HILBERTIAN INVARIANCE PRINCIPLE FOR EMPIRICAL PROCESS ASSOCIATED WITH A MARKOV PROCESS

Chin. Ann. Math.24B:1(2003),1-16.HILBERTIAN INVARIANCE PRINCIPLEFOR EMPIRICAL PROCESS ASSOCIATEDWITH A MARKOV PROCESSJIANG YIWEN* WU LIMING**AbstractThe authors establish the Hilbertian invariance principle for the empirical process of aatationary Markov process, by extending the forward-backward martingale decomposition ofLyons Meyer- Zheng如the Hilbert space valued additive functionals associated with generalnon-reveraible Markov processes.Keywords Forward-backward martingale decomposition, Functional central limittbeorem or Donsker's invariance principle, Empirical process2000 MR Subject Clasification 60F17, 60F25, 60J55, 62G30Chinese Library Classfication O211.4, 0211.6Document Code AArticle ID 0252 9599(2003)01-0001-16g1. Introduction1.1. Motivation and Several Known ResultsLet (8,F,(F)ter, (Xx)tET, (0)teτ, (P. )x∈E) be a Markov process valued in a Polish spaceE, with transition probability semigroup (Pt)ter and with an invariant and ergodic probability measure μ on (E, B), which is unknown. Here T = N (discrete time) or R+ (continuoustime). For any initial measure U, set P():= JgPz()v(dx) and write E() := Ja()dPp.Let f : E→R be a fixed B-measurable function (our observable). A natural questionfrom the point of view of non-parametric statistics is to estimate the distribution functionF(u):= L[f(x) ≤叫= Pp(f(Xo) ≤u) by the observed (X). By an extension of theKolmogorov- Smirnov theorem, we have P-a.s.sup|F砖(u) - F(u)| -→0,as T goes to infinity, whereif 0叶33≈4.373. Further, B. Morel and C. Suquetl2l real-ized that the optimal condition for the weak L2[0, 1] convergence of 5n to a Gaussian randomelementis 2(善- Emax(Xo,Xx)) <∞. This motivates our main result below.1.2. A Main ResultWe are mainly interested in the FCLT of the empirical distribution 5() ( or 5r() ) insorme Hilbert space 88 in [9]. In this paper, we assume F(u) := u[f(x)≤叫] is continuous.Let In(u,u) :=四4ξn(u)En(以) and hu(工) = 1[f(=)S川], we have the fllwing invarianceprinciple:Theorem 1.1. Assume the strong sector condition (1.5) (ie, our Markov process (X)oeris quai-symmetric).号- Emax(F(f)(X), F(f)(X;)) is summable in the sense of Abel,ie., if(E(+()-(*-mma()()(F(x) or。*e“(号- max(F()(Xo),F(f)(X)) ) dt(acconding toT=N or R+) converges in R asε↓0, then we have(a) Ir(u,0) := F"Er(u)Er() weakly converges in L2(R, dF(u)dF()) to r(n,1) asT goesto infinity, andT as a kernel operatorT()w);= (r(u),)f1)dF(0) ，is of the trace class on L2(R, dF(u)dF());(b) for any initial measurer < u, the law of ξn on [2(R, dF) under P, converges weaklyto the Gaussian measure on [2(R, dF) with reproducing kernel given by T(u,v). .In order to prove Theorem 1.1, we shall extend the forward-backward martingale decomposition of Lyon8-Meyer Zhengl71) to the Hilbert space valued additive functionals,The second named authorln] generalized the forward-backward martingale decompositionof Lyon-Meyer-Zbeng's type ftom the symmetric case to the general stationary situationfor the partial sum S:(f) with f stisfying a finite enerey condition for realf.