Supplement to Applications of Distance Correlation to Time Series

Transcription

1 arxiv: arxiv: Supplmt to Applicatios of Distac Corrlatio to Tim Sris RICHARD A. DAVIS,* MUNEYA MATSUI,** THOMAS MIKOSCH 3, ad PHYLLIS WAN, Dpartmt of Statistics, Columbia Uivrsity, 55 Amstrdam Av, Nw York, NY 007, USA. * rdavis@stat.columbia.du; phyllis@stat.columbia.du Dpartmt of Busiss Admiistratio, Naza Uivrsity, 8 Yamazato-cho, Showa-ku, Nagoya , Japa. ** mmuya@gmail.com 3 Dpartmt of Mathmatics, Uivrsity of Cophag, Uivrsittspark 5, DK-00 Cophag, Dmark mikosch@math.ku.dk Kywords: Auto- ad cross-distac corrlatio fuctio, tstig idpdc, tim sris, strog mixig, rgodicity, Fourir aalysis, U-statistics, AR procss, rsiduals. This supplmt provids th omittd tchical dtails to th proofs of Thorm 4. ad Lmma 4., i Davis t al. 06). Th sttig, otatio, quatio rfrc umbrs ar rtaid from th mai papr. For covic, th corrspodig rsults i Davis t al. 06) usig th sam rfrc umbrs ar statd bfor th proofs. Th Rmarks 3.4 ad 3.9 from th mai papr hav also b duplicatd hr. Proof of Thorm 4. For th covic of th radr, w rcall two rmarks from Davis t al. 06) which ar mtiod i th proof. Rmark 3.4. If X j ) ad Y j ) ar two idpdt iid squcs th th statmt of Thorm 3. ) rmais valid if for som α 0, ], E[ X α ] + E[ Y α ] < ad s α ) t α ) µds, dt) <. 3.8) R p+q Rmark 3.9. From th proof of Thorm 3. th ctral limit thorm for th multivariat mpirical charactristic fuctio) it follows that G h has covariac fuctio Γs, t), s, t )) = covg h s, t), G h s, t )) = E [ i s,x0 φ X s) ) i t,yh φ Y t) ) j Z i s,x j φ X s ) ) i t,y j+h φ Y t ) )]. 3.9) imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

2 Davis R.A., Matsui M., Mikosch T. ad Wa P. I th spcial cas wh X t ) ad Y t ) ar idpdt squcs G h is th sam across all h with covariac fuctio Γs, t), s, t )) = φ X s s ) φ X s)φ X s ) ) φ Y t t ) φ Y t)φ Y t ) ). Sic G h is ctrd Gaussia its squard L -orm G h µ has a wightd χ -distributio; s Kuo 975), Chaptr. Th distributio of G h µ is ot tractabl ad thrfor o ds rsamplig mthods for dtrmiig its quatils. Thorm 4.. Cosidr a causal ARp) procss with iid ois Z t ). Assum R [ s ) t ) µds, dt) + s + t ) s t > )µds, dt) <. 4.7). If σ = VarZ) <, th T Ẑ,µh) d G h + ξ h µ, ad RẐ,µh) d G h + ξ h µ T Z µ 0), 4.8) whr G h, ξ h ) ar joitly Gaussia limit radom filds o R. Th covariac structur of G h is spcifid i Rmark 3.9 abov for th squc Z t, Z t+h )), ξ h ad th joit limit structur of G h, ξ h ) ar giv i Lmma 4. blow.. If Z is i th domai of attractio of a stabl law of idx α 0, ), i.., P Z > x) = x α Lx) for x > 0 ad L ) is a slowly varyig fuctio at, ad PZ > x) P Z > x) p ad PZ < x) P Z > x) p as x for som p [0, ] Fllr 97), p. 33). Th w hav T Ẑ,µh) d G h µ ad RẐ,µh) d G h µ T Z µ 0), 4.9) whr G h is a Gaussia limit radom fild o R. Th covariac structur of G h is spcifid i Rmark 3.9 for th squc Z t, Z t+h )). W proof th rsult for th rsiduals calculatd from last squar stimats LSEs). O may show that th sam rsult holds for maximum liklihood ad Yul-Walkr stimats. W rcall th rlvat asymptotic rsults for th last squars stimator from Sctio 4 i Davis t al. 06); w us th sam rfrc umbrs for mathmatical formulæ as i Davis t al. 06). Th last-squars stimator ϕ of ϕ satisfis th rlatio ϕ ϕ = Γ,p t=p+ X t Z t, imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

