Liudas Giraitis, Masanobu Taniguchi, Murad S. Taqqu Asymptotic normality of quadratic forms of martingale differences

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1 Liudas Giraitis, Masaobu Taiguchi, Murad S. Taqqu Asymptotic ormality of quadratic forms of martigale differeces Article Published versio Refereed Origial citatio: Giraitis, Liudas, Taiguchi, Masaobu ad Taqqu, Murad S. 016 Asymptotic ormality of quadratic forms of martigale differeces. Statistical Iferece for Stochastic Processes. ISSN DOI: /s Reuse of this item is permitted through licesig uder the Creative Commos: 016 The Authors CC BY 4.0 This versio available at: Available i LSE Research Olie: August 016 LSE has developed LSE Research Olie so that users may access research output of the School. Copyright ad Moral Rights for the papers o this site are retaied by the idividual authors ad/or other copyright owers. You may freely distribute the URL of the LSE Research Olie website.

2 DOI /s Asymptotic ormality of quadratic forms of martigale differeces Liudas Giraitis 1 Masaobu Taiguchi Murad S. Taqqu 3 Received: 31 May 016 / Accepted: 3 Jue 016 The Authors 016. This article is published with ope access at Sprigerlik.com Abstract We establish the asymptotic ormality of a quadratic form Q i martigale differece radom variables η t whe the weight matrix A of the quadratic form has a asymptotically vaishig diagoal. Such a result has umerous potetial applicatios i time series aalysis. While for i.i.d. radom variables η t, asymptotic ormality holds uder coditio A sp = o A,where A sp ad A are the spectral ad Euclidea orms of the matrix A, respectively, fidig correspodig sufficiet coditios i the case of martigale differeces η t has bee a importat ope problem. We provide such sufficiet coditios i this paper. Keywords Asymptotic ormality Quadratic form Martigale differeces Mathematics Subject classificatio 6E0 60F05 B Liudas Giraitis L.Giraitis@qmul.ac.uk Masaobu Taiguchi taiguchi@waseda.jp Murad S. Taqqu murad@bu.edu 1 School of Ecoomics ad Fiace, Quee Mary Uiversity of Lodo, Mile Ed Road, Lodo W 3PR, UK Research Istitute for Sciece ad Egieerig, Waseda Uiversity, Okubo, Shijuku-ku, Tokyo , Japa 3 Departmet of Mathematics ad Statistics, Bosto Uiversity, 111 Cummigto Mall, Bosto, MA 015, USA

3 1 Mai results We study here quadratic forms Q = a ;tk η t η k 1.1 t,k=1 where {η k } is a statioary ergodic martigale differece m.d. sequece with respect to some atural filtratio F t, with momets Eη k = 0, Eη k = 1 ad Eη4 k <. The real-valued coefficiets a ;tk i 1.1 are etries of a symmetric matrix A = a ;tk t,k=1,...,.wedeoteby 1/ A = the Euclidea orm ad by a;tk t,k=1 A sp = max A x x =1 the spectral orm of the matrix A. For coveiece, we set a ;tk = 0fort 0, t > or k 0, k >. The asymptotic ormality property of the quadratic form Q has bee well ivestigated whe the radom variables η j are i.i.d. If A has vaishig diagoal: a ;tt = 0forallt,the asymptotic ormality is implied by the coditio A sp = o A, 1. see Rotar 1973, De Jog 1987, Guttorp ad Lockhart 1988, Mikosch 1991adBhasali et al. 007a. The aim of this paper is to exted these results to the m.d. oise η j. We will eed the followig additioal assumptios o the m.d. oise η t : E η j F j 1 c > 0, c > The assumptio 1.3 bouds the coditioal variace of η j away from zero. We also assume that A has a asymptotically vaishig diagoal i the sese: a ;tt =o A,. 1.4 Relatio 1.4 implies a;tt = o A,. 1.5 The followig theorem shows that i case of m.d. oise {η k }, the coditio A sp / A 0 above eeds to be stregtheed by icludig the assumptios 1.8 ad1.9 o the weights a ;ts. Its proof is based o the martigale cetral limit theorem.

