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March 15, 2010 10:46 WSPC/INSTRUCTION FILE OfBm-sd-rev Operator fractional Brownian motion as limit of polygonal lines processes in Hilbert space? ALFREDAS RA?KAUSKAS Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-2006 Vilnius, Lithuania. Institute of Mathematics and Informatics, Akadem?os str. 4, LT-08663, Vilnius, Lithuania. CHARLES SUQUET Laboratoire P. Painlevé, UMR 8524 CNRS Université Lille I, Bât M2, Cité Scientifique, F-59655 Villeneuve d'Ascq Cedex, France. In this paper we study long memory phenomenon of functional time series. We consider an operator fractional Brownian motion with values in a Hilbert space defined via oper- ator valued Hurst coefficient. We prove that this process is a limiting one for polygonal lines constructed from partial sums of time series having space varying long memory. Keywords: Fractional Brownian motion; Hilbert space; functional central limit theorem; long memory; linear processes. Mathematics Subject Classifications (2000): 60F17; 60B12. 1. Introduction Long memory phenomenon have played an important role since the 50's when dis- covered by Hurst in certain hydrologycal data sets. Historically this paradigm has been associated with slow decay of long-lag autocorrelation of a stochastic process and certain type of scaling properties embodied in a concept of self-similarity.

• operator fractional


• defined via oper- ator valued


• mean zero


• self- adjoint operator

Sujets

• Vilnius University

Informations

• Publié par: rasum
• Nombre de lectures: 10
• Langue: English

Extrait

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Operator fractional Brownian motion as limit of polygonal lines processes in Hilbert space ∗

ALFREDAS RAČKAUSKAS

Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-2006 Vilnius, Lithuania.

Institute of Mathematics and Informatics, Akademijos str. 4, LT-08663, Vilnius, Lithuania.

CHARLES SUQUET

Laboratoire P. Painlevé, UMR 8524 CNRS Université Lille I, Bât M2, Cité Scientifique, F-59655 Villeneuve d’Ascq Cedex, France.

In this paper we study long memory phenomenon of functional time series. We consider an operator fractional Brownian motion with values in a Hilbert space defined via oper- ator valued Hurst coefficient. We prove that this process is a limiting one for polygonal lines constructed from partial sums of time series having space varying long memory.

Keywords: Fractional Brownian motion; Hilbert space; functional central limit theorem; long memory; linear processes.

Mathematics Subject Classifications (2000): 60F17; 60B12.

1. Introduction

Long memory phenomenon have played an important role since the 50’s when dis- covered by Hurst in certain hydrologycal data sets. Historically this paradigm has been associated with slow decay of long-lag autocorrelation of a stochastic process and certain type of scaling properties embodied in a concept of self-similarity. The- oretical investigations go back to Mandelbrot and his co-authors (see[17],[18]). In the past two decades, the interest in long memory (equally named as long-range dependence) has increased especially in financial mathematics and econometrics, mostly due to the availability of precise empirical measurements such as tick-by- tick observations in stock markets for example. For multivariate data, the theory of long range dependence and self-similarity of processes are studied through operator scaling almost in parallel with the theory of operator stable distributions and their generalized domain of attraction. We

∗ The research was partially supported by Lithuanian State Science and Studies Foundation, Grant No. T-68/09

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refer to Doukhanet al.[8], Jurek and Mason[12], Marinucci and Robinson[19], Dolado and Marmol[7], Meerschaert and Scheffler[21]and references therein for a state-of-the-art of this field of research. This paper is devoted to the long memory phenomenon in connection with functional data analysis. We consider a stationary process(X k ), where eachX k in a real separable Hilbert space (finitetakes values or infinite dimensional), sayH. Different criteria exist to define long memory of univariate time series. The most used are related to the asymptotic decay of the autocovariance function: (i) lack of summability of autocovariance function, (ii) regular variation of the autocovariance function at infinity with an exponent of variation−1< d≤0classical scheme we consider a space varying. Following this decay of the autocovariance operators(Q k )of(X k )in a sense that there exist a nuclear operatorQand a self-adjoint operatorDonHsuch thatQ k ∼k D Qand −I < D≤0, whereIdenotes the identity. In this case we say that the process(X k ) hasa space varying memoryWe discuss this phenomenon together with limiting. properties of partial sums. In Section 2 we introduce a continuous time model with space varying memory, namely an operator fractional Brownian motion which is defined via an operator valued Hurst exponent. In section 3 we consider linear processes in Hilbert space with regularly varying filters that are not summable. These processes have space varying long memory. Polygonal lines build from their partial sums converge in distribution to an operator fractional Brownian motion. The proof is exposed in Section 5, after providing in Section 4 an auxiliary result, which may be of indepen- dent interest, on the convergence of some Hilbert space valued stochastic processes build from linear processes whose coefficients are operators.