3 Supplmt to Applicatios of Distac Corrlatio to Tim Sris 3 whr Γ,p = X T t X t. t=p+ If σ = varz t ) <, w hav by th rgodic thorm, Γ,p a.s. Γ p = γ X j k) ) j,kp, whr γ Xh) = covx 0, X h ), h Z. 4.) Causality of th procss implis that th partial sum t=p+ X t Z t is a martigal ad applyig th martigal ctral limit thorm yilds ϕ ϕ ) d Q, 4.) whr Q is N0, σ Γ p ) distributd. Kpig this i mid, w start with a joit ctral limit thorm for C Z Lmma 4... For vry h 0, Cosidr a iid squc Z t ) with fiit variac. C Z, ϕ ϕ ) d Gh, Q), ad ϕ. whr th covrgc is i CK) R p, K R is a compact st, G h is th limit procss of C Z with covariac structur spcifid i Rmark 3.9 for th squc Z t, Z t+h )), Q is th limit i 4.), G h, Q) ar ma-zro ad joitly Gaussia with covariac matrix covg h s, t), Q) = φ Zs) φ Zt) Γ p Ψ h, s, t R, 4.4) whr Ψ h = ψ h j ),...,p ad φ Z is th first drivativ of φ Z.. For vry h 0, C Z, CẐ C Z ) d Gh, ξ h ), whr G h, Q) ar spcifid i 4.4) ad ξ h s, t) = tφ Z t) φ Zs)Ψ T h Q, s, t) K, 4.5) th covrgc is i CK, R ), K R is a compact st. I particular, w hav C Ẑ d Gh + ξ h. 4.6) imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

4 4 Davis R.A., Matsui M., Mikosch T. ad Wa P. Proof of part ). W obsrv that, uiformly for s, t) K, C Z s, t) = +itz j+h isz j = φ Z s) ) itz j+h φ Z t) ) φ Z s) ) itz j+h itz j φ Z t) ) + O P ). I viw of th fuctioal ctral limit thorm for th mpirical charactristic fuctio of a iid squc s Csörgő 98a,98b)) w hav uiformly for s, t) K, C Z s, t) = φ Z s) ) itz j+h φ Z t) ) + O P / ) = I s, t) + O P / ). Thrfor it suffics to study th covrgc of th fiit-dimsioal distributios of I, ϕ ϕ) ). I viw of 4.) it suffics to show th covrgc of th fiitdimsioal distributios of I, / ) X ) j Z j. This covrgc follows by a applicatio of th martigal ctral limit thorm ad th Cramér-Wold dvic. It rmais to dtrmi th limitig covariac structur, takig ito accout th causality of th procss X t ). W hav cov I, X j Z j ) = E [ k= φ Z s) ) itz j+h φ Z t) ) X k Z k ]. By causality, X k ad Z j ar idpdt for k < j. Hc E[ φ Z s)) itz j+h φ Z t))x l k Z l ] is o-zro if ad oly if l = j + h ad k h, rsultig i E [ iszj φ Z s) ) itz j+h φ Z t) ) X l k Z l ] This implis 4.4). = E [ X j+h k φ Z s) )] E [ Z j+h itz j+h φ Z t) )] = ψ h k E [ Z isz φ Z s) )] E [ Z itz φ Z t) )] = ψ h k ie [ Z isz] ie [ Z itz] = ψ h k φ Zs) φ Zt). imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

