −1 ∗ . −1 ∗. . −1 ∗ . sup τn I 31 [nξ ] = max τn I 31 (m) = max τn θj Xt. 1mn 1mn. 0ξ 1 j =1 t =m− j +1. s m −1 m s. . p. −1 ∗. −1 ∗. X m−l θ j τn max | X t | → 0. = max τn θj −. 1mn. 1t n l =0. j =l+1. l =0. (2.13). j =l+1. Furthermore, note that. sup. 0ξ 1. . . . . ∗ ∗ τn−1 I 31 [nξ ] − I 31 (m). [nξ ] = max τn−1 I 31 (m) − I 31 1mn. m. m. . . θ j − θ ∗ ·. τn−1 max Xt. j. 1mn j =1 t =m− j +1. m. − j . m m. . . m. −1 ∗. −1 ∗. τn max X t + τn max Xt. θj − θj ·. θj − θj ·. 1mn 1mn t =1 t =1 j =1 j =1. m. ∞. . Xt. |θ j |. 2τn−1 max. 1mn. t =1. j = s +1. Therefore, by Markov’s inequality and Lemma 2.1, we have that for any.

(10) lim sup P. τn. sup. n→∞. −1. 0ξ 1. I 31. . ε>0. . −p ∗ [nξ ] − I 31 [nξ ] > ε lim sup 2 p ε − p τn. . n→∞. C ε− p. ∞ . p |θ j |. m. . p. E max. Xt. 1mn. p . ∞ . |θ j |. j = s +1. t =1. → 0 as s → ∞.. (2.14). j = s +1. Thus it follows from (2.13) and (2.14) that. . lim sup P n→∞. . . sup I 31 [nξ ] >. 0ξ 1. ε 2. . τn = 0.. (2.15). Define positive integer ln such that ln → ∞ and ln = o(n) as n → ∞. Applying Lemma 2.1, we have. P. ∞. m . . ε. θj X t > τn. 2 1mn. t =1 j =m+1. ∞. m m ∞. . . . ε. P max. θj X t + max. θj X t > τn. ln +1mn. 2 1mln. . ε. sup I 32 [nξ ] > τn = P max. 0ξ 1. 2. j =m+1. t =1. j =m+1. t =1.

(11) J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. P. 379. m. m. ∞ ∞. ε. ε . |θ j | max. X t > τn + P |θ j | max. X t > τn. 4. 4 1mn. 1mln. . j =0. . −p. 4 p ε − p τn. −p. C ε − p τn. t =1. ∞ . p |θ j |. j =0. ∞ . j =ln +1. p. p. |θ j |. t =1. m. m. ∞ p. p. p . E max. Xt + |θ j | E max. Xt. 1mn. 1mln. ln 2 +. j =0. t =1. ∞ . |θ j |. t =1. j =ln +1. p. p. n2. → 0 as n → ∞.. (2.16). j =ln +1. Therefore, from (2.12), combining (2.15) and (2.16) yields (2.11). This completes the proof of Theorem 2.1.. 2. From Theorem 2.1, we notice that the strong near-epoch dependence property provides a convenient procedure of proving the FCLT for linear processes with a range of dependent innovations. Therefore, it is of importance that we verify whether the innovations are strong L p -NED sequences. 3. Examples of strong L p -NED sequences 3.1. ARMA model Consider the general ARMA(l, q) model. X t = λ0 +. l . λi X t −i + V t +. i =1. q . ψi V t −i ,. (3.1). i =1. where { V t , −∞ < t < ∞} is a mean zero, L p -bounded sequence with p 2. We can write (3.1) in the form. . . 1 − λ1 L − · · · − λl Ll X t = λ0 + V t +. q . ψi V t −i ,. (3.2). i =1. where L is the lag operator such that L X t = X t −1 . Therefore, we have. . X t = 1 − λ1 L − · · · − λl L. . l −1. . = μ + 1 − λ1 L − · · · − λl L. λ0 + V t + l −1. q i =1. Vt +. q . ψi V t −i . ψi V t −i ,. (3.3). i =1. where. μ = λ0 /(1 − λ1 − · · · − λl ). Suppose that the characteristic roots of the equation zl − λ1 zl−1 − · · · − λl = 0. (3.4). all lie inside the unit circle. Note that this is the usual stability condition to ensure the process to be covariance stationary. Lemma 3.1. Let the characteristic roots of (3.4) all lie inside the unit circle. Any ARMA model X t defined by (3.1) can be expressed as ∞ j Xt = j =0 θ j V t − j , where |θ j | = O (ρ ) for some 0 < ρ < 1. Theorem 3.1. Let { X t , −∞ < t < ∞} be an ARMA(l, q) process defined by (3.1) and the characteristic roots of (3.4) all lie inside the unit circle, where the shock process { V t , −∞ < t < ∞} is a zero mean, L p -bounded, p 2, sequence of random variables. Then {Y t , −∞ < t < ∞} is a strong L p -NED sequence on { V t , −∞ < t < ∞}, with dt = 2ρ (1 − ρ )−2 supt V t p and ν (m) = ρ m , where 0 < ρ < 1. Remark 3.1. Note that in the verification of SNED property for ARMA model, we do not impose any dependence structure on the shock processes { V t , −∞ < t < ∞}. Therefore, to apply Theorem 2.1 for linear processes with ARMA innovations, we may assume the shock process in the ARMA model to be ϕ -mixing with the mixing coefficients satisfying (1.1), or more particular sequences, martingale differences or i.i.d. random variables..