2. Operator fractional Brownian motion

Fractional Brownian motion is a Gaussian process with stationary but dependent increments. The dependence structure is modeled by its Hurst parameterH∈(0,1) via the covariance functionR H defined by R H (s, t=) 2 1  t 2H +s 2H − |t−s| 2H  , s, t≥0. The fractional Brownian motion originated by Kolmogorov[13], has been studied in connection with many applications, e.g., financial time series, hydrology, telecom- munications to name a few. Moreover a number of generalizations was suggested, from stable fractional motion to that one with time varying Hurst index. The ex- istence of a fractional Brownian motion with values in a separable Hilbert space His proved in[9]. Namely it is shown that for any self-adjoint nuclear operator QonH, and a Hurst indexH∈(1/2,1)there exists a GaussianH-valued process (B H,Q (t), t≥0)on a probability space(Ω,F, P)that satisfies (i)EB H,Q (t) = 0for allt≥0; (ii)R H,Q (s, t) := cov(B H,Q (t),B H,Q (t)) = 2 −1 (t 2H +s 2H − |t−s| 2H )Q, for all s, t≥0;

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(iii)(B H,Q (t), t≥0)hasH-valued continuous pathsP-a.s. In this section we consider a Hilbert space valued fractional Brownian motion with space varyingHurst parameter. The main aim of introducing such a process is to understand a space varying long memory phenomenon of infinite dimensional time series. Empirical evidences show a big interest in such models, see e.g.,[4],[2],[22]. To be more precise we need to introduce some notations. LetHbe a real sepa- rable Hilbert space of infinite or finite dimension with the inner producth. , .iand the corresponding norm|| ∙ ||,||x|| 2 =hx, xiThe space of bounded linear operators. u:H→His denoted byL(H). We considerL(H)as a Banach space with the usual uniform norm||u||= sup ||x||≤1 ||ux||. The adjoint operator of an operatoru∈L(H) isu ∗ andtr(u)means the trace ofu. LetL 0 (H)denote the space of compact oper- atorsu:H→H, endowed with the usual operator norm. For the definition and the main algebraic and analytic properties ofL 0 (H)we refer to Dunford-Schwartz[10]. ForT∈L 0 (H)letλ k (T ∗ T)be thek’th positive eigenvalue ofT ∗ T. Setµ k = √ λ k , thek’th singular value of operatorT. Then define: L 1 (H) = n T∈L 0 (H) : X µ k <∞ o . k The nuclear normν 1 onL 1 (H)is defined byν 1 (T) = P k ∞ =1 µ k .Several properties ofL 1 (H)are presented in[10]. For an operatorT∈L(H)we sete T = exp(T) = P k ∞ =0 T k /k!provided the series converge inL(H)and we setλ T = exp(Tlogλ), forλ >0. We also denote

m T = inf ||x||=1 hT x, xi, M T = ||x s | u | p =1 hT x, xi.

We refer to[1]all information concerning the spectral theory of linear operatorsfor on Hilbert spaces. For a self-adjoint operatorT∈L(H)let(E λT ,λ∈R)be a spectral decomposition ofTa family of orthoprojectors such that, that is (i)E λT = 0forλ < m T ,E λ =Iforλ≥M T ; (ii) the functionλ→E λT is left continuous in strong topology; (iii)T= R ∞ λdE λT . −∞ For any continuous functionφon[m T , M T ], we haveφ(T) = R φ(λ) dE λT and for anyx, y∈H,hφ(T)x, yi= R φ(λ) dhE λT x, yi. An operatorA∈L(H)is called non- negative (brieflyA≥0) ifhAx, xi ≥0for allx∈H. An operatorA∈L(H)is positive (denotedA >0) providedA≥0andA6= 0. For operatorsA, B∈L(H) the notationA > B(A < B) means thatA−B >0(B−A >0). Ifxandyare two vectors inH, we denote byx⊗ythe rank one operator defined for allu∈Hby(x⊗y)(u) =hx, uiy. IfX,Yare zero mean random elements inHwithE||X|| 2 <∞,E||Y|| 2 <∞then the covariance operator is cov(X, Y) :=EX⊗Y∈L 1 (H)(we refer to Vakhania, Tarieladze, Chobanyan[26] for probability distributions on Banach spaces).

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Now we are prepared to define the operator fractional Brownian motion which is considered in this paper.

