Asymptotics of empirical processes of long memory moving averages with innite variance

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1 Stochastic Processes and their Applications 91 (2001) Asymptotics of empirical processes of long memory moving averages with innite variance Hira L. Koul a;, Donatas Surgailis b a Department of Statistics & Probability, Michigan State University, East Lansing, MI 48824, USA b Vilnius Institute of Mathematics & Informatics, USA eceived 15 November 1999; received in revised form 5 June 2000; accepted 14 June 2000 Abstract This paper obtains a uniform reduction principle for the empirical process of a stationary moving average time series {X t} with long memory and independent and identically distributed innovations belonging to the domain of attraction of symmetric -stable laws, 1 2. As a consequence, an appropriately standardized empirical process is shown to converge weakly in the uniform-topology to a degenerate process of the form fz, where Z is a standard symmetric -stable random variable and f is the marginal density of the underlying process. A similar result is obtained for a class of weighted empirical processes. We also show, for a large class of bounded functions h, that the limit law of (normalized) sums n h(xs) is symmetric -stable. s=1 An application of these results to linear regression models with moving average errors of the above type yields that a large class of M-estimators of regression parameters are asymptotically equivalent to the least-squares estimator and -stable. This paper thus etends various well-known results of Dehling Taqqu and Koul Mukherjee from nite variance long memory models to innite variance models of the above type. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 62G05; secondary 62J05; 62E20 Keywords: Non-random designs; Unbounded spectral density; Uniform reduction principle; M-estimators 1. Introduction and summary A strictly stationary second-order times series X t ;t Z := 0; ±1; ±2;:::; is said to have long memory if its lag t covariances are not summable and decrease as t 2d 1, where 0 d 1=2. The eistence of long memory data has been manifested in numerous scientic areas ranging from climate warming to stock markets (Beran, 1992; obinson, 1994b; Baillie, 1996). One of the most popular models of long memory processes is AFIMA (p; d; q) dened by the autoregressive equation (L)(1 L) d X t = (L) t ; (1.1) Corresponding author. Fa: address: koul@stt.msu.edu (H.L. Koul) /01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S (00)00065-X

2 310 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where t ;t Z; is an independent and identically distributed (i.i.d.) sequence, L is the backward shift operator, (1 L) d is the fractional dierencing operator dened for 1=2 d 1=2 by the corresponding binomial epansion (see e.g. Granger and Joyeu (1980) or Hosking (1981)), and (L); (L) are polynomials in L of degree p; q, respectively, ( ) satisfying the usual root requirement for stationarity of the process. The stationary solution to (1.1) for d 0 can be written as a causal innite moving average process X t = b t s s ; (1.2) s6t whose weights b j satisfy the asymptotic relation b j c 0 j ; (j ); =1 d; (1.3) with the constant c 0 c 0 ( ;;d)= (1) = (1) (d) (see Hosking, 1981). Thus, in case of 0 having zero mean and nite variance, X t (1.1) is well-dened strictly stationary process for all d 1=2 and has long memory in the above sense, provided 1=2 1. An important problem in the contet of long memory processes from the inference point of view is the investigation of the asymptotic behavior of a class of statistics of the type S n;h = h(x t ); where h(); ; is a real valued measurable function, usually assumed to have nite second moment Eh 2 (X 0 ). A special case (up to the factor n 1 ) of utmost interest is the empirical distribution function ˆF n ()=n 1 I(X t 6); ; in which case the above question usually etends to the weak convergence of the corresponding random process indeed by ; in the Skorokhod space D( ); = [ ; ] with the sup-topology. For Gaussian long memory processes X t ;t Z; (including AFIMA (p; d; q) as a special case), the study of limit distributions of S n;h has a long history, starting with osenblatt (1961) and culminating in the papers of Dobrushin and Major (1979) and Taqqu (1979). The weak convergence of the empirical process of Gaussian and their subordinated long memory sequences was obtained in Dehling and Taqqu (1989). For non-gaussian linear processes (1.2) and (1.3) with nite variance, these problems were studied by Surgailis (1982), Giraitis and Surgailis (1989, 1999), Ho and Hsing (1996,1997), Koul and Surgailis (1997). It is well known that in the case the r.v. s X t are i.i.d. with continuous distribution function (d.f.) F, the normalized process n 1=2 ( ˆF n F) converges weakly in the space D( ) with the sup-topology, which we denote by D( ) in the sequel, to a Gaussian process Z(); ; with zero mean and covariance E[Z()Z(y)]=F( y) F()F(y) (see e.g. Billingsley, 1968; Doukhan et al., 1995; Shao and Yu, 1996). In the case of long memory, the asymptotic behavior of ˆF n is very dierent. Assuming a moving average structure of X t (1.2) and some additional regularity and moment

3 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) conditions on the distribution of 0 (which are satised of course in the case the latter are Gaussian), one has n =2 ( ˆF n () F()) D( ) cf()z; (1.4) where Z N(0; 1) is the standard normal variable, f is probability density of the marginal d.f. F of X 0, and c is some constant (see e.g., Dehling and Taqqu, 1989; Ho and Hsing, 1996; Giraitis and Surgailis, 1999). The dierence between (1.4) and the classical Brownian bridge limit is not only in the rate of convergence, which is much slower in (1.4) compared to the classical n 1=2, but, more importantly, in the asymptotic degeneracy of the limit process of (1.4) which shows that the increments of standardized ˆF n over disjoint intervals, or disjoint observation sets, are asymptotically completely correlated. Similar asymptotic behavior is shared by weighted residual empirical processes which arise in the study of multiple regression models with long memory errors (Koul and Mukherjee, 1993; Giraitis et al., 1996). These asymptotic degeneracy results provide the main basis of many surprising results about the large sample behavior of various inference procedures in the presence of long memory (Dehling and Taqqu, 1989; Beran, 1991; Koul, 1992a; Koul and Mukherjee, 1993; obinson, 1994a,b; Csorgo and Mielniczuk, 1995; Giraitis et al., 1996; Ho and Hsing, 1996; Koul and Surgailis, 1997 and the references therein). The aim of the present paper is to etend the functional limit result (1.4) and some of the above mentioned inference results to linear models (1.2), (1.3) with innite variance, in particular, to AFIMA (p; d; q) time series, with i.i.d. innovations t ;t Z, belonging to the domain of attraction of a symmetric -stable (SS) law, 1 2. More precisely, we shall assume in the sequel that X t ;t Z; is a moving average process (1.2), where j ;j Z are i.i.d. r.v. s with zero mean and satisfying the tail regularity condition lim G() = lim (1 G()) = c 1 (1.5) for some 1 2 and some constant 0 c 1, where G is the d.f. of 0. In addition, the weights b j ;j 0 satisfy the asymptotics (1.3), where c 0 0and 1= 1: (1.6) Without loss of generality, we assume c 1 = 1 of (1.5) in the sequel. Under these assumptions, b j = ; B:= b j ; (1.7) j=0 j=0 the linear process X t of (1.2) is well dened in the sense of the convergence in probability, and its marginal d.f. F satises lim F() = lim (1 F()) = B: (1.8) Note that (1.8) implies E X 0 = and E X 0 r for each r, in particular EX 2 0 = and E X 0. Because of these facts and (1.7), this process will be called long memory moving average process with innite variance in this paper. In the sequel,

