Tightness in Schauder decomposable Banach spaces
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Tightness in Schauder decomposable Banach spaces

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Tightness in Schauder decomposable Banach spaces? Ch. Suquet Laboratoire de Statistique et Probabilites, Bat. M2 Universite des Sciences et Technologies de Lille F-59655 Villeneuve d'Ascq Cedex France Abstract We characterize the tightness of a set of probability measures in a large class of Banach spaces including those having a Schauder basis. We give various applications to sequences of stochastic process viewed as random elements in the spaces Lp(0, 1), Lp(IR) or in some Holder or Besov spaces. AMS classifications: 60B10, 60F17, 60G30. Key words: empirical process, Haar basis, multiresolution analysis, Schauder decomposition, strong mixing, tightness, wavelets. ?Preprint

  • ?zj ?

  • projection

  • sup p?f

  • closed finite

  • faber-schauder basis

  • ?x? ejx?

  • lim t?

  • analysis background

  • schauder decomposition



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Tightness in Schauder decomposable Banach spaces
Ch. Suquet LaboratoiredeStatistiqueetProbabilite´s,Bˆat.M2 Universit´edesSciencesetTechnologiesdeLille F-59655 Villeneuve d’Ascq Cedex France
Abstract We characterize the tightness of a set of probability measures in a large class of Banach spaces including those having a Schauder basis. We give various applications to sequences of stochastic process viewed as random elements in the spaces Lp(0,1), LpeHomld¨o)oIRnsri(svapec.sreroeBos
AMS classifications: 60B10, 60F17, 60G30. Key words: empirical process, Haar basis, multiresolution analysis, Schauder decomposition, strong mixing, tightness, wavelets.
1 Introduction Relative compactness in the space of probability measures is a key tool in the study of weak convergence. A familyFof probability measures on the general metric spaceSis said to be tight if for each positiveε, there is a compact set Ksuch thatP(K)>1εfor allPinF. According to Prohorov’s theorem, tightness is always a sufficient condition for relative compactness and is also necessary ifSis separable and complete. The Skorohod spaceS=D(0,1) is the usual framework of many limit the-orems for stochastic processes. This is so because it supports processes that contain jumps and weak convergence inD(0,1) provides results about some useful functionnals of paths like those involving the suprema. Nevertheless this space presents some drawbacks. First, tightness inD(0,1) is sometimes difficult to check. Second, under the pointwise addition of functions, the spaceD(0,1) is not a topological group and hence not a topological vector space. In many cases, it seems very convenient to treat a stochastic process as a random element in a functional Banach space. The best known case is certainly theC(0, As1) one (see Billingsley [4]). for the Hilbert space case, sufficient conditions for tigthness are given by Prohorov [18], Parthasarathy [17] and Gihman Skorohod [8]. The recents developments in the theory of wavelets and their applications in probability and statistics show the interest of using more sophisticatedfunctionsspacesliketheHo¨lder,SobolevorBesovspaces. In this paper we will present an unified approach to tightness problems in a large class of Banach spaces includingC(0,1), theLpdlreS,bolovenad,H¨o Besov spaces. Our starting point is the Hilbertian case: Theorem 1 (Suquet [21])LetHbe a separable Hilbert space and(ei, iIN) an orthonormal basis ofH. Define for eachhinHand each positive integer N: s2N(h) =Xhh, eii2, r2N(h) =Xhh, eii2. i<N iN The familyFof probability measures onHis tight if and only if: (i)N1tli+mPsupFP({hH:s2N(h)> t}) = 0, (ii)t >0Nlim+PsupFP({hH:r2N(h)> t}) = 0. Our aim is to generalize this theorem. Let us denote byVjthe finite dimensional subspace ofHgenerated by{e0, . . . , ej1}and byEjthe orthogonal projection onVj. With these notations, the conditions (i) and (ii) can be recast as: (i)EjF={PEj1, P∈ F }is tight for eachj1, (ii)ε >0,jli+mPsupFP({hH:khEjhk ≥ε}) = 0. 2
