A note on weak convergence of the sequential multivariate empirical process under strong mixing SFB 823. Discussion Paper.

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1 SFB 823 A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig Discussio Paper Axel Bücher Nr. 17/2013

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3 A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig Axel Bücher Uiversité catholique de Louvai & Ruhr-Uiversität Bochum April 18, 2013 Abstract This article ivestigates weak covergece of the sequetial d- dimesioal empirical process uder strog mixig. Weak covergece is established for mixig rates α = O a, where a > 1, which slightly improves upo existig results i the literature that are based o mixig rates depedig o the dimesio d. Keywords ad Phrases: Multivariate sequetial empirical processes; weak covergece; strog alpha mixig; Ottaviai s iequality. AMS Subject Classificatio: 60F17, 60G10, 62G30. 1 Itroductio Let U i i Z, U i = U i1,..., U id, be a strictly statioary sequece of d- dimesioal radom vectors whose margials are stadard uiform. Deote the joit cumulative distributio fuctio of U i by C. The correspodig empirical process is defied, for ay u = u 1,..., u d [0, 1] d, by D u = 1 {IU i u Cu}. i=1 Uiversité catholique de Louvai, Istitut de statistique, Voie du Roma Pays 20, 1348 Louvai-la-Neuve, Belgium. axel.buecher@rub.de. 1

4 Uder various types of weak depedece coditios, the process D is kow to coverge weakly i the space of bouded fuctios o [0, 1] d equipped with the remum-orm, deoted by l [0, 1] d,, to a tight, cetered Gaussia process D C with covariace Cov{D C u, D C v} = i Z Cov{IU 0 u, IU i v}, see for istace Arcoes ad Yu 1994 ad Doukha et al for β- mixig, Rio 2000 for α-mixig, Doukha et al for η-depedece or Durieu ad Tusche 2012 for multiple mixig codtios, amog others. Here ad throughout, weak covergece is uderstood i the sese of Defiitio of va der Vaart ad Weller I this ote, we are iterested i situatios i which the sequece U i i Z satisfies strog α-mixig coditios. Let X i i Z be a sequece of Baachspace valued radom variables. For a b, where a, b Z {, }, let F b a deote the σ-field geerated by X i a i b. The strog mixig coefficiets of the sequece X i i Z are the defied by α 0 = 1/2 ad α = p Z A F p +,B F p+ PA B PAPB, 1. The sequece X i i Z is said to be strogly mixig if α 0 as. I the followig, let α deote the mixig coefficiets of the sequece U i i Z. It has bee show by Rio 2000 that D D i l [0, 1] d, provided that α = O a for some a > 1, thereby improvig previous results by Yoshihara 1975 ad Shao ad Yu I this ote, we are iterested i the slightly more geeral sequetial empirical process B s, u = 1 s {IU i u Cu}, i=1 where s, u = s, u 1,..., u d [0, 1] d+1 ad s deotes the iteger part of s. Note that D u = B 1, u. Ivestigatig the process B is iterestig for several reasos i mathematical statistics. For istace, the process ca be used to derive oparametric tests for chage poit detectio i a d-dimesioal time series, see, e.g., Ioue As a secod applicatio, pose oe is iterested i costructig cofidece bads for some real-valued estimator that ca be writte as a fuctioal of the empirical cumulative distributio fuctio, as for istace its itegral over u [0, 1] d. The, followig the self-ormalizig approach developed i Shao 2010, a 2