4 312 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) we refer to the above assumptions as the standard assumptions about the time series in consideration. The class of moving averages satisfying these assumptions includes AFIMA (p; d; q) with SS-innovations, 0 d 1 1=. See Kokoszka and Taqqu (1995) for detailed discussion of properties of stable AFIMA series. Astrauskas (1983), Avram and Taqqu (1986, 1992), Kasahara and Maejima (1988) have shown that under the standard assumptions the sample mean X n = n 1 n X t is asymptotically -stable: na 1 n X n cz; (1.9) where Z is a standard SS r.v. with E[e iuz ]=e u ;u, and ( (2 )cos(=2) 1 1= 1 c = c 0 2c 1 (t s) + dt ds) : (1.10) 1 0 The normalization A n = n 1 +1= (1.11) grows much faster as compared with the usual normalization n 1= in the case of partial sums of independent random variables in the domain of attraction of -stable law. The latter fact is another indication that the series X t ;t Z; ehibits long memory. Clearly, the usual characterization of this property in terms of the covariance s decay is not available in the case of innite variance. In the case when the i.i.d. innovations j ;j Z; are SS, the moving average X t ;t Z; has SS nite dimensional distributions and the role of the covariance is played, to a certain etent, by the (Levy spectral measure) quantities such as covariation and=or codierence (see Samorodnitsky and Taqqu, 1994). A related characteristic of dependence in the innite variance case is Cov(e iu1x0 ; e iu2xt )=E[e i(u1x0+u2xt) ] E[e iu1x0 ]E[e iu2xt ]; u 1 ;u 2 : In the particular case when X t ;t Z is a fractional SS noise, for any u 1 ;u 2 Cov(e iu1x0 ; e iu2xt ) k u1;u 2 E[e iu1x0 ]E[e iu2x0 ]t 1 ; (1.12) as t, where k u1;u 2 is a constant depending on u 1 ;u 2 (Astrauskas et al., 1991). In section 6 we etend the asymptotics (1.12) to an arbitrary (not necessarily SS) moving average X t of (1.2) satisfying the standard assumptions above. We shall now summarize the contents of the remaining sections. Theorem 2.1 below contains the main result of the paper about the uniform reduction principle for weighted residuals empirical processes of an innite variance moving average observations X t ; t =1;:::;n. It yields in particular, an analog of (1.4) where now Z is a standard SS r.v. These results involve some additional regularity assumptions about the probability density of the innovations. Corollary 2.3 below shows that the weak limit of A 1 n S n;h, for a bounded h, isan SS r.v. This result itself is surprising, as it shows that an -stable (1 2) limit law may arise from sums of bounded random variables h(x t ). It is well known that in the case of i.i.d. or weakly dependent summands such limit laws require a long tailed summands distribution and the contribution of the maimal summand to be comparable

5 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) to the sum itself. These results further reconrm the deep dierences between long and short memory. Section 3 discusses the asymptotic distribution of (robust) M-estimators of the underlying regression parameters in linear regression models with innite variance longmemory moving average errors. We show that the least-squares estimator converges in distribution to a vector of SS r.v. s and asymptotically these M-estimators are equivalent to the least-squares estimator in probability (Theorem 3.1). These ndings should be contrasted with those available in the i.i.d. errors linear regression models with innite variance. In these models the asymptotic distribution of an M-estimator of the regression parameter vector is known to be Gaussian with zero mean and an asymptotic variance that depends on the given score function (Knight, 1993), a fact that is in complete contrast to the above ndings. The proofs of Theorems 2.1 and 3.1 appear in Sections 4 and 5, respectively. Finally, Section 6 discusses the asymptotics (1.12). 2. Uniform reduction principle for weighted empiricals We assume below that X t ;t Z; satises the standard assumptions of section 1 and, furthermore, that G is twice dierentiable with the second derivative G satisfying the inequalities G () 6C(1 + ) ; (2.1) G () G (y) 6C y (1 + ) ; ;y ; y 1: (2.2) These conditions are satised if G is SS d.f., which follows from asymptotic epansion of stable density (see e.g. Christoph and Wolf (1992, Theorem 1:5) or Ibragimov and Linnik (1971)). In this case, (2.1) (2.2) hold with + 2 instead of. Under the standard assumptions, the d.f. F of X 0 is shown to be innitely dierentiable in Lemma 4.2 below. Now, let f denote the density of F and introduce the weighted empirical process: S n ()= n;s (I(X s 6 + n;s ) F( + n;s )+f( + n;s )X s ); ; (2.3) s=1 where ( n;i ; n;i ;16i6n) are non-random real-valued sequences. We are ready to state Theorem 2.1. Assume; in addition to the standard assumptions and conditions (2:1) and (2:2); that (a:1) ma n;i = O(1); 16i6n (a:2) ma n;i = O(1): 16i6n Then; there eists 0 such that; for any 0, [ ] P sup A 1 n S n () 6Cn ; where A n is as in (1:11).

6 314 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) In the special case n;i 1; n;i 0; S n ()=n( ˆF n () F()+f() X n ) and Theorem 2.1 implies Corollary 2.1. There is 0 such that; for any 0; [ ] P sup na 1 n ˆF n () F()+f() X n 6Cn : This fact and (1.9) readily imply the following two corollaries: Corollary 2.2. na 1 n ( ˆF n () F()) D( ) cf()z; where Z is a standard SS random variable and the constant c given in (1:10). Corollary 2.3. Let h be a real valued measurable function of bounded variation; such that Eh(X 0 )=0. Then; A 1 n S n;h ch 1 Z; where h 1 = f()dh()= h()f ()d. emark 2.1. Corollaries etend the uniform reduction principle and some other results of Dehling and Taqqu (1989) to the case of long memory processes with innite variances. As mentioned earlier, Corollary 2.3 is surprising in the sense it shows that an -stable (1 2) limit law may arise from sums of bounded random variables h(x t ). This is unlike the case of i.i.d. or weakly dependent summands, where such limit laws require a long tailed summands distribution and the contribution of the maimal summand to be comparable to the sum itself. emark 2.2. In the case h 1 = 0, Corollary 2.3 implies A 1 n S n;h 0 only. The question whether in this case it is possible to obtain a nondegenerate limit for S n;h with some other normalization o(a n ), is open. It is possible that the situation in the innite variance case is quite dierent in this respect from (say) the Gaussian case, in the sense that higher order epansions of the empirical distribution function (the analogs of the Hermite epansion in the case of a Gaussian underlying process) may not eist at all. emark 2.3. Corollary 2.3 contradicts the recent result of Hsing (1999, Theorem 2) which claims, under similar assumptions on X t and h, that (var(s n;h )) 1=2 S n;h converges to a nondegenerate Gaussian limit. Note the normalization (var(s n;h )) 1=2 =O(n (3 )=2 ) grows faster that A n = n 1 +1=. The proof of the above mentioned theorem in Hsing (1999) uses an approimation of S n;h by a sum of independent (but not identically distributed) random variables, whose normal limiting behavior is deduced by the classical Lindeberg Feller central limit theorem. However, Lindeberg s condition ((38) of Hsing (1999)) actually does not hold, as we shall now show. For convenience we shall use the notation of Hsing (1999) in the rest of this remark. Also, the letter H below will stand for Hsing (1999). Note that N and K in H play the roles of our n and h, respectively.