We would like to generalize this result in two directions: dropping the finite dimensionality of theVj(as in the case of multiresolution analysis) and dropping the orthogonality of theEj(and hence the hilbertian character of the space). Of course, we have to keep some control on the norms of the projectorsEj. This leads us naturally to deal with Schauder decomposable Banach spaces. The next section will present the functional needed analysis package . The third section exposes the proof of our main result: the extension of the theorem 1 for these spaces. In the following section, we use our main theorem and a multiresolution analysis to obtain a sufficient condition for tightness in Lp(IR), 1< p <+. We study also the relative compactness in Lp(0,1) of the random step func-tions involved in Donsker’s theorem (without any assumption on the dependence structure of the underlying random variables). We rederive and clarify a previ-ous condition for tightness inL2(0,1) due to Jacob, Oliveira and Suquet ([14], [15]). Next we are interested in the convergence of the empirical process based on strong mixing uniform variables (Xi)i1on [0,1]. In theD(0,1) setting, the best result up to now is due to Yoshihara [23] who proved the weak convergence of the empirical process to a gaussian process under the conditionαn=O(n3ε) (theαnbeing the strong mixing coefficients of the sequence (Xi)i1). Recently, Oliveira and Suquet proved the same convergence inL2(0,1) under the weaker assumptionPαn<+. Of course the convergence inL2(0,1) is weaker than inD(0,1). Here we obtain the convergence in Lp(0,1) (2p <6) under a weaker condition (depending onp) than Yoshihara’s one. In the fifth section we are concerned with the spaces H0α-cniredufnaofolH¨ tions on [0,1] (i.e.f(0) = 0,|f(t)f(s)| ≤C|ts|αand|f(t)f(s)|= o(|ts|α)). We obtain a tightness criterion and a sufficient condition very similar to theC(0,1) case. Some examples are discussed. The last section presents an easy application to some sequences spaces and their isomorphic Besov functional spaces. This application could be useful in the study of weak convergence for stochastic processes known by their wavelets coefficients. 2 The functional analysis background We refer to Singer [20] for the Schauder decompositions and to Meyer [13] for wavelets and multiresolution analysis. Let (X,k k A) be a Banach space. system{xn, nIN}of elements ofXcalled a Schauder basis if for everyis elementx∈ X, there is a unique series: +x=Xanxn, an=an(x)IR,(1) n=0 which converges toxin the norm ofX. We define the associated coordinate projectionsvnbyvn(x) =anxn. These projections are continuous and there is 3
a constantCdepending only on the basis{xn, nIN}such that: N NIN,x∈ X,Xvn(x)Ckxk.(2) n=0 The basis is said unconditional if the series (1) is unconditionally convergent, that is, for every permutationσ={σ(n), nIN}of the indexes, the series Pn+=0aσ(n)xσ(n)converges toxin the norm ofX the multiresolution analysis. In setting defined below, we are dealing with bases of wavelets indexed by IN×Z, Z×Z,Z×Zd for many functional spaces, the decomposition on. Fortunately these bases is an unconditionally convergent series so that neither the order of summation nor the groupings of terms do matter. Here the groupings of terms are usually made according to the level