Definition 2.1.LetQ∈L 1 (H), Q≥0andH∈L(H), H≥0. AH-valued Gaus- sian process(B H,Q (t),t≥0)on probability space(Ω,F, P)is called an operator fractionalQ-Brownian motion with Hurst parameterH(shortly ofBm with param- eters(H, Q)), if this process satisfies

(i)EB H,Q (t) = 0for allt≥0; (ii)R H,Q (s, t) := cov  B H,Q (s), B H,Q (t)  = 2 −1 (t 2H +s 2H − |t−s| 2H )Q, for all s, t≥0.

In the rest of the paper we shall frequently use the notation r A (s, t=:) 2 1  t 2A +s 2A − |t−s| 2A  for an operatorAas well as for a real numberA. The meaning ofAinr A (s, t) should be everytime clear from the context. The important question now is the existence of ofBm with parameters(H, Q). A partial answer is given in the following proposition. Proposition 2.1.Let the operatorH∈L(H), be self-adjoint and satisfy 21 I < H < I(equivalently1/2< m H , M H <1). LetQ∈L 1 (H)be a non-negative operator commuting withH. Then there exists a fractionalQ-Brownian motion with Hurst parameterH.

Proof.For anys, t≥0,R H,Q (s, t)is a linear bounded operator. SinceH andQcommute, it is a nonnegative operator. Indeed, consider the spec- tral measureE λH ,λ∈R. AsH= R −∞ ∞ λdE λH we havehr H (s, t)Qx, xi= R ∞ ∞ − r λ (s, t) dhE λH Q 1/2 x, Q 1/2 xi ≥0for eachx∈H, sincer λ (s, t)>[min{s, t}] 2λ ≥ 0for alls, t∈[0,1]andλ∈R. Moreover the operatorR H (s, t)is nuclear as the product of the bounded operatorr H (s, t)by the nuclear operatorQ. Hence, for any s, t≥0,R H,Q (s, t)is the covariance operator of some mean zeroHvalued random element. As any non-negative definite function(s, t)7→T(s, t), with values in the set of self-adjoint nuclear operators onHdefines uniquely the distribution of a zero mean Gaussian process, we have now to check that(s, t)7→R H,Q (s, t)is positive definite, that is n n X X hR Q,H (t i , t j )x i , x j i ≥0,(2.1) i=1j=1 for anyt 1 , . . . , t n ≥0,x 1 , . . . , x n ∈H,n∈N. Sincer H (s, t) = R ∞ ∞− r λ (s, t) dE λH , we have

N r H (s, t) = N li → m ∞ X r λ k (s, t)(E λH −E λH k−1 ) k k=1

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for any partition(λ k )of the interval[m H −ε, M H ]with diameter tending to zero. Then (2.1) follows from

n n X X r λ k (t i , t j )h(E λH k −E λH k−1 )Qx i , x j i ≥0,(2.2) i=1j=1 for anyt 1 , . . . , t n ≥0,x 1 , . . . , x n ∈H,n∈N. Since the operatorE λH k −E λH k−1 :H→ His a projector and commutes withQ, the left-hand side of (2.2) is equal to n n X X r λ k (t i , t j )hQy k,i , y k,j i, i=1j=1 wherey k,j = (E λH k −E λH k−1 )x j . From the existence of the fractionalQ-Brownian motion with values in Hilbert space and Hurst indexλ k ∈(1/2,1)as proved in[9], Prop. 2.1, it follows that

n n X X r λ k (t i , t j )hQz i , z j i ≥0, i=1j=1

(2.3)

for anyt 1 , . . . , t n ≥0,z 1 , . . . , z n ∈H,n∈N. Hence, (2.1) is proved. Throughout we consider only operator fractional Brownian motionsB H,Q with commuting operatorsHandQand we shall assume that 1 2 I < H < I. As mentioned in the introduction, self-similarity of processes introduced by Lam- perti[16]is one of the sources of long memory. Operator self-similar processes appeared later in the paper by Laha and Rohatgi[15]and were investigated by Matache and Matache[20]. Let us recall that a stochastic process{X(t) >, t0} with values in a Banach spaceEis operator self-similar if there exists a family of linear bounded operators{A(a) >, a0}onEsuch that for eacha >0, {X(at) >, t0}= D {A(a)X(t) >, a0},

where= D means equality in distribution. The family{A(a), a >0}is refered to as the scaling family of operators.