7 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Eq. (38) of H claims the Lindeberg condition: for each 0, lim N 1 var(t N ) N 1 k=1 M N [ E 2 N;kI( N;k ] var(t N )) =0: (38) Here, M N is a sequence of positive integers satisfying M N N, for some 1. See page 1583 of H for the denition of T N. To prove the invalidity of (38), it suces to show lim N 1 var(t N ) N=2 k=1 From denitions around (6) and (38) of H, (k+m N ) N N;k = n=(k+1) 1 [ E 2 N;kI( N;k ] var(t N )) 0: (2.4) [ K (a n k k ) EK (a n k k ) ] ; K ()=EK( + X n;1; ); X n;1; = a i n i = X n : i 1 From the stationarity of {X n }, we thus obtain that K ()=EK( + X 0 ): Now, let K();, be bounded, strictly increasing and antisymmetric: K() = K( ); K(0)=0: As SS are symmetric, and a j = j 0, so the distribution of X 0 is also symmetric, which implies that K is bounded, strictly increasing and antisymmetric. Consequently, K ( )=EK( + X 0 )=EK( X 0 )= EK( + X 0 )= K (); : Thus, EK (a n k k )=0;K (0)=0; and K () 0( 0): Therefore, (k+m N ) N N;k = n=(k+1) 1 Hence, (2.4) follows from lim N K (a n k k ) 0 (if k 0); 0 (if k 0): 2 N=2 1 (k+m N ) N K (a n k ) var(t N ) k=1 0 n=(k+1) 1 (k+m N ) N I K (a n k ) var(t N ) F(d) 0; (2.5) n=(k+1) 1 where F is the d.f. of SS law. Observe the integral in (2.5) only decreases if we replace K () by a smaller function, say K () ci( 1); ( 0); (2.6)

8 316 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where c := K (1) 0: Clearly, we may take c = 1 in the sequel. Thus, (2.5) follows from 2 N=2 1 N lim I(a n k 1) N var(t N ) k=1 0 n=(k+1) 1 N I I(a n k 1) var(t N ) F(d) 0: (2.7) n=(k+1) 1 Here, we used the fact that for k 1 and M N N, (k + M N ) N = N. Now, since a j = j, we obtain for all 0, N N k ( 1 I(a n k 1) = I j 1 ) =(N k) 1= n=(k+1) 1 j=1 (N=2) 1= ; (16k6N=2): Hence, (2.7) follows from var(t N )=O(N 3 ) (see (14) of H) and N lim sup N N 3 (N 1= ) 2 I(N 1= N (3 )=2 )F(d) 0: (2.8) 0 Because 1 2, (2.8) in turn is implied by N lim N 2 2= d= +1 0: (2.9) N N (3 )=2 Now, the last integral equals ( (2=) ) 1 ((N ) (2=) (N (3 )=2 ) (2=) ) : But, N 2 (N ) (2=) = 1, for all N 1, while lim N N 2 (N (3 )=2 ) (2=) =0 because of 1. This proves (2.9), thereby, also proving the invalidity of Hsing s conclusion (38). Note that (2.6) holds for any strictly monotone bounded antisymmetric function K on. In particular K() [() 1=2]= [(1 + y) 1=2]F(dy), where denotes the d.f. of a standard Gaussian r.v., satises all of these conditions. emark 2.4. The only place where we need conditions (2.1) (2.2) on the second derivative of G is to prove Lemma 4.2 below, which gives a similar bound for the second derivative of F and its nite memory approimations. Note that the latter d.f. is innitely dierentiable provided G satises -Holder condition with arbitrary 0 (see Giraitis et al. (1996), which suggests that (2.1) and (2.2) probably can be relaed). Furthermore, it seems that our results can be generalized to the case of innovations belonging to the domain of attraction of non-symmetric -stable distributions, 1 2. However, the case 0 61 is ecluded by (1.6) and is quite open. 3. Limit behavior of M-estimators Consider the linear regression model: Y n;t = C n;t + t ; t =1;:::;n;

9 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where p is an unknown parameter vector, C n;t is the tth row of the known n p nonsingular design matri V n ; 16t6n, and the errors t ;t Z; follow an innite variance long-memory process: t = b t j j ; (3.1) j6t where b i ;i 0 and t ;t Z; satisfy the assumptions of Theorem 2.1. Let be a real valued nondecreasing right continuous function on such that E ( 0 ) for all. Put ()=E ( 0 ): Note that is nonincreasing, continuously dierentiable and we assume (0) = E ( 0 )=0: The corresponding M-estimator ˆ of the parameter is dened as { } ˆ = argmin C n;t (Y n;t C n;tb) : b p : In the particular case () =, the corresponding M-estimator is known as the least-squares estimator which we denote by ˆ ls : ˆ ls =(V n V n ) 1 C n;t Y t : We shall consider the particular case when the designs are of the form C n;t = C(t=n); (3.2) where C(t) =(v 1 (t);:::;v p (t));t [0; 1] is a given continuous p -valued function on [0; 1]. In such a case, n 1 V n V n V; where V = 1 0 C(t)C(t) dt is the p p-matri with entries V ij = 1 0 v i(t)v j (t)dt; i; j =1;:::;p: We shall assume, as usual, that the matri V is nondegenerate. Then n( ˆ ls )=V 1 C(i=n) i (1+o P (1)) and it follows from Kasahara and Maejima (1988) (under the standard assumptions on b j and j ) that na 1 n ( ˆ ls ) c 1 V 1 Z(C); where Z(C) isanss random vector, whose characteristic function is { } 1 E ep{iu Z(C)} = ep u C(t)(t s) + dt ds ; u p : 0 Let 1 := d()=d =0. Theorem 3.1. Assume; in addition to the above conditions; that is bounded and 1 0.Then na 1 n ( ˆ ) c 1 V 1 Z(C); na 1 n ( ˆ ˆ ls )=o P (1):