中国煤化工MYHCNMHG4CHIN. ANN. MATH.Vol.24 Ser.BThis paper is organized as fllows. The next two sections are devoted to extendingforward-backward martingale decomposition of Lyons Meyer Zheng's type in the discretetime and continuous time from the real valued case to the Hilbertian valued case. In section4 we discuss the quasi-symmetric case. We complete the proof of Theorem 1.1 in the lastsection.Throughout this paper, <, ) and I .1 denote respectively the inner product and norm inL2(E,u:; R), E() the expectation w.r.t P= Pp.功(E,x;R):= {g∈L2(E,x;R);(g,1> =0}.We say that (At)teris an additive functional, if for all s,t∈T,As+t(w) = A。(山) + As(0。w), Pu -a.8.(i.e., in the loose sense). A typical additive functional is (S(f)) given in (1.3).82.Forward-Backward Martingale Decompositionand Invariance Principle: the Discrete Time Case2.1. Some Preliminary Lemmas in the Real Valued CaseWe begin by recalling some results in [17].Let T= Nand write P= P. Let P* be the adjoint operator of P in L2(E, ;R) and .P°= PtF-, the synmetrization of P. Let Wo = LE(E,x;R)= {J∈L2(E,;R);(f)p = 0}equipped with norm |fIlo = I|l|l2(u). It is easy to see that Vu∈LE(E,u;R) := Wo,((I-P")u,u)=0→u=0.(2.1)Then (eee [17)I-P°:Wo→Wo is injective, its invere R :D(RE)(C Wo)→Wo isaself-adjoint operator with domain D(RE)= Ran(I - P").DeAnition 2.1. Let W1 be the completin of the pre-Hibert space (Wo = L好(E,;R),<)1) where the inner product is given by他,):= ((I- P)u,v).(2.2)We define (W_,I. HI-1) as the dual Hilbert space of (W,I . |I1) w.r.t. the canonical duarelation Wo = Wo.Lemma 2.1.1 WoC w, W_1 C Wo are both continuous and dense imbedding, and .forf∈e(E,;R),f∈W_s if (1.3) holds; and the minimal constant C in (1.3) equals toHf-1.By the ergodicity of P we can define the potential (or Poisson) operatorsRo=(I- P)-1: D(Ro)= Ran(I - P)(c W)→wo,R=(I-P*)-1 :D(R)= Ran(I- p*)(c Wo)→W.Lemma 2.2.17] D(Ro)UD(R)C W_1 andIfiI_IS√lPoflo，Vf e D(Ro), and If-1 ≤√1|fR1lo,Yf∈D(R);(2.3)IRif1l, Vf∈D(R), and Illfl_1，, Vf∈ D(RE).(2.4)2.2. Forward-Backward Martingale Decomposition for H-valued AdditiveFunctionalsLet H be a Beparabl Hibert space with iner product (,)m and norm | . lax; (e)ie1(I = [I,dim(EN)} nN denotes an othonormal basis for H The H valued function will be中国煤化工MYHCNMHGNo.1JIANG, y. W.& wu, L. M. HILBERTIAN INVARIANCE PRINCIPLEdenoted by the bold letters f, g, ... Let LK(E,μ;H):= {f∈L2(E,;H);IE"(f)=0}. Thedomain of operator Ro on L(E,r; H) is denoted by D:(Ro). We denote the norm and innerproduct in L2(E,p;H) by |. 1 and《, .》respectively. We defineDefinition 2.2. Letf∈LE(E,x;H) and f;:= (f,e:)u. We say thatf∈W1, iflve :=2.<+o.Lemma 2.3. Letf∈ LK(E,p;H). Thenf∈Wtl i讲 there erists a constantC 2 0, suchthat《f,g)》S CV(Ag,g)i，Vg∈ Da(4) = L2(E,x;H)(2.5)(recaling that A=I- P) and(fw = inf{C≥0| (2.) holds for C}.