(12) 380. J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. 3.2. GARCH model GARCH model, which was proposed by Bollerslev [1], is widely used in financial econometric time series analysis. Consider the following GARCH(l, q) model. Ut =. . ht εt ,. (3.5). where {εt , −∞ < t < ∞} is a sequence of L p -bounded, p 2, i.i.d. random variables with mean 0 and variance 1, and ht is a Gt −1 -measurable process defined as. ht = α0 +. l i =1. αi U t2−i +. q . β j ht − j ,. (3.6). j =1. where Gt = σ (εs , s t ), α0 > 0, αi > 0 for 1 i l and β j > 0 for 1 j q. Define Suppose l > q, then (3.6) can be written as. ht = α0 +. l i =1. where. (αi + βi )U t2−i +. q . β j ηt − j ,. αi = 0 for i > l and β j = 0 for j > q.. (3.7). j =1. η j = h j − U 2j . So we have U t2 = α0 +. q l (αi + βi )U t2−i + ηt + β j ηt − j . i =1. (3.8). j =1. Furthermore, note that. . . . . . E(η j |G j −1 ) = E h j 1 − ε 2j |G j −1 = h j E 1 − ε 2j |G j −1 = 0. Therefore, from (3.8), we see that {U t2 , −∞ < t < ∞} is an ARMA(l, q) sequence with martingale difference shock process {ηt , −∞ < t < ∞}. Suppose that the characteristic roots of the equation. zl − (α1 + β1 ) zl−1 − · · · − (αl + βl ) = 0. (3.9). l. all lie inside the unit circle, which implies that i =1 (αi + βi ) < 1 because of the fact that αi > 0 for 1 i l and β j > 0 for 1 j q. Then {U t2 } is a covariance stationary process. So by the discussions in Section 3.1, we obtain the following conclusion. Theorem 3.2. Let {U t , −∞ < t < ∞} be a fourth-order stationary GARCH(l, q) process defined by (3.5) and (3.6). Then {U t2 , −∞ < t < ∞} is a strong L p -NED sequence on {ηt , −∞ < t < ∞}, p 2, with ν (m) = ρ m and dt = K supt ηt p , where 0 < ρ < 1, K is a positive number depending only on ρ . And hence, the result of Theorem 2.1 holds for linear processes with GARCH innovations. Remark 3.2. In fact noting that from the derivation of (3.8), {U t2 } can be treated as an ARMA model with shock process {ηt } of any dependence structure. As stated in Remark 3.1, in the verification of SNED property, i.e. Definition 1.2, for ARMA models, there is no requirement on the dependence structure of the shock processes. So the verification of Definition 1.2 for {U t2 } also does not rely on the dependence structure of {ηt }. Therefore, the sequence {εt , −∞ < t < ∞} in (3.5) is not necessarily assumed to be i.i.d. random variables. It might be assumed to be a mixing process or a martingale difference sequence. Most of the literature dealing with linear processes with GARCH innovations supposed {εt } to be i.i.d. random variables (cf., e.g., Wu and Min [19], Davidson [3]). 4. Conclusion The purpose of this paper is twofold. First, it discusses the FCLT for the partial sum process of a linear process with strong L p -NED innovations. The conditions, especially the size requirements of both NED and mixing dependence, are substantially weaker than those in the existing literature. Second, it presents two important examples of strong near-epoch dependence, ARMA models and GARCH models. What merits particular attention is that the shock processes in the ARMA model and GARCH model are not necessarily assumed to be i.i.d. random variables. Acknowledgments The authors are grateful to the anonymous referees for providing several helpful comments..