10 318 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) emark 3.1. Using the weak convergence methods and Theorem 2.1, one can also obtain analogous results for the classes of the so called -estimators of in the present case, as in Koul and Mukherjee (1993) for the nite variance case. 4. Proof of Theorem 2.1 Throughout the proofs below, C stands for a generic constant not depending on n, and for any real function g() and any y, let g(; y)=g(y) g(). Let the function H t (z) H t (z; ; y);z be dened, for any y; t Z, by H t (z)=i( + n;t z6y + n;t ) F( + n;t ;y+ n;t )+f( + n;t ;y+ n;t ) z; so that S n (; y)= n;t H t (X t ): (4.1) Our aim is to prove the following crucial Lemma 4.1. There eist 1 r ; 0; and a nite measure on the real line such that for any y; E S n (; y) r 6(; y)a r nn : Proof. We use the martingale decomposition as in Ho and Hsing (1996) and Koul and Surgailis (1997). Let U t;s (; y)=e[h t (X t ) F t s ] E[H t (X t ) F t s 1 ]; where F t = { s : s6t} is the -algebra of the past. Then, we can rewrite H t (X t )= s 0 U t;s (; y): (4.2) Observe that the series (4.2) converges in L r L r (), for each r. Namely, the series (E[I(X t 6 + n;t ) F( + n;t ) F t s ] s 0 E[I(X t 6 + n;t ) F( + n;t ) F t s 1 ]) converges in L 2 by orthogonality and hence in L r (r 2) as well, while t F t s ] E[X t F t s 1 ]) = s 0(E[X b s t s = X t s 0 converges in L r ( r ). For s 0, introduce the truncated moving averages: s X t;s = b i t i ; X t;s = b i t i ; i=0 i=s+1 X t =X t;s + X t;s. Let F s ()=P[X t;s 6]; F s ()=P[ X t;s 6] be the corresponding marginal d.f. s. Note that, for each 1= r, E X t;s r 6C b i r 6C i r 6C(1 + s) 1 r : (4.3) i=s+1 i=s+1

11 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) To proceed further, we need some estimates of the derivatives of F s ; F s and their dierence, similar to the estimates obtained in Giraitis et al. (1996), Koul and Surgailis (1997) for the case of a moving average with nite variance. Lemma 4.2. For any k 0 one can nd s 1 such that the d.f. F; F s ;s s 1 are k times continuously dierentiable. Furthermore; for any 1 r r ;r 1= and any suciently large s 1 there eists a constant C = C r;r such that for any ; y ; y 61; s s 1 ; F () + F s () 6C(1 + ) r ; (4.4) F (; y) + F s (; y) 6C y (1 + ) r ; (4.5) F () F s () 6C s (1=r ) (1 + ) r : (4.6) Proof. Assumption (2:1) implies that E ep{iu 0 } 6C=(1 + u ); for all u. This alone implies the dierentiability of F and F s as in Koul and Surgailis (1997, Lemma 4.1). We shall now prove (4.4). Assume b 0 = 1 without loss of generality. Then F()= G( u)d F 0 (u): According to condition (2.1), F () 6 G ( u) d F 1 (u)6c (1 + u ) d F 0 (u): As E X 0 r for any 1= r, the required bound (4.4) for F now follows from Lemma 5.1(i) below. The proof of the remaining bounds in (4.4) and (4.5) is eactly similar. To prove (4.6), write 3 F () F s () = (F s ( y) F s ()) d F s (y) 6 J i (); where J 1 ()= F s ( y) F s () d F s (y); y 61 J 3 ()= F s () d F s (y): y 1 By (4.3) and (4.5), J 1 () 6C(1 + ) r E X 0;s 6 C(1 + ) r E 1=r [ X 0;s r ] 6 C(1 + ) r s (1=r ) : Net, using (4.3) and Lemma 5.1 (i) below, we obtain J i () 6C(1 + ) r E X 0;s r 6 C(1 + ) r E r=r [ X 0;s r ] 6 C(1 + ) r s r((1=r ) ) ; i =2; 3. This proves the Lemma 4.2. J 2 ()= F s ( y) d F s (y); y 1

12 320 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Proof of Lemma 4.1 (continued). As in Ho and Hsing (1996), Giraitis and Surgailis (1999), write U t;s (; y)=u t;s (y) U t;s (), where U t;s ()=F s 1 ( + n;t b s t s X t;s ) Observe that n;t H t (X t )= where M j;n (; y)= F s 1 ( + n;t b s u X t;s )dg(u)+f( + n;t )b s t s : j= M j;n (; y); n;t U t;t j (; y) j is F j -measurable and E[M j;n (; y) F j 1 ] = 0, i.e. {M j;n (; y): j Z} are martingale dierences. We use the following well-known inequality (von Bahr and Esseen, 1965): for any martingale dierence sequence Y 1 ;Y 2 ;:::;E[Y i Y 1 ;:::;Y i 1 ]=0 i, and any 16r62 m r m E Y i 6 2 E Y i r ; m 1: Applying this inequality to the martingale dierences M j;n (; y); j6n one obtains, for any 1 r 2, r E M j;n (; y) 62 E M j;n (; y) r : (4.7) j= j= Write F s =f s. As in Ho and Hsing (1996) or Koul and Surgailis (1997), decompose U t;s ()= 3 i=0 U (i) t;s (); and, respectively, M j;n (; y)= 3 i=0 M (i) (0) j;n (; y), where U t;s (; y)= U t;s (; y) I(06s6s 1 ); =U (i) t;s (; y)=0 for i =1; 2; 3; 06s6s 1 and, for s s 1, U (1) t;s ()=F s 1 ( + n;t b s t s X t;s ) F s 1 (y + n;t b s u X t;s )dg(u) +f s 1 ( + n;t X t;s )b s = t s ; U (2) t;s ()=b s t s (f( + n;t ) f( + n;t = X t;s )); U (3) t;s ()=b s t s (f( + n;t X t;s )= f s 1 ( + n;t X t;s )): Lemma 4.3. For each 1 r ;r suciently close to ; and any 0 satisfying the inequality (1 + )r ; (4.8) one can nd a nite measure = r; on such that E M (0) E M (i) j;n (; y) r 6(; y) I( s 1 6j6n); j;n (; y) r 6(; y) [1 (t j) (1+) ] ; y; ; 2; 3: j