of resolution. At one given level, say 2j, we have a countable family of functions (ψj,k,kZ) of the bas is which is dense in a closed subspace of the involved functional space. In the important case of the Faber-Schauder basis of the spaceC(0,1), which do not define a multiresolution analysis, we have the groupings{Δj,k,0k <2j}which span closed finite dimensional subspaces ofC(0,1). Banach spaces having a Schauder decomposition are the natural framework unifying all these situations. Definition 1An infinite sequence(Gj, jIN)of closed linear subspaces of a Banach spaceXsuch thatGj6={0}(jIN)is called a Schauder decomposition ofXif for everyx∈ Xthere exists an unique sequence(yn, nIN)with yjGj(jIN)such that: +x=Xyj j=0 and if the coordinate projections defined byvn(x) =yn, are continuous onX. In other words, (Gj, jIN) is a decomposition ofXif and only if,Xis the direct topological sum of the subspacesGj should be noticed here that some. It Banach space do not possess a Schauder decomposition, for instance the space `a Schauder decomposable Banach space need not(see Singer [20]) and that be separable. Let us denoteVj=LijGiandEj=Pijvithe continuous projections ofXontoVj follows from proposition 15.3 p. 488 in Singer [20] that:. It C= supkEjk<+.(3) jIN By the orthogonality relations between the coordinate projections (i.e.:vivj= δijvi=δijvj) we have: Ej0Ej=EjEj0=Ejj0 j, j,0IN.(4)
This implies: kxEixk ≤(1 +C)kxEjxk, x∈ X, i > j.(5) In separable Banach spaces having a Schauder decomposition, we have a very simple criterion for relative compactness. This criterion is a generalization of the Hilbert space case (see for instance Akhiezer and Glazman [1]). We did not find this result in its general form in the literature, so we give a detailed proof. Theorem 2LetXseparable Banach space having a Schauder decomposi-be a tion. A subsetKis relatively compact inXif and only if: (i)For eachjIN,EjKis relatively compact inVj, (ii) supkxEjxk →0asj+. xK Proof : Sufficiency of(i)and(ii): Let (zn, nIN) be a sequence inK. We have to check that (zn, nIN) contains a convergent subsequence (zn, nI) whereI is some infinite subset of IN. Using repeatedly (i), we can construct a sequence (JjINt:ucNshahtsbteosIfninetus)ofiJ0J1⊃ ∙ ∙ ∙and for eachj, (Ejzn, nJj) converges inVj. Moreover we can require that min(Jj)<min(Jj+1), j us define thenIN. LetI={min(Jj), jIN}. By construction we have {nI, nmin(Jj)} ⊂Jjfor eachjIN and hence the sequence (Ejzn, nI) converges inVjtowards someyj we show that (. Nowzn, nI) is a Cauchy sequence inX fixed positive. Forεthere is by (ii) an integerjsuch thatkxEjxk< εfor allxinK the construction of. ByI, there is an integer n0such that for alln > n0inI,kEjznyjk< ε for every. Sonandplarger thann0inI,kznzpk<4ε. Necessity of(i)and(ii necessity of (): Thei) follows obviously from the con-tinuity of the projectionsEj prove (. Toii), we can assume without lose gen-erality thatKis closed and hence compact. Thus the continuous function x7→ kxEjxktakes its maximum overKfor somezjK. Put: yj=zjEjzj,kyjk= supkxEjxk. xK It suffices then to prove the convergence to zero of (yj, jIN). First we observe that (yj, jIN) is relatively compact inX. Indeed, taking subsequences it suffices to check that if (zj, jJ) converges toz, (Ejzj, jJ) is a convergent sequence. Writing: kEjzjzk ≤ kEjk ∙ kzjzk+kEjzzk, this follows from (3) and the definition 1. Moreover we have limEjzj= limzj, so that the only possible limit for a subsequence of (yjjIN) is zero. Hence yjconverges to 0, which ends the proof. 5