5 weak covergece result for B ca used to obtai cofidece bads for this estimator that do ot require a tuig-parameter-depedet estimatio of the asymptotic covariace. Regardig weak covergece results for B, it is agai kow for various types of weak depedece coditios that B B C i l [0, 1] d+1,, 1.1 where B C deotes a tight, cetered Gaussia process with covariace Cov{B C s, u, B C t, v} = s t i Z Cov{IU 0 u, IU i v}, see for istace Dedecker et al for β-mixig or Dehlig et al for multiple mixig properties, amog others. To the best of our kowledge, the best rate available i the literature for strogly mixig sequeces follows from the strog approximatio result established i Dhompogsa 1984: if α = O b with b > 2+d, the 1.1 holds. It is the purpose of the preset ote to improve this rate to α = O a for ay a > 1, idepedetly of the dimesio d, which is the same rate as established i Rio 2000 for the o-sequetial process D. The proof of this result is ispired by the proof of Theorem i va der Vaart ad Weller 1996 ad is based o a adapted versio of Ottaviai s iequality, see Propositio A.1.1 i the last-amed referece, to strogly mixig sequeces. This iequality might be of idepedet iterest i other applicatios where oe wats to trasfer a weak covergece result from the o-sequetial to the sequetial settig. 2 Mai result Theorem 1. If α = O 1+η for some η > 0, the, as, B B C i l [0, 1] d+1,. For the proof of this Theorem, we eed to establish weak covergece of the fiite-dimesioal distributios fidis ad asymptotic tightess. Regardig weak covergece of the fidis, we ca for istace apply Theorem 2.1 i Peligrad The details are omitted for the sake of brevity. Let us cosider the tightess part. For some fuctio f l [0, 1] p, p 1, ad δ > 0 let w δ f = fx fy x y δ 3

6 deote the modulus of cotiuity of f. By the results i va der Vaart ad Weller 1996, Sectio 1.5, the followig Lemma completes the proof of Theorem 1. Lemma 2 Asymptotic tightess of B. Let α = O 1+η for some η 0, 1. The lim lim Pw δ B > ε = 0. δ 0 Proof. First, ote that, by the results i Sectio 7 i Rio 2000 ad Theorem ad its addedum i va der Vaart ad Weller 1996, we have lim δ 0 By the triagle iequality w δ B s 1 s 2 δ The secod summad is equal to k=1 u v δ lim Pw δ D > ε = B s 1, u B s 2, u u [0,1] d + B k/, u B k/, v = k=1 u v δ 0 s 1 u v δ k {D ku D k v} = B s, u B s, v. 2.2 k k=1 w δd k. Defie G i u, v = IU i u Cu IU i v + Cv. Set κ = η/8 ad l = 1/2 κ. For istace by observig that both D : Ω D[0, 1] d ad w δ : D[0, 1] d R are ball-measurable, we ca apply Ottaviai s iequality uder strog mixig, see Lemma 3, with T = {t = u, v [0, 1] 2d : u v δ} ad Y i t = 1/2 G i t, where t = u, v. Let ε > 0. The we obtai P k/ wδ D k > 3ε k=1 where A 1 = Pw δ D > ε ad A 2 = P j<k {1,...,} k j 2l A 1 + A 2 + /l α l 1 k=1 P k/ w δ D k > ε, 2.3 u v δ 1 k i=j+1 G i u, v. > ε 4

7 For sufficietly large, we have l 1 2 1/2 κ, whece /l α l /2 κ 1/2 κ1+η = 1 2 2κ η/2+κη = 1 2 η/4η/2 1 = o1 as. Next, A 1 coverges to 0 as followed by δ 0 by 2.1. Moreover, 1 k G i u, v 8l / 8 κ = o1, j<k {1,...,} k j 2l u v δ i=j+1 as, whece A 2 = o1. To complete the treatmet of the secod summad i 2.2, it remais to be show that the deomiator i 2.3 is bouded away from zero for sufficietly large ad small δ. By 2.1, there exists δ 0 > 0 such that lim Pw δ D > ε < 1/2 for all δ δ 0. The, there exists 0 = 0 δ 0 such that Pw δ0 D > ε < 1/2 for all 0. Therefore, for all δ < δ 0, k= 0 P k/ w δ D k > ε k= 0 Pw δ D k > ε O the other had, for k < 0 ad arbitrary δ > 0, we have w δ D k 2 u [0,1] d D k u 4 k 4 0, k= 0 Pw δ0 D k > ε < 1/2. which implies that 0 1 k=1 P k/ w δ D k > ε = 0 for sufficietly large ad all δ > 0. Therefore, the deomiator i 2.3 is bouded from below by 1/2. Fially, cosider the first rema o the right of 2.2. It suffices to show that, for every ε > 0, P 0 jδ 1 j N jδ s j+1δ B s, u B jδ, u > 3ε u [0,1] d coverges to 0 as followed by δ 0. By statioarity of the icremets of B i s, the at most 1/δ terms i the imum are idetically distributed. Therefore, the probability ca be bouded by 1/δ P = 1/δ P 0 s δ δ k=1 B s, u > 3ε u [0,1] d k/ D k u > 3ε u [0,1] d 1/δ B 1 + B 2 + δ /l α l k/, δ k=1 P u [0,1] d D k u > ε 5