13 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) The proof of Lemma 4.3 will be given below. We now use this lemma, (4.2) and (4.7) to prove Lemma 4.1. By (4.7), it suces to show that one can nd r and 0 such that A r n j= By Lemma 4.3, where j= n = E M (i) j;n (; y) r 6Cn (; y): (4.9) { E M (i) n if i =0; j;n (; y) r 6C(; y) n if i =1; 2; 3; j= [1 t j (1+) ] j n ( n r 6 C (t s) dt) (1+) ds 0 s ( 1 r 1 6 Cn 1+r (1+)r (t s) dt) (1+) ds6cn 1+r (1+)r ; (4.10) 0 s provided (1 + ) 1 and (1 + )r 1 hold. Observe that the last conditions (which guarantee the convergence of the double integral in (4.10)), with the preservation of (4.8), can always be achieved by choosing r suciently close to, as this implies 0 suciently small by (4.8) and thus (1 + ) 1 because of 1, and also implies (1 + )r r 1 because of 1. Now for i =1; 2; 3 (4.9) follows from (4.10) and A r n = n r r+r= by choosing (1 + )r = and taking 0 suciently small; indeed, in such a case =(r r + r=) (1 + r (1 + )r) ( 1)( r) =( 1) (r=)( 1) = 0: Finally, for i = 0 (4.9) follows from n =O(A r nn ), or 16r(1 +1=), by taking 0 r and suciently small. Lemma 4.1 is proved. 5. Some proofs Proof of Lemma 4.3 (case i = 1). Similarly as Giraitis and Surgailis (1999), write U (1) t;s (; y)=w (1) t;s (; y) W (2) t;s (; y);

14 322 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where, for y, W (1) t;s (; y)= bs t s dg(u) dz b su bs W (2) t s t;s (; y)= dg(u) dz b su Net, introduce certain truncations: y y (0) t;s = I( X t;s 61; b s u 61; b s t s 61); (1) t;s = I( X t;s 1; b s u 61; b s t s 61); (2) t;s = I( b s u 1; b s t s 61); (3) t;s = I( b s t s 1): Accordingly, E M (1) j;n (; y) r = E Dene 6 C C j 3 E i=0 3 i=0 g (z)=(1+ z ) 1 ; r n;t U (1) t;t j (; y) n;t U (1) t;t j (; y)(i) j f s 1(w n;t + z X t;s )dw; f s 1(w n;t X t;s )dw: t;t j r D (i) j;n (; y) say: (5.1) (; y)= y g (z)dz; 0; (5.2) We shall use Lemma 4.2 and the nice majorizing properties of the measures to estimate the above epectations D (i) j;n (; y);i=0; 1; 2; 3: The following lemma will be often used in the sequel. Lemma 5.1. (i) For any 0 there eists a constant C ; depending only on ; such that g (z + y)6c g (z)(1 y ) 1+ ; y; z : (ii) Moreover; for any nonnegative function l of bounded variation on ; with lim u l(u)=0; g (z + u)l(u)du6c g (z) (1 u ) 1+ dl(u) ; z :

15 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Proof. (i) It suces to consider y 1. We have g (z + y) =g (z)+ y 0 y g (z + u)du ( ) 1+ 2y g 1+u (z + u) du 6 g (z)+ 0 6 g (z)+cy 1+ (1 + u) 1 g (z + u) du: The last integral in the above bound is equal to (1 + ) (1+u) 1 (1 + z + u ) 2 du6c g (z): Hence, g (z + y)6(1 + C y 1+ )g (z)6c (1 y) 1+ g (z): 1 (ii) Assume again without loss of generality l(u) =0;u 1. Then [ u ] g (z u)l(u)du = l(u)d g (z w)dw [ u ] 6 g (z w)dw dl(u) 6 (z) 1 1 u 1+ dl(u) ; where u ] (z) sup [u 1 g (z w)dw 6C g (z): (5.3) u 1 1 Indeed, write (z)6sup 16u6 z =2 [ :::] + sup u z =2 [ :::] (z)+ + (z); then + (z) clearly satises (5.3) as g (w)dw C. On the other hand, for 16u6z=2, u 1 z 1 g (w z)dw 6 z u = 1 (z u) ( d = 1 1+ ( 1 1 (z u) 1 ( z u z 1 (z 1) ) ) u 6C z 1+ ; implying (z)6c ( sup16u6z=2 u ) z 1 6C g (z). The case z 0; 16u6 z=2 is similar. This proves Lemma 5.1. Proof of Lemma 4.3 (case i = 1)(continued). Consider the terms on the right-hand side of (5.1). Consider D (0) n;j (; y). From Lemma 4.2 and condition (a.2) of Theorem 2.1 we obtain f s 1(w n;t + z X t;s ) f s 1(w n;t X t;s ) (0) t;s 6C min(1; z )g (w n;t ) (0) t;s 6C z g (w) (0) t;s ; )

16 324 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where 0 1 will be specied later, and we have used the inequality min(1; )6 which is valid for any. Then bs U (1) t;s (; y) (0) t;s 6 C ( + n;t ;y+ n;t ) (0) t s t;s dg(u) z dz b su 6 C (; y) (0) t;s b s 1+ ( t s 1+ + u 1+ )dg(u) 6 C (; y) b s 1+ (1 + t s 1+ ); assuming 1 +. Consequently, j;n (; y) 6 C( (; y)) r E[(1 + j 1+ ) r ] D (0) 6 C (; y) b t j 1+ j b t j 1+ j (5.4) provided (1 + )r, or (4.8) holds. Net, consider the term 2 n;j (; y)6 E D (1) j n;t W (i) t;t j (; y)(i) t;j r 2 D (1;i) n;j (; y): By Lemma 4.2(4.4), f s 1 () 6Cg ();, with C independent of s s 1, where 0 will be chosen below. Furthermore, z 6ma( b s t s ; b s u )61 on the set { (1) t;s = 1}. According to Lemma 5.1(i), implying W (i) t;s (; y) (1) t;s 6 C b s I( X t;s 1) dg(u) u t s D (1;i) j;n (; y) 6 C( (; y)) r E 6 C b s (1 + t s ) I( X t;s 1) y y g (w X t;s )dw g (w X t;s )dw 6 C (; y) b s (1 + t s ) X t;s 1+ I( X t;s 1); (5.5) 6 C (; y) j [( 1 b t j X t;t j 1+ I( X t;t j 1) ) r ] b t j E 1=r [ X t;t j r(1+) I( X t;t j 1)] ;