(13) J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. 381. Appendix A Proof of Theorem 3.1. Put Fst = σ ( V s , . . . , V t ). Then by applying Lemma 3.1, Minkowski’s inequality and Jensen’s inequality, we obtain that. k+n. k+n k+n p. . . k+n+m E. Y t − Ek+1−m Y t = E. t =k+1. t =k+1. ∞ . . t =k+1 j =t −(k−m). n+m θ j V t − j − Ekk+ +1−m V t − j. p . p. k−m k+n . . n+m = E. θt − j V j − Ekk+ Vj. + 1 − m. j =−∞. . k −m. . t =k+1. k+n . p 1 / p . |θt − j | E V j − Ek+n+m V j. k+1−m. j =−∞ t =k+1. . 2. k −m. p. p. k+n . ρ. t− j. V j p. 2. p. j =−∞ t =k+1. =2. p. p sup V t p t. p. . ρ (1 − ρ )2. ρ. m. . 1−ρ. n. . p sup V t p t. . p ρ. pm. k+n . ∞ . p. ρ. t+ j. t =k+1 j =m−k. n. 4ρ 2. (1 − ρ. sup V t 2p )4 t. p /2 .. Therefore, it follows that. k+n k+n k+n 1/2 k+n+m 2 Y t − Ek+1−m Y t ν (m) dt , t =k+1. t =k+1. 2ρ (1 − ρ )−2 sup. where dt = obtained. 2. t V t p. t =k+1. p. and. ν (m) = ρ → 0 as m → ∞. By the definition of strong L p -NED, the conclusion is m. Proof of Lemma 3.1. Case 1. If Eq. (3.4) has l distinct roots. 1. =. 1 − λ1 z − · · · − λl zl. A1. + ··· +. 1 − ρ1 z. Al 1 − ρl z. ρ1 , . . . , ρl , then we have the following decomposition (A.1). ,. where. ρil−1. Ai = l. (A.2). ρ − ρk ). k=1, k =i ( i. (cf., e.g., Hamilton [8]). So (3.3) can be expressed as. . −1. X t = μ + A 1 (1 − ρ1 L ). = μ + A1. ∞ . = μ + A1 + Al. + · · · + Al (1 − ρl L ). j j 1L. ρ. Vt +. j =0. . −1. q . j 1 V t− j. ρ. + ψ1. j =0. ∞ j =0. j. ρl V t − j + ψ1. Vt +. q . ψi V t −i. i =1. ψi V t −i + · · · + A l. i =1. ∞ . . ∞ . j. ρl L. j. Vt +. j =0. ∞ . j 1 V t −1 − j. ρ. + · · · + ψq. j =0. ∞ j =0. j. ρl V t −1− j + · · · + ψq. q . ψi V t −i. i =1. ∞ . . ρ. j =0. ∞ . j 1 V t −q− j. j. + ···. . ρl V t −q− j .. (A.3). j =0. Note that when ρ1 , . . . , ρl are all real numbers, A 1 , . . . , Al are also real. On the other hand, if some roots of (3.4) are complex, then the complex roots will appear as complex conjugate quantity pairs. Without loss of generality, we assume (3.4) has a pair of complex roots ρ1 = r (cos θ + i sin θ) and ρ2 = r (cos θ − i sin θ), where the modulo r < 1. Then by (4.2), A 1 and A 2 in (4.1) must be a pair of complex conjugate quantities. Thus we can denote A 1 = a + bi and A 2 = a − bi. So we can rewrite (4.3) as follows.