5 Supplmt to Applicatios of Distac Corrlatio to Tim Sris 5 Proof of part ). W obsrv that, uiformly for s, t) K, CẐ s, t) C Z s, t) = isz j+itz j+h i ϕ ϕ) T sx j +tx j+h ) ) + + iϕ ϕ) T sx j ) iϕ ϕ) T sx j + itz j+h ) iϕ ϕ) T tx j+h itz j+h + O P ) S.) = E s, t) + E s, t) + E 3 s, t) + O P ). S.) Writ Ẽ s, t) = i ϕ ϕ) T sx j + t X j+h ) +itz j+h. I viw of th uiform rgodic thorm, 4.) ad th causality of X t ) w hav Ẽ s, t) d iq T E [ sx 0 + tx h ) isz+tz h+) ] S.3) = tφ Z t)φ Zs)Ψ T h Q = ξ h s, t), whr th covrgc is i CK). By virtu of part ) ad th mappig thorm w hav th joit covrgc C Z, Ẽ) d G h, ξ h ) i CK, R ). Dotig th sup-orm i CK) by, it rmais to show that E + E 3 + E Ẽ ) P 0. Th proof for E ad E 3 is aalogous to S.3) by obsrvig that th limitig xpctatio is zro. W hav by a Taylor xpasio for som positiv costat c, E s, t) Ẽs, t) c ϕ ϕ) sup s,t) K 3/ I th last stp w usd th uiform rgodic thorm ad 4.). sx j + tx j+h P 0. Proof of Thorm 4.). W procd as i th proof of Thorm 3.. By virtu of 4.6) ad th cotiuous mappig thorm w hav CẐ s, t) µds, dt) d Gs, t) + ξ h s, t) µds, dt),. K δ K δ Thus it rmais to show that lim lim sup P δ 0 K c δ ) CẐ s, t) µds, dt) > ε = 0, ε > 0. S.4) imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

6 6 Davis R.A., Matsui M., Mikosch T. ad Wa P. Followig th lis of th proof of Thorm 3., w hav lim lim sup E[ C Z s, t) ] µds, dt) = 0 ; δ 0 K c δ s also Rmark 3.4. Thus it suffics to show lim lim sup P ) CẐ s, t) C Z s, t)) µds, dt) > ε = 0, ε > 0. δ 0 K c δ For covic w rdfi C Z = iszj+itz j+h This vrsio dos ot chag prvious rsults for C Z. Usig tlscopig sums, w hav for = p h, CẐ s, t) C Z s, t)) = + A j B j U j B j + A j V j A j =: B j 6 I j s, t), iszj U j whr, supprssig th dpdc o s, t i th otatio, U j = iszj φ Z s), V j = itz j+h φ Z t), itz j+h. B j V j A j A j = isϕ ϕ) X j ), B j = itz j+h isϕ ϕ) X j+h ). Writ K = ϕ ϕ) ad c > 0 for ay positiv costat which may diffr from li to li. By Taylor xpasios w hav I s, t) c c mi s t K 3/ ) A j B j s ϕ ϕ X j ) t ϕ ϕ X ) j+h ) X j X j+h, s K t K X j, X j+h )). imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