(2.6)In particular, |f\w does not depend on the choice of the ONB (ei).Proof. We prove it only in the case where dimnHl= +∞(I= N").Necessary part: Note that for all g∈Dr(A),《f,g》) = Z(4ng)≤Ilif-√(A9.,9n)EIs 11(/2(49gng)= I吧V(AB,B).Hence inequality (2.5) holds with some c≤l/wy,,Suficient part: Letn≥1 and ()1....∈解such that乙X≤1. For eachJ{;= (f,ei), .and for any g∈Wo, we have by the suficiency assumption (2.5),(feg) =《f,ge;)》≤CV(Ag,9),anditfollowsthatf;∈W_1.Foranyε>0,wecanfind9;∈Wosuchthat|i|-1≤.(1 +e)(fu9>) and (Ag,9;)= 1. Thus by (2.5) again,二λI/1l_ (1 +e)2 ; \:(f,gi)=(1 + a)《f,λig;e:》=1-1i二1≤(1 +e)c2 ; \?0. So |fl}wm≤c bylettingε→0, n→∞.Theorem 2.1. Let T = N. There erist three bounded linear mappingsG:W!I→唔=功(E,;H)，(Gf,Gf)x ≤2()fr )2;M; :W4→L(A,F,B)日L2(Q,Fo,P)，E(M1 ()|l)2≤2({f1\we,); (2.7)MT :W41→L(Q,Go,P) θ L2(2,9,P)，E(|M$ ()|x)2≤2(f[|we, )只,中国煤化工MYHCNMHGCHIN, ANN. MATH.Vol.24 Ser.Bwhere Gk = o(Xm;m≥k) is the future o-field, such that the fllowing foruward-backuardmartingale decomposition holds P-a.s. for everyf∈WI,2Zf(Xx)= M(f)+ M()+ Gf(Xo)- Gf(Xn), hn∈N,(2.8)k=0whereM2(f) ='26% -1M+(), M5()=':S0n_ 1M5(f).k=1In particular for each f∈W,(a) the maximal inequality below holds:4 sup |Sl()脂≤(24n + 3)(f;(2.9)0≤kSn(b) the family of the laws oft→一Smg(f)∈D([0,1],H)，n≥1,on D([0,1],H) under Pu is precompact for the weak convergence topology;(c) if moreoverf belongs to the closure of Rapx(I- P) = {(I-P)f :f∈谮=L2(E, u;HI)} = Dr(Ro) in wl, then there is an additive square integrable H -valued mar-tingale (Fn)- (M()n20 and an H -valued additive functional (On())n2o such thatSn(f) = An(f) + Mn(f),(2.10)二回4 maxl|Oe(f)临→0.(2.11)In prticular, for any itial measurev《), asn goes to ininity, the law of (Sn()eouunder P, conuerges weakly in DX([0,1],H) to the law of an H-valued BM (Bt) where the co-variance of B1 i8 given by政(Br,h1)m(B,b2)x) = (Th,hz2)H, Vh1,h2∈H,(Tbs, bz2)a := 8(M(),h)r(Mr(f),bz)m)(2.12)= lim. ((),hI)e(Sn(f),b2)x)，v h;,h2∈HHere D(0,1],H) is the space of all H-valsed cddlag functions on [0, 1] equipped with theSkorokhod topology.Proof, When H= R parts (a), (b), (c) are due to [17, Theorem 2.5].For everyf= 2fer∈H, putG(f) :=:艺cG(f)e;, M7(f) :='Mi(f)e;, M[(f) :=Mi (f)e,51中国煤化工MYHCNMHGNo.1JIANG, Y. w.& wU, L. M. HILBERTIAN INVARIANCE PRINCIPLEwhere G, Mi", M个are defined in [17, Theorem 2.5]. They are convergent, becauseE|G(f)临=ES(G(1)2≤2ZIUAI2l≤2(fp+ 0, (M$_ ()tE(O,)) is a (Gr. t)eE0.,n] -martingale (though it is leftcontintous).In particular we have(a) the maximal inequality below holds:郾。sup IS(f)脂≤1T(fw), VT> 0;(3.6)OSiST(b) the family of the laws oft-→六Snt(f)∈C(0,1],H), n≥1, on the Banach space :C([0, 1],H) under Pp is precompact;(c) assume moreover that f belongs to the closure of W41 ∩Dr(Ro) in W1r , thenthere are an additive square integrable cidlag martingale (M({f)) and an additive cidligfunctional (f)) such thatS{(f) = M(f)+OQe(f)，Vt≥0， Pp-a.s.(3.7)suplQ,(f)|2 →0 (ast→+oo).