Let us now have a more detailed look at the examples of Schauder decom-positions referred above. The first instance is provided by the Schauder bases (xn, nIN), takingGj= span[xj some bases it is more convenient to]. For have theGjas span of a finite number of vectors of the basis. This is the case of the Haar and Faber-Schauder bases we are now recalling the definition. The Haar basis (en, nIN) is an unconditional basis for the spaces Lp(0,1), (1< p <+). Putψ(t)=1I[0,1/2[(t)1I[1/2,1[(t). The Haar basis is defined bye0(t) = 1 anden(t) =ej,k(t) = 2j/2ψ2jtkwheren= 2j+kwith 0k <2j the Schauder decomposition of L. Herep(0,1) we are interested in is given byG0= span[e0],Gj= span[ej,k,0k <2j us recall that the]. Let projection offontoVj=L0ijGiis its approximation by a step function equal to the mean value offover each intervallk2j,(k+ 1)2j, 0k <2j. The Faber-Schauder basis (Δn, nIN∪ {−1}) is a Schauder basis for the spaceC(0,1) of continuous functions on [0, space have no unconditional1]. This basis. Put Δ(t) = 2t1I[0,1/2[(t) + 2(1t)1I[1/2,1[(t). Then Δ1(t) = 1, Δ0(t) =t, Δ1(t) = Δ0,0(t) = Δ(t), Δn(t) = Δj,k(t) = Δ(2jtk) wheren= 2j+kwith 0k <2j. Here we define theGjas in the Haar basis case (andG1= span[Δ1]). The projection of a continuous functionfontoVjis simply its approximation by linear interpolation with knotsk2j, f(k2j). The Faber SchauderbasisisalsoaSchauderbasisintheHo¨lderianspacesH0α(see section 5 below). Finally, we recall some useful facts about wavelets and multiresolution anal-ysis (in a reduced version adapted to our purpose, the general definitions can be found in Meyer [13] or Daubechies [7]). In what follows, forgL2(IR), we writegj kfor the functiongj,k(t) = 2j/2g2jtk,j, kZ. By multiresolu-, tion analysis with scaling functionϕ, we mean a ladder of closed subspaces (Vj, jZ) ofL2(IR) such that: a)\Vj={0},[Vj=L2(IR), jZjZ b)VjVj+1, c)Vj [= spanϕj,k, kZ], d) (ϕ0,k, kZ) is an orthonormal basis ofV0. The multiresolution analysis is calledr-regular (rIN) ifϕis ofCrclass and for each integermthere is a constantamsuch that: ϕ(α)(t)am1 +|t|m, tIR, αr.(6) DefineWjas the orthogonal complement ofVjinVj+1, then we have for each jZthe decomposition: L2(IR) =VjMWi.(7) ij 6
One can construct a functionψsuch that (ψ0,k, kZ) is an orthonormal basis ofW0and (ψj,k, j, kZ) is an orthonormal wavelets basis ofL2(IR). If the multiresolution analysis isr-regular,ψ write Weverify also the property (6). Ej(resp.Dj) for the orthogonal projection fromL2(IR) ontoVj(resp.Wj) and its associated integral kernel: Ejf=ZIREj(. , s)f(s)ds, Ej(t, s) =Xϕj,k(t)ϕj,k(s),(8) ¯ kZ Djf=ZIRDj(. , s)f(s)ds, Dj(t, s) =kXZψj,k(t)ψ¯j,k(s).(9) The kernelsEjandDjverify: Ej(s, t) = 2jE0(2js,2jt), Dj(s, t) = 2jD0(2js,2jt) t, s,IR.(10) Using (6), it is easily verified thatE0andD0are majorized by convolution kernels. More precisely, there exists two rapidly decreasing functionsKandL such that: E0(s, t)K(st),D0(s, t)L(st).(11) As shown in Meyer [13], the usefulness of the wavelets bases associated to a regular multiresolution analysis goes far beyond theL2(IR) space. They provide unconditional bases for many functions spaces as Lp(IR) (1< p <+), Sobolev, H¨lder and Besov spaces. In each case, the function spaceXis the topological o direct sum ofV0and theWi(i0) (these subspaces being redefined in an adapted way). We have then a Schauder decomposition ofXgiven byG0=V0, Gj=Wj1(j1). 3 Main result We give now the characterization of the tightness for separable Schauder de-composable Banach spaces. Theorem 3LetXseparable Banach space having a Schauder decomposi-be a tion: +j X=MGi, Vj=MGi, j= 0,1,2, . . . i=0i=0 and denote byEjthe continuous projection fromXontoVj. LetFbe a family of probability measures onXandEjF={µEj1, µ }∈ F. ThenFis tight if and only if: (i)EjFis tight,j= 0,1,2, . . . (ii)For each positiveε,jli+msµupFµx∈ X:kxEjxk> ε= 0. 7