7 Supplmt to Applicatios of Distac Corrlatio to Tim Sris 7 Th quatitis K ar stochastically boudd. From rgodic thory, X j = O P ) ad 3/ X j X j+h = o P ). Hc I s, t) mis, t, st) ) O P ) s ) t ) + s + t ) s t ) ) O P ), whr th trm O P ) dos ot dpd o s ad t. Thus w coclud for k = that ) lim lim sup P I k s, t) µds, dt) > ε = 0, ε > 0. S.5) δ 0 A similar argumt yilds I s, t) c j,k=p+ j,k=p+ K c δ A j B k mi st K 5/ s ϕ ϕ X j ) t ϕ ϕ X k+h ) s K mis, t, st) ) O P ). j,k=p+ X j X k+h, X j, t K Th S.5) holds for k =. Taylor xpasios also yild I 3 s, t) c j,k=p+ j,k=p+ U j B k mit, st) ) O P ). k=p+ ) ) X k+h s Z j + E Z )) t ϕ ϕ X k+h ) This provs S.5) for k = 3. By a symmtry argumt but with th corrspodig boud imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

8 8 Davis R.A., Matsui M., Mikosch T. ad Wa P. mis, st) ) O P ), S.5) for k = 4 follows as wll. By Taylor xpasio, w also hav I 5 s, t) c U j B j mit, st) ) O P ). s Z j + E Z )) t ϕ ϕ X j+h ) W may coclud that S.5) holds for k = 5. Th cas k = 6 follows i a similar way with th corrspodig boud mis, st) ) O P ). Proof of Thorm 4.). W follow th proof of Thorm 4.) by first showig that C Ẑ d Gh S.6) i CK) for K R compact, ad th S.4). Th covrgc C Z d G h i CK) cotius to hold as i th proof of Thorm 4.) sic th coditios i Csörgő 98a,98b) ar satisfid if som momt of Z is fiit. For S.6) it suffics to show that C Ẑ C Z ) p 0 S.7) i CK). Rcallig th dcompositio S.), w ow ca show dirctly that p sup s, t M Ei s, t) 0 for ay M > 0 ad i =,, 3, which implis S.7). W focus oly o th cas i = to illustrat th mthod; th cass i =, 3 ar aalogous. W obsrv that for δ > 0, sup E s, t) sup ϕ ϕ s, t M s, t M M δ ϕ ϕ δ sx j + t X j+h X j. S.8) O th othr had, udr th coditios of Thorm 4.) Haa ad Katr 977) showd for δ > α, /δ a.s. ϕ ϕ) 0. For α, ), E[ X ] < ad sic w ca choos δ = such that /δ + / =. Th rgodic thorm fially yilds that th right-had sid i S.8) covrgs to zro a.s. As rgards th cas α 0, ], w hav E[ X α γ ] < for ay small γ ad [ E /δ / X j α γ] α γ)/δ+/)+ E[ X α γ ] 0. imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

9 Supplmt to Applicatios of Distac Corrlatio to Tim Sris 9 If w choos δ clos to α ad γ clos to zro th right-had sid i S.8) covrgs to zro i probability. Usig th sam bouds as i part ), but writig this tim K = /δ ϕ ϕ, w hav I s, t) c mi s t K / /δ t K /δ / j=0 X j X j+h, s K /δ / X j, X j )) c mi s t, s, t ) max K / /δ X j X j+h, K /δ / j=0 X j ). Th sam argumt as abov shows that /δ / j=0 X j = O P ) for δ clos to α. Sic X j X j+h Xj + X j+h a similar argumt shows that / /δ X j X j+h = O P ). Ths facts stablish S.5) for k =. Th sam argumts show that bouds aalogous to part ) ca b drivd for I k s, t) for k =,..., 6. W omit furthr dtails. j=0 Rfrcs Aaroso, J., Burto, R., Dhlig, H., Gilat D., Hill, T. ad Wiss, B. 996) Strog laws of L- ad U-statistics. Tras. Amr. Math. Soc. 348, Bickl, P.J. ad Wichura, M.J. 97) Covrgc critria for multiparamtr stochastic procsss ad som applicatios. A. Statist. 4, Billigsly, P. 999) Covrgc of Probability Masurs, d d. Wily, Nw York. Brockwll, P. ad Davis, R.A. 99) Tim Sris: Thory ad Mthods. Sprigr, Nw York. Csörgő, S. 98a) Limit bhaviour of th mpirical charactristic fuctio. A. Probab. 9, Csörgő, S. 98b) Multivariat charactristic fuctios ad tail bhaviour. Z. Wahrsch. vrw. Gbit 55, Csörgő, S. 98c) Multivariat mpirical charactristic fuctios. Z. Wahrsch. vrw. Gb. 55, Davis, R.A., Matsui, M., Mikosch, T. ad Wa, P. 06) Applicatios of distac corrlatio to tim sris. Submittd to Broulli. Doukha, P. 994) Mixig: Proprtis ad Exampls. Sprigr-Vrlag, Nw York. Duck, J., Edlma, D., Gitig, T., ad Richards, D. 04) Th affily ivariat distac corrlatio. Broulli 0, imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07