4 Theorem 1.1 Let Q be as i 1.1, where {η j } is a statioary ergodic m.d. oise such that Eη 4 j < ad 1.3 hold. Suppose that the a ;ts s are such that, as, The there exist c 1, c > 0 such that If i additio, ad A sp / A c 1 A VarQ c A, : t s L a ;ts = o A,, L, 1.8 a ;t,t k a ;t 1,t 1 k = o A, t=k+ k the the followig ormal covergece holds: VarQ 1/ Q EQ d N0, As usual, d N0, 1 deotes covergece i distributio to a ormal radom variable with mea zero ad variace oe. Theorem 1.1 plays a importat istrumetal role i establishig asymptotic properties of various estimatio ad testig procedures i parametric ad o-parametric time series aalysis where the object of iterest ca be writte as a quadratic form Q,X = of a liear movig-average process e t sx t X s X t = a j η t j j=0 of ucorrelated oise η t ad the weights e s may deped o. I the case of i.i.d. oise η t, the asymptotic ormality for Q,X is established by approximatig it by a simpler quadratic form Q,η = b t sη t η s with some differet weights b t ad the derivig the asymptotic ormality for Q,η,asi Bhasali et al. 007b. For example, oe sets b t = π π u x f xe itx dx where f x is the spectral desity of the sequece X t,adwhereu x is some coveiet fuctio related to e t, typically such that e t = π π u xe itx dx.

5 I geeral, obtaiig simple asymptotic ormality coditios for Q,X is a hard theoretical problem but of great practical importace, which for a i.i.d. oise η t was solved i Bhasali et al. 007b. I additio, i Sect. 6. i Giraitis et al. 01 oe cosiders discreet frequecies ad shows that a sum / S = b j I u j of weighted periodograms j=1 I u j = π 1 e iku j X k of the sequece X t at Fourier frequecies u j ca be also effectively approximated by a quadratic form Q,η. This allows, by theorem like Theorem 1.1, to establish the asymptotic ormality for such sums S. However, assumptio of i.i.d. oise is restrictive ad may be ot satisfied i practical applicatios ad i some theoretical, i.e. ARCH, settigs. I a subsequet paper we will derive correspodig ormal approximatio results for Q,X ad S whe η t is a martigale differece process. The followig Corollary 1.1 displays situatios where the coditios of Theorem 1.1 are easily satisfied. For a Toeplitz matrix A, that is with etries k=1 a ;ts = b t s, the assumptio 1.9 is clearly satisfied, sice a ;t,t k a ;t 1,t 1 k = b k b k = 0. The followig lemma provides a useful boud that ca be used to prove 1.6. Lemma 1.1 Suppose that b t = π π e itx g xdx, t = 0, 1,..., where g x, x π is a eve real fuctio. If there exists 0 α<1/ ad a sequece of costats k > 0 such that the g x k x α, x π, A sp Ck α, For the proof see Theorem.i i Bhasali et al. 007a. Suppose ow, i additio, that g x gx, 1ad gx C x α, x π.the π g xdx <, b t = bt ad b t < π t= ad, i additio, k = 1i1.11. Hece A = b t s = k= b k k t= b t as

6 ad which implies 1.6. Moreover, : t s L A sp C α = o 1/ = o A a ;ts = : t s L b t s b k. k L Sice k L b k 0asL,weobtai1.8. This together with Theorem 1.1 implies the followig corollary. Corollary 1.1 Let Q = bt kη t η k, t,k=1 where bt = b t, b0 = 0 are real weights ad {η j } is a statioary ergodic m.d. oise such that Eη 4 j < ad 1.3 hold. i If t=0 bt <,the c 1, c > 0 : c 1 VarQ c, 1, 1.1 VarQ 1/ Q EQ d N0, ii If bt = π π eitx gxdx, t = 0, 1,..., where gx, x π is a eve real fuctio such that for some 0 α<1/ad C > 0, the 1.1 ad 1.13 hold. Next we cosider two quadratic forms gx C x α, x π 1.14 Q 1 = Q = a 1 ;ts η tη s, ad a ;ts η tη s, 1.15 with correspodig matrices A 1, A ad a m.d. sequece η t which satisfy the assumptios of Theorem 1.1,sothat 1/ VarQ i Q i d EQi N0, 1, i = 1,. The ext corollary provides additioal sufficiet coditio that implies asymptotic ormality of their sum. Corollary 1. Suppose that the quadratic forms Q 1 of Theorem 1.1.Set,Q i 1.15 satisfy the assumptios A = A 1 + A.