Proof : Sufficiency of(i)and(ii): For fixed positiveη, putηl= 2l,l= 1,2, . . .and choose a sequence (εl By) decreasing to 0. (ii), there is an integerjlsuch that: µ∈ F, µx∈ X:kxEjlxk> εl< ηl.(12) By (i), there is a compact subsetKlofXsuch that: µ∈ F, µx∈ X:EjlxKl>1ηl.(13) From (12) and (13) we deduce: µ∈ F, µ\x∈ X:EjlxKlandkxEjlxk ≤εl>12η.(14) l1 It remains to check the compacity inX Thisof the intersection in (14). follows easily from the continuity of theEj, (4), (5) and the theorem 2. Necessity of(i)and(ii tightness is preserved by continuous mappings, the): As necessity of (i) follows from the continuity of theEj. To prove the necessity of (ii), we need the following lemma. Lemma 1LetFcompact family (for the topology of weak convergence)be a of probability measures on the separable metric spaceS. Let(Fl, lIN)be a sequence of closed subsets ofSdecreasing to. Define the functionsul(lIN) by:ul:P7ul(P) =P(Fl) the sequence. Then(ul)uniformly converges to zero onF. Proof :For positiveε, let us defineDl,ε={P∈ F:ul(P)ε} first. We verify thatDl ε The topology of weak convergence on probabilityis closed. , measures overSbeing metrizable, this can be done by means of sequences. Let (Pn, nIN) be a sequence inDl,ε, weakly convergent to someP the. By portmanteau theorem we have: ul(P) =P(Fl)lim supPn(Fl) = lim supul(Pn)ε, n+n+soPis inDl,ε, which is then closed. The monotone continuity of probability measures implies clearlyTlINDl,ε=. In view of the compactness ofF, we can find anl0such thatTll0Dl,ε= the sequence (. AsDl,ε) decreases, we have:ul(P)< εfor allll0,and allP∈ F. Now we apply this lemma withFjtaken as the closure of: Aj={x∈ X,supkxEixk ≥ε}. ij Clearly (Aj, jIN) is decreasing and so is (Fj, jIN). Using (5), we have: xAj,kxEjxk ≥1 +εC. 8
By continuity ofIdEj, this remains true for allxinFj. As for eachxinX, limj+Ejx=x, this implyTjINFj=. Sinceµ(x∈ X,kxEjxk ≥ε)µ(Fj), the lemma 1 give us the expected conclusion. When the subspaces of the Schauder decomposition are finite dimensional, the theorem 3 has the following more tractable version: Theorem 4Assume the Banach spaceXhas the Schauder decompositionX= LjINGjwhere eachGjis of finite dimension. ThenFis tight if and only if the condition(ii)of theorem 3 and the following condition(i0)hold: (0)Ali+mµ∈Fµx∈ X:kxk> A= 0. isup Proof :Clearly (i0) holds ifFis tight. On the other hand,kxkand supjkEjxk being equivalent norms inX b) p. 488), ((Singer [20], prop. 15.3i0) implies: Alim+sµupFµx∈ X:kEjxk> A= 0, j= 0,1,2, . . . As theVjare finite dimensional, the tightness ofEjFfollows. The theorem 4 applies in particular to the decompositions coming from a Schauder basis. As a result, combining theorems 3 and 4, we can treat the case of multiresolution analysis. Indeed, by an elementary topological argument, tightness inVjreduces to tightness inV0and in theWi(0i < j these). And spaces have a wavelets Schauder basis. By Markov’s inequality, the sufficient conditions for tightness in theorem 4 admit the following moment form. Theorem 5Assume the Banach spaceXhas the Schauder decompositionX= LjINGjwhere eachGj Thenis of finite dimension.Fis tight if: (i)α >0,sup IEµkξkα<+, µ∈F (ii)β >0,jli+pusmEIµkξEjξkβ= 0, µ∈F whereξdenotes a random element inXwith distributionµandIEµthe expec-tation with respect toµ. This last theorem is a generalization of the theorem 1.13 of Prohorov [18] for the Hilbertian case. According to this theorem (see also Parthasarathy [17] th. 2.2 p. 154), whenXis a separable Hilbert space,Fis tight if: jli+mµ∈FZXj2(x)µ(dx) = 0,(15) supr whererj2(x) =Pijhx, eii2and (ei, iIN) is an orthonormal basis ofX. That is exactly the condition (ii) above withβ= 2 for the Schauder decomposition associated to the basis (ei, iIN). 9