10 0 Davis R.A., Matsui M., Mikosch T. ad Wa P. Fllr, W. 97) A Itroductio to Probability Thory ad its Applicatios. Vol. II, d d. Wily, Nw York. Furvrgr, A. ad Murika, R.A. 977) Th mpirical charactristic fuctio ad its applicatios. A. Statist. 5, Furvrgr, A. 993) A cosistt tst for bivariat dpdc. It. Stat. Rv. 6, Fokiaos, K. ad Pitsillou, M. 06) Cosistt tstig for pairwis dpdc i tim sris. Tchomtrics, to appar. Haa, E.J. ad Katr, M. 977) Autorgrssiv procsss with ifiit variac. J. Appl. Probab. 4, Hasltt, J. ad Raftry, A.E. 989) Spac-tim modllig with log-mmory dpdc: Assssig Irlad s wid powr rsourc. J. R. Stat. Soc. Sr. C. Appl. Stat. 38, 50. Hlávka, Z., Hušková, M. ad Mitais, S.G. 0) Tsts for idpdc i oparamtric htroscdastic rgrssio modls. J. Multivariat Aal. 0, Ibragimov, I.A. ad Liik, Yu.V. 97) Idpdt ad Statioary Squcs of Radom Variabls. Woltrs Noordhoff, Groig. Krgl, U. 985) Ergodic Thorms. With a supplmt by Atoi Brul. Waltr d Gruytr, Brli. Kuo, H.H. 975) Gaussia Masurs i Baach Spacs. Lctur. Nots i Math Sprigr, Brli. Lyos, R. 03) Distac covariac i mtric spacs. A. Probab. 4, Mitais, S.G. ad Iliopoulos, G. 008) Fourir mthods for tstig multivariat idpdc. Comput. Statist. Data Aal. 5, Mitais, S.G., Ngatchou-Wadji, J. ad Taufr, E. 05) Goodss-of-fit tsts for multivariat stabl distributios basd o th mpirical charactristic fuctio. J. Multivariat Aal. 40, 7 9. Politis, D.N., Romao, J.P. ad Wolf, M. 999) Subsamplig. Sprigr, Nw York. Rémillard, B. 009) Discussio of: Browia distac covariac. A. Statist. 3, Sjdiovic, D., Sriprumbudur, B., Grtto, A. ad Fukumizu, K. 03) Equivalc of distac-basd ad RKHS-basd statistics i hypothsis tstig. A. Statist. 4, Székly, G.J., Rizzo, M.L. ad Bakirov, N.K. 007) Masurig ad tstig dpdc by corrlatio of distacs. A. Statist. 35, Székly, G.J. ad Rizzo, M.L. 009) Browia distac covariac. A. Appl. Stat. 3, Székly, G.J. ad Rizzo, M.L. 04) Partial distac corrlatio with mthods for dissimilaritis. A. Statist. 4, Zhou, Z. 0) Masurig o liar dpdc i tim-sris, a distac corrlatio approach. J. Tim Sris Aal. 33, imsart-bj vr. 04/0/6 fil: DistCorBroulli_sup_r.tx dat: Fbruary 5, 07