8 by the Ottaviai-type iequality i Lemma 3, where δ / B 1 = P D δ u > ε u [0,1] d k B 2 = P 1/2 {IU i u Cu} > ε. j<k {1,..., δ } k j 2l u [0,1] d i=j+1 Here, we were allowed to apply Lemma 3 by a similar argumet as before. It remais to be show that B 1, B 2 ad δ /l α l, multiplied with 1/δ, coverges to zero as followed by δ 0 ad that the deomiator i 2.4 is bouded away from zero. First of all, for δ 1, we have 1/δ δ /l α l 1 δ + 1 δαl l 2 α l l = o1 as, by the same argumets as above. Secod, by the Portmateau- Theorem, lim B 1 lim P D u > εδ 1/2 u [0,1] d lim P D u εδ 1/2 u [0,1] d P u [0,1] d D C u εδ 1/2 = P u [0,1] d D C u > εδ 1/2 sice u [0,1] d D C u is a cotiuous radom variable. Sice additioally u [0,1] d D C u possesses momets of ay order cf. Propositio A.2.3 i va der Vaart ad Weller 1996, the latter probability coverges to zero faster tha ay power of δ, as δ 0. Third, regardig B 2, we have j<k {1,..., δ } k j 2l u [0,1] 1/2 d i=j+1 k {IU i u Cu} 4l 1/2 = o1, as, whece B 2 coverges to zero as followed by δ 0 as asserted. Fially, let us cosider the deomiator i 2.4. By a similar argumet as before, we have from the Portmateau Theorem that lim P D u > ε P D C u > ε u [0,1] d u [0,1] d 6

9 by cotiuity of u [0,1] d D C u. Also, sice ε > 0, we obtai that p = P u [0,1] d D C u > ε < 1, whece we ca choose ζ > 0 such that 0 < ζ < 1 p. It follows that there exists 0 N such that Hece, P D k u > ε p + ζ. k 0 u [0,1] d δ P k/ D k u > ε δ P D k u > ε p + ζ. k= 0 u [0,1] d k= 0 u [0,1] d O the other had, for k < 0, we have k/ u [0,1] d D k u k/ 2 k 2 0 /, which implies that, for large eough, 0 1 k=1 P k/ u [0,1] d D k u > ε = 0. Hece, the deomiator i 2.4 is bouded from below by 1 p ζ > 0 for large eough. This completes the proof. 3 A auxiliary Lemma Let X i i Z be a sequece of radom elemets i some Baach space E. Let T be some arbitrary idex set ad, for i Z, let G i l E T. For t T, set Y i t = G i X i, t ad S t = i=1 Y it, for 1 ad S 0 0. Fially, for f l T, let f = t T ft. Lemma 3 A Ottaviai-type iequality uder strog mixig. Suppose that S m S is measurable for each 0 < m. The, for each ε > 0 ad 1 l <, { } P S k > 3ε 1 P S S k > ε k=1 k=1 P S > ε + P S k S j > ε j<k {1,...,} k j 2l + /l α l, where α deotes the sequece of mixig coefficiets of the sequece X i i Z. Proof. For k = 1,...,, defie the evet B k by B k = { S k > 3ε, S 1 3ε,..., S k 1 3ε}. 7