Proposition 2.2.The ofBm with parameters(H, Q)is operator self-similar with the scaling family{a H , a >0}. Proof.Since{a H , a >0}is a multiplicative group of operators andQcommutes withHwe have cov(B H,Q (as), B H,Q (at)) = 2 −1 ((at) 2H + (as) 2H − |at−as| 2H )Q =a 2H 2 −1 ((t) 2H + (s) 2H − |t−s| 2H )Q = cov(a H B H,Q , a H B H,Q ). This yields self-similarity of the Gaussian process(B H,Q (t), t≥0).

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To investigate the smoothness of the paths of the Gaussian process(B H,Q (t)) t≥0 , we use the following estimate of its increments where the parameterm H plays the main role.

Proposition 2.3.For anys, t >0such that|t−s| ≤1it holds E||B H,Q (t)−B H,Q (s)|| 2 ≤ |t−s| 2m H tr(Q).(2.4) Proof.For anyx∈Hwe have EhB H,Q (t)−B H,Q (s), xi 2 =hR H,Q (t, t)x, xi −2hR H,Q (t, s)x, xi+hR H,Q (s, s)x, xi =h|t−s| 2H Qx, xi(2.5) = Z |t−s| 2λ dhE λH Qx, xi.(2.6) As|t−s| 2H Qis a nuclear operator, choosing any orthonormal basis(x j ) j≥1 inH, we deduce from (2.5) that ∞ E||B H,Q (t)−B H,Q (s)|| 2 = X h|t−s| 2H Qx j , x j i= tr(|t−s| 2H Q). j=1 Now recalling that|t−s| ≤1and thatE λH = 0forλ < m H we deduce from (2.6) that tr(|t−s| 2H Q) = j= X∞ 1 Z |t−s| 2λ dhE λH Qx j , x j i ≤ |t−s| 2m H X∞ Z dhE λH Qx j , x j i j=1 =|t−s| 2m H tr(Q), so (2.4) is established.

The estimate (2.4) enables us to obtain the following.

Theorem 2.1.The space fractionalQ-Brownian motion with Hurst indexHhas a continuous version which satisfies on every bounded interval[a, b]⊂[0,∞) ,Q (s)|| a≤ s s u < p t≤b ( | t |B − H s , ) Q m (t H ) |l − n( B t H −s)| 1/2 <∞,a.s.(2.7)

Proof.AsB H,Q is a Gaussian process with values in the Banach spaceHand satisfying by (2.4) an estimate of the formEkB H,Q (t+h)−B H,Q (t)k 2 ≤σ(h) 2 , we have for some version ofB H,Q : sup kB H,Q (t+h)−B H,Q (t)k <∞,a.s., 0<h≤b−a ρ(h) a≤t≤b for anyρ(h) =h α `(h)(with0< α <1and`slowly varying) such that lim inf h→0 ρ(h)(|lnh| 1/2 σ(h)) −1 >0, see e.g. Corollary 4 (i) in[24]. Then (2.7) follows from the choiceρ(h) =σ(h)|lnh| 1/2 withσ(h)given by (2.4).

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3. Long memory linear processes

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Consider aH-valued linear process(X k )defined by ∞ X k = X u j ε k−j , j=0

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(3.1)

whereu 0 =Iis the identity map,(u j , j≥1)⊂L(H)is a given sequence of operators such that P j ||u j || 2 <∞and(ε j , j∈Z)is a sequence of independent identically distributed (i.i.d.) random elements inHwith mean zero and finite second moment σ 2 0 =E||ε 0 || 2 . For simplicity we assumeσ 02 = 1. LetQdenotes the covariance operator ofε 0 . From the theoretical point of view, one of the most interesting features of difference between short and long memory of the linear process(X k )is in the limit behavior of the corresponding partial sums process. In this section we shall consider a polygonal line process. SetS 0 = 0and

n S n = X X k , n≥1.(3.2) k=1 The polygonal line process based on partial sumsS k ,k≥1, is defined by ζ n (t) =S [nt] + (nt−[nt])X [nt]+1 , t∈[0,1]. We consider this process in the spaceC([0,1];H), the Banach space of continuous functionsx: [0,1]→Hendowed with the norm||x||= sup 0≤t≤1 ||x(t)||. The following result is proved in[5], see also[23]for a more general approach. Proposition 3.1.Assume that the filter(u k )is summable, that is P k ||u k ||<∞. LetA= P k ∞ =1 u k . Then n −1/2 ζ n − D −−→W AQA ∗ inC([0,1];H), n→∞ whereW AQA∗ =B 1/2,AQA ∗ is aHvalued Brownian motion. Autocovariance operator of lagkof time series(X j )is ∞ Q k =EX 0 ⊗X k = X u ∗ j Qu j+k . j=0 Since||Q k || ≤ ||Q|| P j ∞ =0 ||u j || ∙ ||u j+k ||absolute summability of the linear filter (u k )generates short memory of the process(X k )in a sense that the sequence of autocovariance operators(Q k )is absolutely summable, P k ∞ =1 ||Q k ||<∞. Even more, it is summable in the nuclear norm. Indeed, sinceν 1 (uT v)≤ ||u||ν 1 (T)||v|| for any nuclear operatorTand anyu, v∈L(H), we have ∞ ∞ X ν 1 X  u ∗ j Qu j+k  ≤ X X ||u j ||ν 1 (Q)||u j+k ||. k j=0k j=1