7 If i additio lim A 1 1 A 1 tr A 1 A = the the quadratic form Q := Q 1 + Q satisfies A 1 c 1, c > 0 : c 1 + A A 1 VarQ c + ad VarQ 1/ Q EQ d N0, 1. A, 1, Proof We have Q = a ;ts η t η s where a ;ts = a 1 ;ts + a ;ts. Thus, to prove the corollary, it suffices to show that A satisfies assumptios of Theorem 1.1. This easily follows from the fact that both A 1 ad A satisfy assumptios of Theorem 1.1, ad the property A A 1 = + A 1 + o1. The latter follows from A = A 1 + A + tr A 1 A because the matrices A 1 ad A are symmetric so the cross term t,s a 1 ;ts a ;ts = t,s a 1 ;ts a ;st = tr A 1 A. Hece where A A 1 = + r = tr A 1 A A / A r A. Sice A 1 + A A 1 A we get r = o1 by Corollary 1. idicates that we eed the additioal coditio 1.16 i order to obtai the asymptotic ormality of Q. It does ot imply that i this case the compoets Q 1 ad Q are asymptotically ucorrelated ad hece asymptotically idepedet. We cojecture that Q 1 ad Q will be asymptotically idepedet i the case whe η t is a i.i.d. oise. Proof of Theorem 1.1 I the proof of Theorem 1.1 we shall use the followig result. Lemma.1 Dalla et al. 014, Lemma 10.

8 i Let T = c j V j, j Z where {V j }, j Z ={, 1, 0, 1, }is a statioary ergodic sequece, E V 1 <, ad c j are real umbers such that for some 0 <α <, 1, c j =Oα, c j c, j 1 =oα..1 j Z j Z The E T ET =oα. I particular, if α = 1, the T = ET + o p 1. ii If the m.d. sequece η t satisfies max t E η t p <, forsomep, the E d p p/, j η j C j Z j Z d j. for ay d j s such that j Z d j <,wherec< does ot deped o d j s. For the coveiece of the reader we provide the proof of the followig lemma. Lemma. Oe has max,..., a;ts A sp, max a ;ts A sp..3,..., Proof We drop the idex ad let A = a ts. The secod iequality a ts A sp follows from the first sice max a t,s ts max a t ts A sp. Turig to the first iequality, we have A sp = sup x =1 Ax where x = x 1,...,x ad Ax = a 1s x s,..., a s x s = a 1s x s + + a s x s. Set y = 0,...,0, 1, 0,...,0 where 1 is at the t 0 positio. Note that y = 1. The sice A is symmetric. Hece A sp Ay = a 1t 0 + +a t 0 = A sp max t 0 =1,..., ast 0 = at 0 s. a t 0 s

9 Proof of Theorem 1.1 Usig 1.6, the secod claim of.3 implies max ;ku =o A, 1 k,u L L 1fixed..4 Relatio.4 implies that o sigle a ;ku domiates. Proof of 1.7 Below we write a ts istead of a ;ts.let The t 1 z t = η t ad z t = a tt Q EQ = z t + t= η t Eηt..5 z t = S + S..6 Observe that Eη t η s = 0fort > s ad hece ES = 0siceη s is a m.d. sequece. I additio, [ ES = 4 t 1 E ηt ]..7 Usig Eη 4 t C ad 1.4, E S C Nowweshowthat t= a tt =o A, ES C a tt = o A..8 c 1 A ES c A. The lower boud follows by usig 1.3ad1.5 because of the fact that c > 0: ES = 4 4c = c t= [ E ηt E t= ] = 4 t 1 t 1 = 4c t= t 1 t= [ E E[ηt F t 1] a ts t 1 ] ats c att = A o A A,.9 for large. To prove the upper boud, otice that ES = 4 4 t= t= [ E ηt t 1 ] Eη 4 t 1/ E t 1 4 1/ ] C ats = C A.10

10 by. ad assumptio Eηt 4 = Eη1 4 <. To obtai 1.7, ote that VarQ ES + ES C A + o A C A by.8ad.10. I additio,.6.10imply Ideed, by.6, VarQ = ES 1 + o1,..11 VarQ VarS = VarS + CovS, S VarS + VarS VarS 1/ = o A + O A o A 1/ = o A so that VarQ = VarS + o A ad by.9 wehavees A, which leads to.11. Proof of 1.10 We ow prove the asymptotic ormality of Q.LetB = VarQ, X t = B 1z t ad X t = B 1 z t. The, by.6 B 1 Q EQ = X t + t= X t..1 Observe that by 1.7ad.8, E X t =B 1 E z t C A 1 a tt = o1. Therefore, to prove 1.10 it remais to show that d X t N0, t= Sice X t is a m.d. sequece, the by Theorem 3. of Hall ad Heyde 1980, it suffices to show ae max 1 j X j 0, b max X j p 0, 1 j To verify a ad b, it suffices to show that for ay ε>0, c X j=1 j p EXj I X j ε 0,.15 j=1 which clearly implies a, while b follows from.15 otig that P max X j ε ε EXj I X j ε 0. 1 j To prove.15, let K > 0 be large. We cosider two cases: η t K ad η t > K. The, EXt I X t εi ηt K ε 1 EXj 4 I ηt K ε 1 4 K B 4 E j=1 Cε 1 K B 4 t 1 ats Cε 1 K B 4 t 1 4 A sp t 1 ats