17 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) where in the last part we used niteness of and the norm (Minkowski) inequality in L r. Therefore, by (4.3), D (1;i) j;n (; y) 6 C (; y) b t j (1 t j ) (1=r) (1+) 6 C (; y) j (1 t j ) (1+) ; ; 2; (5.6) j provided r 1; 0 satisfy (1 + )r ; i.e. inequality (4.8). The last inequality in (5.6) follows from r 1, which follows from (1.6) provided r is chosen suciently close to. We shall now discuss the most delicate case r D (3) j;n (; y) =E n;t U (1) t;t j (; y)(3) t;j 6 j 2 E 2 W (i) j D (3;i) j;n (; y) say: t;t j (; y) I( b t j j 1) Now, recall the denition of W (i) t;t j (; y) and using Lemma 4.2, to obtain where 2 W (i) t;t j (; y) 6C k=1 W (1;1) t;t j(; y)= dg(u) dz W (1;2) t;t j(; y)= dg(u) dz W (2;1) t;t j(; y)= W (2;2) t;t j(; y)= W (i;k) t;t j(; y); ; 2; dg(u) dg(u) y y dz dz r g (w + z X t;t j )I( z b t j j ; j u )dw; y g (w + z X t;t j )I( z b t j u ; j u )dw; y Then apply Lemma 5.1(i) and (ii), to obtain W (1;1) t;t j(; y) I( b t j j 1) j 6 dz y g (w X t;t j )I( z b t j j ; j u )dw; g (w X t;t j )I( z b t j u ; j u )dw: g (w + z X t;t j )I( z b t j j ; b t j j 1) dw j

18 326 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) y 6C dz g (w + z)(1 X t;t j ) 1+ I( z b t j j ; b t j j 1) dw j 6C (; y) j (1 X t;t j ) 1+ b t j j 1+ I( b t j j 1); where in the last inequality we used Lemma 5.1(ii) with l(z)=i( z b t j j ) implying y g (w + z)i( z b t j j )dzdw6c (; y)(1 b t j j ) 1+. This results in D (3;1;1) j;n (; y) E j t;t j(; y) I( b t j j 1) W (1;1) 6C( (; y)) r E 6C (; y) 6C (; y) (1 X t;t j ) 1+ b t j j 1+ j r b t j 1+ E 1=r [(1 X t;t j ) r(1+) ]E 1=r [ j r(1+) ] j b t j 1+ ; j provided r 1; 0 satisfy (1 + )r, or (4.8). In a similar way, consider j W (1;2) t;t j(; y) I( b t j j 1) 6 dg(u) dz 6C y dg(u) dz g (w + z X t;t j ) I( z b t j u ; b t j u 1) dw g (w + z)(1 X t;t j ) 1+ j y j I( z b t j u ; b t j u 1) dw 6C (; y) dg(u) j (1 X t;t j ) 1+ b t j u 1+ ;

19 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) implying again, under condition (4.8), (; y) E W (1;2) t;t j(; y)i( b t j j 1) D (3;1;2) j;n j 6 C (; y) b t j 1+ : j As D (3;1) j;n (; y)6c 2 D(3;1;i) j;n (; y), one obtains the same bounds for D (3;1) j;n (; y), and, eactly in the same way, for D (3;2) j;n (; y). This yields, nally, D (3) j;n (; y)6c (; y) b t j 1+ ; (5.7) j provided r ; 0 satisfy (4.8) and r is suciently close to. Thus, we have shown in (5.4),(5.6) and (5.7) the same bound for D (i) j;n (; y); =0; 1; 3, respectively, and the case i = 2 is completely analogous to that of the estimation of D (3;1;2) j;n (; y). Then we have the required bound for i = 1, which completes the proof of Lemma 4.3 in the case i =1. Proof of Lemma 4.3 (cases i =2; 3 and i =0). Consider the case i =2, or M (2) j;n (; y). We have M (2) j;n (; y)= j n;t b t j (f( + n;t ;y+ n;t ) j f( + n;t X t;t j ;y+ n;t X t;t j )) so that, by the independence of j and X t;t j, for any 1 r we can write E M (2) j;n (; y) r = E 0 r E n;t b t j (f( + n;t ;y+ n;t ) j r f( + n;t X t;t j ;y+ n;t X t;t j )) where H (1) j;n 6 C (; y)=e 2 ; H (i) j;n (; y) say; (5.8) b t j f( + n;t ;y+ n;t ) t=j 1 f( + n;t X t;t j ;y+ n;t X t;t j ) I( X t;t j 61) ;

20 328 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) H (2) j;n (; y)=e b t j f( + n;t ;y+ n;t ) t=j 1 f( + n;t X t;t j ;y+ n;t X t;t j ) I( X t;t j 1) : Now by Lemma 4:2(4:4), f( + n;t ;y+ n;t ) f( + n;t X t;t j ;y+ n;t X t;t j ) I( X t;t j 61) 6I( X t;t j 61) y+n; t + n; t (f (z) f (z X t;t j )) dz 6C X t;t j 0 ( + n;t ;y+ n;t )6C X t;t j 0 (; y); where 0 = 1 2, say. Therefore, j;n (; y) 6 Cr 0 (; y)e H (1) 6 C 0 (; y) 6 C 0 (; y) b t j X t;t j j b t j E 1=r [( X t;t j r ] j (1 t j ) (1 t j ) (1=r) : (5.9) j Consider H (2) n;j (; y). From Lemma 4.2(4:4) and (a:2), f( + n;t X t;t j ;y+ n;t X t;t j ) 6C y g (z X t;t j )dz; implying f( + n;t ;y+ n;t ) 6C H (2) j;n (; y) 6 CE j 6 (; y) y g (z)dz; b t j I( X t;t j 1) y r (1 t j ) (1+) j (g (z)+g (z X t;t j )) dz (5.10) eactly as in (5.5) and (5.6). Lemma 4.3 in the case i = 2 now follows from (5.9) and (5.10).

21 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Consider the case i = 3, or M (3) j;n = j n;t b t j (f( + n;t X t;t j ;y+ n;t X t;t j ) j f s 1 ( + n;t X t;t j ;y+ n;t X t;t j )) : Similarly as in (5.8), we have E[ M (3) j;n r ]6C 2 K (1) j;n (; y)=e j b t j I( X t;t j 61) K (i) j;n y f s 1(z + n;t X t;t j ) dz ; (; y), where f (z + n;t X t;t j ) K (2) j;n (; y)=e j b t j I( X t;t j 1) y + f s 1(z + n;t X t;t j ) )dz : ( f (z + n;t X t;t j ) By Lemma 4.2(4.6), (; y) 6 0 (; y) K (1) j;n 6 C 0 (; y) b t j (1 t j ) (1=r) j (1 t j ) (1+) ; j where 0 0 r 1 and =1 1=(r) 0 provided r is suciently close to. On the other hand, by Lemma 4.2(4:4), r y (; y)6ce b t j I( X t;t j 1) g (z X t;t j )dz K (2) j;n j and the right-hand side can be estimated eactly as in (5.10). This proves Lemma 4.3 (case i = 3). It remains to prove the case i=0. As the sum M (0) j;n ()= n (j+s 1) j U t;t j (; y) consists of a nite number (6s 1 ) of terms, for each j6n, and vanishes for j6 s 1, it suces to show for any t Z and any 06s6s 1 the inequality: E U t;s (; y) r 6C(; y): (5.11)