(14) 382. J. Qiu, Z. Lin / J. Math. Anal. Appl. 376 (2011) 373–382. X t = μ + (a + bi ). q j r cos( j θ) + i sin( j θ) L V t + ψi V t −i. ∞ j j =0. + (a − bi ). ∞ j =0. + A3. ∞ . . . j j 3L. ρ. = μ + 2a. Vt +. ∞ . − 2b + Al. ψi V t −i + · · · + A l. j. r cos( j θ) V t − j + ψ1. j. r sin( j θ) V t − j + ψ1. j. ρl L. j. Vt +. q . ψi V t −i. i =1. j. r cos( j θ) V t −1− j + · · · + ψq. ∞ . j 3 V t− j. ρ. + ψ1. j =0. ∞ . ∞ . j. r sin( j θ) V t −1− j + · · · + ψq. j 3 V t −1 − j. ρ. ρl V t − j + ψ1. ∞ . + · · · + ψq. j =0. ∞ . r cos( j θ) V t −1− j. . j. r sin( j θ) V t −1− j. j 3 V t −q− j. ρ. j =0 j. j. j =0. j =0 j. ∞ j =0. j =0. ∞ . j =0. ∞ . . j =0. j =0. ∞ . ∞ j =0. j =0. . + A3. . q. i =1. . q. i =1. ∞ . i =1. r cos( j θ) − i sin( j θ) L j V t + ψi V t −i j. j =0. . . ρl V t −1− j + · · · + ψq. ∞ . + ···. j. ρl V t −q− j .. (A.4). j =0. Case 2. If Eq. (3.4) has at least one multiple root, (3.3) can be expressed in a form similar to that of (4.3) or (4.4). defined by (3.1) can always be expressed as linThat is, under the conditions of Lemma 3.1, ∞ ∞the jARMA(l, q) model ∞ j j ear combinations of the items j =0 ρ V t − j , j =0 ρ cos( j θ) V t − j and j =0 ρ sin( j θ) V t − j and so on, which implies the conclusion. 2 References [1] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (1986) 307–327. [2] J. Davidson, Stochastic Limit Theory, Oxford University Press, Oxford, 1994. [3] J. Davidson, Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes, J. Econometrics 106 (2002) 243–269. [4] J. Dedecker, F. Merlevède, D. Volný, On the weak invariance principle for non-adapted sequences under projective criteria, J. Theoret. Probab. 20 (2007) 971–1004. [5] R.M. De Jong, Central limit theorems for dependent heterogeneous random variables, Econometric Theory 13 (1997) 353–367. [6] R.M. De Jong, J. Davidson, The functional central limit theorem and weak convergence to stochastic integral I: Weakly dependent processes, Econometric Theory 16 (2000) 612–642. [7] A.R. Gallant, H. White, A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models, Basil Blackwell, New York, 1988. [8] J.D. Hamilton, Time Series Analysis, Princeton University Press, 1994. [9] I.A. Ibragimov, Some limit theorems for stationary processes, Theory Probab. Appl. 7 (1962) 349–382. [10] T. Kim, J. Baek, A central limit theorem for stationary linear processes generated by linearly positively quadrant-dependent process, Statist. Probab. Lett. 51 (2001) 299–305. [11] Z.Y. Lin, Strong near-epoch dependence, Sci. China Ser. A 47 (4) (2004) 497–507. [12] Z.Y. Lin, J. Qiu, A central limit theorem for strong near-epoch dependent random variables, Chin. Ann. Math. Ser. B 25 (2) (2004) 1–12. [13] C.R. Lu, The invariance principle for linear processes generated by a negative associated sequence and its applications, Acta Math. Appl. Sin. Engl. Ser. 19 (4) (2003) 641–646. [14] H.J. Moon, The functional CLT for linear processes generated by mixing random variables with infinite variance, Statist. Probab. Lett. 78 (14) (2008) 2095–2101. [15] M. Peligrad, S. Utev, Central limit theorem for stationary linear processes, Ann. Probab. 34 (2006) 1608–1622. [16] J. Qiu, Z.Y. Lin, The functional central limit theorem for strong near-epoch dependent random variables, Prog. Nat. Sci. 14 (2004) 16–21. [17] J. Qiu, Z.Y. Lin, The variance of partial sums of strong near-epoch dependent variables, Statist. Probab. Lett. 76 (17) (2006) 1845–1854. [18] Q.Y. Wang, Y.X. Lin, C.M. Gulati, The invariance principle for linear processes with applications, Econometric Theory 18 (2002) 119–139. [19] W.B. Wu, W. Min, On linear processes with dependent innovations, Stochastic Process. Appl. 115 (2005) 939–958..

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