11 by.ad.3. Recall that by 1.7, B C A. Thus, for ay ε>0adk > 0, t= EXt I X t εi ηt K Cε 1 K B 4 A sp t 1 t= a ts Cε 1 K A sp / A 0.16 by 1.6as for ay fiite K. We ow focus o the case ηt K.SiceEηt 4 < ad, by statioarity of η t, δ K := Eη1 4I η 1 > K 0asK, this implies EXt I Xt ε I ηt > K EXt I ηt > K B E ηt I ηt > K t 1 by.. Hece, t= C A δ 1/ K E 4 t 1 EXt I Xt ε I ηt > K Cδ 1/ K A 1/ C A δ 1/ K t 1 t= a ts t 1 ats Cδ 1/ K 0, K..17 Sice.16 holds for ay fixed K as, ad sice.17 holds as K uiformly i, weget.15. The verificatio of c i.14 is particularly delicate. We wat to show that s p 1. Recall that x t = B 1 z t where z t is defied i.5. We shall decompose s = Xs ito two parts ivolvig L > 1. Write where The, s = 4B s,1 := 4B η t η t t 1 t 1 s=t L = s,1 + s,,.18, s, := s s,1. We show that as, s = Es + s,1 Es,1 + s, Es,. i Es 1; ii s,1 Es,1 p 0, L 1; iii E s, 0,, L.19 which proves.14c sice E s 0 implies s P 0as ad L. The claim.19i follows from.11, ES /VarQ = B ES 1,

12 otig that B ES = Es, which holds by defiitio of s ad.7. To show.19ii, ope up the squares, set s = t k ad s = t u,toget { s,1 Es,1 =4 L k,u=1 B [ a t,t k a t,t u η t η t k η t u Eηt η ] } t kη t u = 4 It suffices to verify that for ay fixed k ad u, g,ku = o p 1. Settig ad c t := B a t,t ka t,t u V t := η t η t kη t u Eη t η t kη t u, L k,u=1 g,ku. write g,ku = c t V t. Sice the oise {η t } is statioary ergodic ad such that Eη1 4 <, by Theorem i Stout 1974, the process {V j } is statioary ad ergodic, ad E V 1 <. Because of the ceterig, Eg,ku = 0. Thus, by Lemma.1i, to prove g,ku = o p 1, it remais to show that c t s satisfy.1 with α = 1. Observe that t Z c t =B by 1.7. O the other had, t Z a t,t k a t,t u B a t,s = B A C, +1 c t c,t 1 =B a t,t k a t,t u a t 1,t 1 k a t 1,t 1 u +1 B { a t,t k a t 1,t 1 k a t,t u + a t 1,t 1 k a t,t u a t 1,t 1 u } B +1 1/ at,t k a t 1,t 1 k + at,s 1/ +1 1/ at,t u a t 1,t 1 u = o B A = o1, by 1.9,.3ad1.7. Hece.1 holds. By Lemma.1i we coclude that g,ku = o p 1 ad, thus, s,1 Es,1 = o p 1. Hece.19ii holds. To verify E s, 0i.19iii, write s, = s s,1 = 4B η t t 1 t 1 s=t L.

13 We use the idetity a b = a b + a bb, to obtai s, =4B t 1 t 1 ηt a ts η s s=t L = 4B t L 1 t L 1 t 1 ηt + s=t L t L 1 1/ 4q,1 + 4 where 4B B η t η t t 1 s=t L q,1 := B η t 1/ t L 1 4 q,1 + q 1/,1 s1/,1,. Hece, E s, 4Eq,1 +4Eq,1 Es,1 1/. To boud Eq,1, we argue partly as i.10: by 1.8. We also have Eq,1 C A t L 1 Es,1 C A a ts 0,, L t 1 s=t L a ts C. Hece E s, 0as ad L. This completes the proof of.19iii ad the theorem. Ackowledgmets Liudas Giraitis ad Murad S. Taqqu would like to thak Masaobu Taiguchi for his hospitality i Japa ad support by the JSPS grat 15H0061. Murad S. Taqqu was partially supported by the NSF grat DMS at Bosto Uiversity. Ope Access This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made. Refereces Bhasali R, Giraitis L, Kokoszka P 007a Covergece of quadratic forms with ovaishig diagoal. Statistics & Probability Letters 77: Bhasali R, Giraitis L, Kokoszka P 007b Approximatios ad limit theory for quadratic forms of liear processes. Stochastic Processes ad their Applicatios 117:71 95 Dalla V, Giraitis L, Koul HL 014 Studetizig weighted sums of liear processes. Joural of Time Series Aalysis 35:151 17

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