SFB 823 A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig Discussio Paper Axel Bücher Nr. 17/2013
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. E-mail: axel.buecher@rub.de. 1
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. 1995 for β- mixig, Rio 2000 for α-mixig, Doukha et al. 2009 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 1.3.3 of va der Vaart ad Weller 1996. 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 1996. 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 2001. 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
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. 2013 for β-mixig or Dehlig et al. 2013 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 2.12.1 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 1996. 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
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 1.5.7 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 = 0. 2.1 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
For sufficietly large, we have l 1 2 1/2 κ, whece /l α l 1 2 1 1/2 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/, 2.4 1 δ k=1 P u [0,1] d D k u > ε 5
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
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
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.
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 1994. Cetral limit theorems for empirical ad U-processes of statioary mixig sequeces. J. Theoret. Probab. 7 1, 47 71. Dedecker, J., F. Merlevède, ad E. Rio 2013. Strog approximatio of the empirical distributio fuctio for absolutely regular sequeces i R d. http://hal.archives-ouvertes.fr/hal-00798305. workig paper. Dehlig, H., O. Durieu, ad M. Tusche 2013. A sequetial empirical CLT for multiple mixig processes with applicatio to B-geometrically ergodic markov chais. arxiv:1303.4537. Dhompogsa, S. 1984. A ote of the almost sure approximatio of the empirical process of weakly depedet radom vectors. Yokohama Mathematical Joural 32, 113 121. Doukha, P., J.-D. Fermaia, ad G. Lag 2009. A empirical cetral limit theorem with applicatios to copulas uder weak depedece. Stat. Iferece Stoch. Process. 12 1, 65 87. 9
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