(3.8)In particular, for any initial measure v《μ，a8 T goes to infinity, the law of(+r()e(0,1) under P converges weakly in C(0,1,H) to the law of an H- valued BM(Bt), where the covariance of B1 is given byE(B,h1)a(B,h2)u) = (Thr,h2)n, Vhr,h2∈H,with(Th, hz2)u := B(M(f),h1)x(M;(f), bz2)u)(3.9)= lim. (0.For the same reason there exists an Hvalued cadlag backward martingale (M; (f)) suchthatsplIM4-"(f)- M(f)|x-→0(n→∞), VT>0.Thus, for eacht≥0 we have Pp-a.8.,M=()= EM+(s)e, M(f)= EMt(f)e,台which satisfyElM+()脂= E2 (M(F)2 =2tSf1.< +o,二1+∞EIM+ ()临=EE(M:()2 =2[I1w1l24<+∞o.Moreover, for eacht∈R+ , we have Pp_-a.s,2S()= E25(f)e,=(M+(f;) + Mt ()e;= M:(f) + M?(f).By the right continuity of t→S:(), MZ(f) + Mt (), we get with Pp-probablit one,2S(f) = Mr(f) + M{(f),Vt∈R+.With the same proof as that of Corllary 2.5, we getCorllary 3.1. fininfel|S etr/w. < +o∞, then f belongs to the closure of叫∩De(Ro) in啊. And all cnclausons in Theorem 3.1 () hold.中国煤化工MYHCNMHG12CHIN. ANN. MATH.Vol.24 Ser.Bs4. Quasi Symmetric CaseThroughout this section, the Markov process (X)tET is quasi-symmetric, i.e, it satisfies(1.5), In the continuous time case, we shall asume that our process (X.) is Hunt (see [10).The main result is Theorem 4.1 below, which is stated in [17] in the real valued case withoutdetailed proof.Let A=(I-P)or A=-L according toT=Nor= IR+,and R:= A-1. Applying(1.4) tou= Rf,v= Rog, we see that Ro is still sectorial. Hence the bilinear formE-_(f,g) := (Rof,9)，Vfg∈ D(R)(4.1)is closable, and its closure will be denoted by (E 1,D(E-1)) By [5), D(R) CD(E_1)is alsoa form core of (E_1,D(E-1)).Lemma 4.1. Ifliminf(f,Ref)< +∞，(4.2)then .f∈D(E-1)，and E_s(f,f) ≤liminf(f, Ref>.(4.3)Proof. In fact, take fe= f-εRef = ARef, Rofe = Ref and thenliminf E-_1(fe,fe)= liminf(f - eRef,Ref)≤liminf(f,Ref)< +∞.By [5] (Chapter VI, Theorwm 1.15, p. 314), it fllws thatf∈D(E_1)，E-(Jf,f) ≤liminf(f,Ref).Lemma 4.2. For anyf∈D(Ro), we havef∈W_1 andVE _1(f,f)≤IflI_1≤KVE (f,f).(4.4)Proof. The left inequality in (4.4) follows from(Rf|1)2 = (Rof,ARof) = (Rof,f)≤|Rolfo1of-Iand the fact that (IFRf1)2} = E_1(U,f). For the right inequality in (4.4), let g= Rof. Wehave for any u∈D(C),($,u) = (Ag,2)≤K√(Ag,9) .√(Au,u) = K√E-1(,). lThus lfl-≤K√E-1(f,F), the desired right side inequality in (4.4).Lemma 4.3. D(Ro)C W-1 and is dense i (W1,H1-1).Proof. By Lemma 4.1, for any f satisfying (4.2),f ∈D(E_1). As D(R) is a form coreofE_1, we can find fr∈D(Ro),h≥1so that{fk-j,fk-f)+E_1(fh-f,jn-J)→0. ByLemma 4.2, we have|fh- f川|+IIfn- f1-1-→+0ask,l→+∞. Consequently f∈W_1 and fh→fin W_1. Then D(Ro) is dense in W_1.Since D(Ro) is dense both in W_1 and D(ε_1), we have D(E-_1)= W_1.Lemma 4.4. For f∈LK(E,x;H), if liminf《f, Rf》>< +∞, then|fl≤K2 liminf(《f,Rf》.中国煤化工MYHCNMHGNo.1JIANG, y. W.