10 Note that these evets are pairwise disjoit ad that their uio is give by { k=1 S k > 3ε}. Furthermore, for m = 1,..., /l 1, let C m = ml k=m 1l+1 B k, ad C /l = k= /l 1l+1 which are also pairwise disjoit ad have the same uio as the B k s. Now, let us first cosider a fixed m /l 1. The P C m mi P S S k ε k=1 P C m P S S m+1l ε P C m, S S m+1l ε + α l ml P C m, k > 3ε, S S m+1l ε k=m 1l+1 + α l. 3.1 Sice S k S k S m+1l + S m+1l S + S for ay k = 1,...,, we have ml S { S k S m+1l S k S m+1l S } k=m 1l+1 { ml S k } { ml S m+1l S k } S m+1l S. k=m 1l+1 k=m 1l+1 Therefore, we ca estimate the right-had side of 3.1 by ml P C m, S > 2ε S m+1l S k + α l k=m 1l+1 ml P C m, S > ε + P C m, S m+1l S k > ε + α l k=m 1l+1 P C m, S > ε + P C m, S k S j > ε j<k {1,...,} k j 2l Next, let us cosider the case m = /l. The P C /l mi P S S k ε k=1 P C /l, S k > 3ε k= /l 1l+1 P C /l, S > 3ε k= /l 1l+1 S S k 8 B k, + α l.

11 P C /l, S > 2ε + P C /l, S S k > ε k= /l 1l+1 P C /l, S > ε + P C /l, S k S j > ε j<k {1,...,} k j 2l Joiig both cases, we have, for ay m = 1,..., /l, P C m mi k=1 P S S k ε P C m, S > ε + P C m, Summatio over m fially yields the assertio. S k S j > ε j<k {1,...,} k j 2l. + α l. Ackowledgemets. The author is thakful to Iva Kojadiovic for thorough proofreadig ad umerous suggestios cocerig this mauscript. This work has bee ported i parts by the Collaborative Research Ceter Statistical modelig of oliear dyamic processes SFB 823 of the Germa Research Foudatio DFG ad by the IAP research etwork Grat P7/06 of the Belgia govermet Belgia Sciece Policy, which is gratefully ackowledged. Refereces Arcoes, M. A. ad B. Yu Cetral limit theorems for empirical ad U-processes of statioary mixig sequeces. J. Theoret. Probab. 7 1, Dedecker, J., F. Merlevède, ad E. Rio Strog approximatio of the empirical distributio fuctio for absolutely regular sequeces i R d. workig paper. Dehlig, H., O. Durieu, ad M. Tusche A sequetial empirical CLT for multiple mixig processes with applicatio to B-geometrically ergodic markov chais. arxiv: Dhompogsa, S A ote of the almost sure approximatio of the empirical process of weakly depedet radom vectors. Yokohama Mathematical Joural 32, Doukha, P., J.-D. Fermaia, ad G. Lag A empirical cetral limit theorem with applicatios to copulas uder weak depedece. Stat. Iferece Stoch. Process. 12 1,

12 Doukha, P., P. Massart, ad E. Rio Ivariace priciples for absolutely regular empirical processes. A. Ist. H. Poicaré Probab. Statist. 31 2, Durieu, O. ad M. Tusche A empirical process cetral limit theorem for multidimesioal depedet data. Joural of Theoretical Probability, Ioue, A Testig for distributioal chage i time series. Ecoometric Theory 17 1, Peligrad, M O the asymptotic ormality of sequeces of weak depedet radom variables. J. Theoret. Probab. 9 3, Rio, E Théorie asymptotique des processus aléatoires faiblemet dépedats. Berli Heidelberg: Spriger. Shao, Q.-M. ad H. Yu Weak covergece for weighted empirical processes of depedet sequeces. A. Probab. 24 4, Shao, X A self-ormalized approach to cofidece iterval costructio i time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 3, va der Vaart, A. ad J. Weller Weak covergece ad empirical processes. New York: Spriger. Yoshihara, K.-i Weak covergece of multidimesioal empirical processes for strog mixig sequeces of stochastic vectors. Z. Wahrscheilichkeitstheorie ud Verw. Gebiete 33 2,

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