22 330 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) But implying U t;s (; y) 6 P[ + n;t X t 6y + n;t F t s ] +P[ + n;t X t 6y + n;t F t s 1 ]+ t f( + n;t ;y+ n;t ); E U t;s (; y) r 6C(F( + n;t ;y+ n;t )+f( + n;t ;y+ n;t )) by the boundedness of f and the fact that 16r. Now, Lemma 4.2 completes the proof of (5.11) and the proof of Lemma 4.3 itself. Proof of Theorem 2.1. We follow the proof in Giraitis et al. (1996) (GKS). Observe, the majorizing measure in Lemma 4.1 above may be taken to be = C (F + ); 0, where is given by (5.2) and C depends only on. Furthermore, according to Lemma 4.2, this constant may be chosen so that, for any 1, and any y, we have the relations F( + ; y + )6(; y); F ( + ; y + )6(; y); where F (; y)= y f (z) dz. For any integer k 1, dene the partition such that =: 0;k 1;k 2 k 1;k 2 k ;k := + ; ( j;k ; j+1;k )=()2 k ; j =0; 1; = :::;2 k 1: Given 0, let K K n := [log 2 (n 1+ =A n )]+1: For any and any k =0; 1;:::;K, dene j k by j k ;k 6 j k +1;k : Dene a chain linking to each point by = j 0 ; 0 6 j 1 ;1 6 6 j K ;K6 j K +1;K: Then S () n ()=S n () ( j 0 ;0 ; j 1 ;1 )+S n () +S n () ( j K ;K;): Similarly as in Koul and Mukherjee (1993), where S () n ( j 1 ;1 ; j 2 ;2 )+ + S n () ( j ( j K ;K;) 6 S () ( j K ;K; j K +1;K) +2B n (); B n () = n n;i (F( j K ;K; j K +1;K)+ X t F ( j K ;K; j K +1;K) 6 C( j K ;K; j K +1;K) K 1 ;K 1 ; j K ;K) (1 + X t )6 C()2 K (1 + X t );

23 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) so that [ ] E sup B n () 6C2 K ne[(1 + X 0 )]6CA n n : Net, similarly as in the above mentioned papers, for any k =0; 1;:::;K 1; and any 0 we obtain [ ] P sup S n () ( j k ; j k+1 ;k+1 ) A n =(k +3) 2 and 2 k P[ S () n ( i;k+1 ; i+1;k+1 ) A n =(k +3) 2 ] 2 k+1 1 6(k +3) r r A r n E[ S n () ( i;k+1 ; i+1;k+1 ) r ] 2 k+1 1 6(k +3) r r n ( i;k+1 ; i+1;k+1 ) 6C(k +3) r r n [ ] P sup S n () ( j K ;K; j K +1;K) A n =(K +3) 2 2 K 1 6 P[ S () n ( i;k ; i+1;k ) A n =(K +3) 2 ] 6C(K +3) r r n : Consequently, for any 0 61, [ ] P sup S n () () A n 6C r n K k=0 6 r n (K +3) r+1 ; [ ] (k +3) r + P sup B n () A n which completes the proof of Theorem 2.1, as (K +3) r+1 = O(log r+1 n). Proof of Theorem 3.1. Drop the subscript n from the notation where no ambiguity arises. We shall rst show that ˆ =O P (A n =n): (5.12) Put n = A n =n. elation (5.12) follows from C t ( t )=O P (1); (5.13) A 1 n and the fact that, for any 0 and any L 0 one can nd K 0 and N such that for all n N [ ] P inf s K A 1 n C t ( t n s C t ) L 1 : (5.14)

24 332 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Here, (5.13) follows from Theorem 2.1; see also (5.16), case i = 3 below, while the relation (5.14) can be proved, using Theorem 2.1, the monotonicity of, and an argument like the one used in Koul (1992b, Lemma 5:5:4). Net, we shall check that for each 0 there eists a K and N such that for all n N, ( ) P inf s 6K A 1 n C t ( t n s C t ) 6 1 : (5.15) To prove this, let 1 denote the rst derivative of () at = 0, and write 4 C t ( t n s C t )= i (s); where 1 (s)= 2 (s)= 3 (s)= C t ( ( t n s C t ) ( n s C t ) ( t )+(0)); C t (( n s C t ) (0) 1 n s C t )); C t ( ( t )+ 1 t ); 4 (s)= 1 C t ( n s C t t ): It suces to show that for each K sup A 1 n i (s) =o P (1); ; 2; 3 (5.16) s 6K and that for any 0 one can nd K; N such that for all n N, P[ inf s 6K A 1 n 4 (s) 6] 1 : But the last relation obviously follows from 4 ( 1 n ( ˆ ls )) = 0 and the fact that ˆ ls =O P ( n ); see Section 3. elation (5.16) for i = 2 follows from A 1 n 2 n = o(1) and the fact that () is twice continuously dierentiable at = 0. To show (5.16) for i = 3, write { } 3 (s)= C t [I( t 6) F()+f() t ] d () and use Theorem 2.1 and the fact that has bounded variation. In a similar way, write 1 (s) { } = C t [I( t 6 + n s C t ) F( + n s C t ) I( t 6)+F()] d () 6C sup C t [I( t 6 + n s C t ) F( + n s C t ) I( t 6)+F()] : Finally, relation (5.16) for i = 1 follows from Theorem 2.1 by employing the argument in Koul (1992b, Theorem 2:3:1).

25 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) According to (5.12), for any 0 there are K; N such that P[ ˆ 6 n K] 1 n N. According to (5.16), on the set ˆ 6 n K, with high probability 1 2 one has the following inequalities A n C t ( t ( ˆ ) C t ) or, equivalently, 1 C t ( ˆ ) C t 1 n 1 V nv n 1 n ( ˆ ) A 1 n C t t 3 C t t 6 + sup s 6K 3 i (s) A 1 n sup s 6K i (s) : Whence and from (5.16), the assumptions about the non-degeneracy of 1 and the matri V=lim n n 1 V nv n, both statements of Theorem 3.1 follow, thereby completing its proof. 6. Asymptotics of the bivariate characteristic function Theorem 6.1. Assume the moving average X t (1:2) satises the standard assumptions of Section 1. Then for any u 1 ;u 2 there eists the limit lim t t 1 cov(e iu1x0 ; e iu2xt )k u1;u 2 E[e iu1x0 ]E[e iu2x0 ]; (6.1) where k u1;u 2 = c 2 c 0 [ u 1 ( s) + +u 2 (1 s) + u 1 ( s) + u 2 (1 s) + ]ds; with c 2 =2 (2 )cos(=2)=(1 ). Proof. Write (u)=e[e iu0 ];(u)=e[e iux0 ]; t (u 1 ;u 2 )=E[e i(u1x0+u2xt) ]. Then, t (u 1 ;u 2 )= (u 1 b j + u 2 b t j ); (u 1 )(u 2 )= (u 1 b j ) (u 2 b t j ); j Z j Z where we put b j =0;j 0. Similarly, as Giraitis et al. (1996, Proof of Lemma 2), write t (u 1 ;u 2 )= =: a 1 a 2 a 3 ; I 1 I 2 I 3 (u 1 )(u 2 )= = =: a 1 a 2 a 3; I 1 I 2 I 3 where I 1 = {j Z: j 6t d }; I 2 = {j Z: t j 6t d }; I 3 = Z \ (I 1 I 2 ) and where d 0 will be chosen below. Then t (u 1 ;u 2 ) (u 1 )(u 2 )=a 1 a 2 a 3 a 1a 2a 3 =(a 1 a 1)a 2 a 3 + a 1(a 2 a 2)a 3 + a 1a 2(a 3 a 3):