& wu, L. M. HILBERTIAN INVARIANCE PRINCIPLE18Proof. Witef= E fei, f;= (f,e;)x. By Fatou's lemma,eEliminf(fr R.sf) ≤lnigf《{R{I〉< +o.Then f;∈D(E-1) by Lemma 4.1. We have by Lemmas 4.2 and 4.1 that for eachN≥1,|fl4 =2 (fu,fl)_1≤K2E8_(f,f)E≤K2 2 liminf(i,Rfh)≤K2 linf(f,Rf).C→+0E→0Lemma 4.5. DE(Ro) c W明. and Dr(Ro) is dense in WH∩W凹Proof. (1) If f∈DH(Ro), then there exists g∈L2(E,p;H), such that Ag = f. Then《{f, Rf>)=《{,Re(- C)g》=《f,(1 - eRe)g》≤|f(IsI + lIeR_gi)≤2|f|. IgI.By Lemma 4.4, f∈w4(2) Givenf∈W醉∩WL,letfN:= S fe. ThenjfN- f|w,→0,asN→∞.By Lemma 4.3, Ve> 0,3 ji,e∈D(Ro) such thatNChoose fe,N= 2 f,e;∈Dr(Ro). We have|fN~-s.Nl =lf-1el2: r小.Then[f(Xx)≤叫= [f(Xe)≤F-()] =[F()(Xk)≤川，u= Ff-(y).Let hw(x) := 1j(&)S叫- F(u). We have (by fllowing [9)EMa(X)r(X))dF(u)=。H.eJF.F///-}04= I Plal(),(Xx))≤dv-j= 8- Emax(F(f)(Xo),F()(Xx)).中国煤化工MYHCNMHGNo.1JIANG, Y. W.& wU, L. M. HILBERTIAN INVARIANCE PRINCIPLE15Let h(x) := (h())ueR∈[2 (R,dF() and (Rth)。= Rrhu. We then have(Rth,h)L2(B,u.= (Rghu,hu)dF(u) ='(e+1)-k-1Ehu(Xo)hu(Xn)dF(u)k=0= De+1)-(5 -Ex(OF)0)(FC(x)),and it fllws thath∈W"l by Theorem 4.1 and by our condition. Thus by part (c) ofTheorem 2.1,max IAr(b)临Sn(h) = Mn(h)+ An(h), E-sn(5.1)where (Mn(h)) is an H valued additive L2-martingale.Step 2. Recalling that In(u,t) := Corv(),Gn()), then \Vf,g∈L2(R,dF(u), we haveby (5.1),lim r(u,v)f(u)g(v)dF(u)dF(u) = lim -Esn(h,f)m)Sn(h, 9)m)= lim E(Mn(h),f)u . (Mr()9)x = E(M (h),{)u (M()9)e,+∞o > E|M(h)脂= lim二E|Sn(h)临= im (s(u)2) dP(u)= lim. [rn(u y)dF(u),+∞JRHence by Cauchy- Schwartz's inequality,sl ()}f()dF() swp f.. rn(u, ur(,)dF(2)dF()= 8upp [r(,u)dF(u)]" <∞.n21 LHence (Tn(4,叭} iseltively compet wrt. the weak topology o (I2(R, dP),I?(R,dF))and any weak limit r of In must verifyfor al,g∈E, where the existence of the last limit is shown above. Thus r is umique. Inother words, In→r weakly in LZ(R, dF).We now see whyr isof trace on H Let (ei)izl denote an ONB of H = [2(R, dF()) andh; := (h,er)z(R,dF). We have(eie)zQ,dP) =fS& (0(e(9()f())=。Is (1(e(e()}F()= n l Esn(:)Sn(hi) = EM(hz)M(h),Eelroaudn)= EM()M(h)= EM(_an<∞台中国煤化工MYHCNMHG6CHIN. ANN, MATHVol.24 Ser.B(b) It fllows directly by Theorem 4.1.Remark. In the discrete time case, let1 (凹n(,2):=元2 (1r(Xa)Su- F(u))√n0and regard m := (t →()e)le/1)0 as a random element in D(0,][L(R,dF)). .Then Theorem 4.1 is applicable (by the proof of part (a) above) and yields the fllowinginvariance principle:for any initial measure v< μ, P(Mn∈) converges weakly on D(0, 1],L2(R,dF)) to thelaw of an L2(R, dF)-valued Brownian Motion withT(u, )f(u)g(v)dF(u)dF().REFERENCES[1] Chen, X., Limit theorems for functionals of ergodic Markov chain with general state space, Memoirs ofthe AMS, 139, 1999.} Djellout, H. & Guillin, A., Moderate deviations of Markov Chains with atom, Stochast. Proc. Appl,203-217.'，95:2, (2001)，203-217P. A, Probabilites et potentiels, Vol. Il1, IV, Hermann 1983, 1987.[3] Dellacherie,& Meyer, P.[4j Gordin, M. I, The central limit theorem for stationary processes, Sovict Math. Dokl, 10:5(1969),1174-1176[5] Kato, T, Perturbation theory for linear operators, 2nd ed. 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Henri Poincare, 35:2(1999), 121-141.中国煤化工MYHCNMHG