26 334 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) Hence the theorem will follow from a i 61; a i 61; ; 2; 3 and that, for any u 1 ;u 2 ed, a i a i =o(t 1 ); ; 2; (6.2) a 3 a 3 = a 3k u1;u 2 t 1 +o(t 1 ): (6.3) Consider (6.2). Using the inequality c i c i 6 c i c i ; for c i 61; c i 61, one obtains a 1 a 1 6 j 6t d (u 1 b j + u 2 b t j ) (u 1 b j ) (u 2 b t j ) : Note u 2 b t j 6Ct for j 6t d and d 1. Therefore, as (y 1 + y 2 ) (y 1 ) (y 2 ) 6C y 2 with C62E[ 0 ], we obtain a 1 a 1 =O =O(t d ); j 6t d t proving (6.2) for i = 1, provided d 0 is chosen so that d 1 which is of course possible. The case i = 2 being analogous, it remains to prove (6.3). It is well-known (Ibragimov and Linnik, 1973, Theorem 2:6:5) that assumption (1.5) (where we put c 1 = 1) is equivalent to the representation log (u)= c 2 u (1 + (u)); in a neighborhood of the origin u = 0, where (u) 0(u 0). Note j I 3 implies u 1 b j + u 2 b t j = o(1). Therefore, for suciently large t, a 3 a 3 = a 3(e c2 Qt(u1;u2) 1); where Q t (u 1 ;u 2 )=Qt 0 (u 1 ;u 2 )+q t (u 1 ;u 2 ) and Qt 0 (u 1 ;u 2 )= [ u 1 b j + u 2 b t j u 1 b j u 2 b t j ]; j t d ; t j t d q t (u 1 ;u 2 )= Therefore, (6.3) follows from [ u 1 b j + u 2 b t j (u 1 b j + u 2 b t j ) j t d ; t j t d u 1 b j (u 1 b j ) u 2 b t j (u 2 b t j )]: lim t t 1 Qt 0 (u 1 ;u 2 )=k u1;u 2 ; (6.4) lim q t t1 t (u 1 ;u 2 )=0: (6.5) To show (6.4), write t 1 Qt 0 (u 1 ;u 2 )= m t;u1;u 2 (s)ds + t (u 1 ;u 2 ); where t (u 1 ;u 2 )=t 1 j t d or t j t d ; [ u 1 b j + u 2 b t j u 1 b j u 2 b t j ] = o(1)

27 H.L. Koul, D. Surgailis / Stochastic Processes and their Applications 91 (2001) by a similar argument as in the proof of (6.2), and where m t;u1;u 2 is a piecewise-constant function on the real line given by m t;u1;u 2 (s)= u 1 t b j + u 2 t b t j u 1 t b j u 2 t b t j if s [j=t; (j +1)=t);j Z. It is easy to note by (1.3) that for each s (and any u 1 ;u 2 ) ed, lim m t;u t 1;u 2 (s)= c 0 [ u 1 ( s) + + u 2 (1 s) + u 1 ( s) + u 2 (1 s) + ] m u1;u 2 (s); where the limit function is integrable on the real line. Furthermore, one can check that the sequence {m t;u1;u 2 } t=0;1;::: is dominated by a integrable function. By the Lebesgue dominated convergence theorem, lim t m t;u 1;u 2 (s)ds = m u 1;u 2 (s)ds = k u1;u 2.Ina similar way, one can verify convergence (6.5). Theorem 6.1 is proved. eferences Astrauskas, A., Limit theorems for sums of linearly generated random variables. Lithuanian Math. J. 23, Astrauskas, A., Levy, J.B., Taqqu, M.S., The asymptotic dependence structure of the linear fractional Levy motion. Lithuanian Math. J. 31, Avram, F., Taqqu, M.S., Weak convergence of moving averages with innite variance. In: Eberlein, E., Taqqu, M.S. (Eds.), Dependence in Probability and Statistics. Birkhauser, Boston, pp Avram, F., Taqqu, M.S., Weak convergence of sums of moving averages in the -stable domain of attraction. Ann. Probab. 20, von Bahr, B., Esseen, C.-G., Inequalities for the rth absolute moment of the sum of random variables, 16r62. Ann. Math. Statist. 36, Baillie,.T., Long memory processes and fractional integration in econometrics. J. Econometrics 73, Beran, J., M-estimators of location for data with slowly decaying correlations. J. Amer. Statist. Assoc. 86, Beran, J., Statistical methods for data with long range dependence. Statist. Sci. 7, Billingsley, P., Convergence of Probability Measures. Wiley, New York. Christoph, G., Wolf, W., Convergence Theorems with Stable Limit Law. Akademie Verlag, Berlin. Csorgo, S., Mielniczuk, J., Density estimation under long-range dependence. Ann. Statist. 23, Dehling, H., Taqqu, M.S., The empirical process of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 17, Dobrushin,.L., Major, P., Non-central limit theorems for non-linear functions of Gaussian elds. Z. Wahrsch. Verw. Geb. 50, Doukhan, P., Massart, P., io, E., The invariance principle for the empirical measure of a weakly dependent process. Ann. Inst. H. Poincare B 31, Giraitis, L., Koul, H.L., Surgailis, D., Asymptotic normality of regression estimators with long memory errors. Statist. Probab. Lett. 29, Giraitis, L., Surgailis, D., Limit theorems for polynomials of linear processes with long range dependence. Lithuanian Math. J. 29, Giraitis, L., Surgailis, D., Central limit theorem for the empirical process of a linear sequence with long memory. J. Statist. Plann. Inference 80, Granger, C.W.J., Joyeu,., An introduction to long-memory time series and fractional dierencing. J. Time Ser. Anal. 1, Hsing, T., On the asymptotic distributions of partial sums of functionals of innite-variance moving averages. Ann. Probab. 27, Ho, H.C., Hsing, T., On the asymptotic epansion of the empirical process of long memory moving averages. Ann. Statist. 24, Ho, H.C., Hsing, T., Limit theorems for functionals of moving averages. Ann. Probab. 25,

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