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If sumability of autocovariance sequence fails, then the limiting distribution needs not to be a Wiener process and norming needs not to be a classical √ n. This phe- nomenon has been understood long ago. We refer to a survey paper by Samorod- nitsky[25]for more information on this phenomenon of long memory. In this section we consider linear process(X k )for which summability of the filter fails but operators(u k )are regularly varying. More precisely we restrict ourselves to the case

u k =k −D , k≥1,(3.3) whereD∈L(H)satisfies 1 2 I < D < I. Moreover we assume that the operatorsQ andDcommute. Then the condition P k ||u k || 2 <∞is satisfied but the absolute summability of autocoavariances operators(Q k , k≥0)fails since for an eigenvector x 0 corresponding to the eigenvalueM D we have N N∞N∞ X ||Q k ||= X  X j −D Q(j+k) −D  ≥ X X j −M D (j+k) −M D hQx 0 , x 0 i → ∞ k=1k=1 j=1 k=1 j=1 asN→ ∞sinceM D <1. Moreover, sinceQandDcommute, one can show that Q k ∼k −2D+I Qthat isk 2D−I Q k tends tocQwhenktends to infinity. The main result of this paper is the following theorem.

Theorem 3.1.Assume that the linear filter(u k )satisfies(3.3)and covariance operatorQcommutes withD. SetH= 3 2 I−D. Then c(D)n −H ζ− D −−→B H,Q inC([0,1];H),(3.4) n n→∞ whereB H,Q is an operator fractionalQ-Brownian motion with operator Hurst index Hand the operatorc(D)∈L(H)is defined by c 2 (D) = Z ∞− ∞ β 3( (2( − λ 2 − λ (11) 1)( − − λ λ )) ) dE λD .

The proof of this result is given in Section 5. It is deduced from more general functional central limit theorem stated and proved in the next section.

Remark 3.1.In the scalar case (H=R) where theu k ’s are real numbers, more general results are known. Konstantopoulos and Sakhanenko[14]proved the weak convergence of a step partial sums process build on theX k ’s to a fractional Brown- ian motion with Hurst indexH∈(1/2,1], assuming thatVarS n is regularly varying with exponent2H, a condition which is also necessary. Recently, Dedecker, Mer- levède and Peligrad[6]extended this result to a large class of linear processes with dependent innovations.

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4. Auxiliary result

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Assume that the sequence(Z n ) n≥1 of random elements in the spaceC([0,1];H), has a representation Z n (t) = X a nk (t)ε k , t∈[0,1], n≥1,(4.1) k∈Z where(ε j , j∈Z)is a sequence of i.i.d. random elements inHwith mean zero, finite second momentσ 2 0 =E||ε 0 || 2 = 1and covariance operatorQand for eacht∈[0,1] and eachn≥1,(a nk (t), k∈Z)is a sequence inL(H). Let(Z Q (t),t∈[0,1])be aC([0,1];H)-valued mean zero Gaussian random pro- cess with covariance kernelK Q (s, t): EZ Q (s)⊗Z Q (t) =K Q (s, t), s, t∈[0,1]. Set K n (s, t) := X a nk (t)Qa ∗ nk (s), s, t∈[0,1]. k∈Z We are interested in the convergence in distribution of the sequence(Z n , n≥1)to the processZ Q . Theorem 4.1.Assume that the following conditions are satisfied:

(C0)lim n→∞ sup j∈Z ||a nj (t)||= 0for eacht∈[0,1]; (C1)lim sup n→∞ P j∈Z ||a nj (t)|| 2 <∞for eacht∈[0,1]; (C2)there are constantsβ∈(1/2,1]andc >0such that lim sup X ||a nk (t)−a nk (s)|| 2 ≤c|t−s| 2β for all n→∞ k∈Z (C3)lim n→∞ ν 1 (K n (s, t)−K Q (s, t)) = 0for alls, t∈[0,1].

s, t∈[0,1];

Then Z n −− D −→Z Q inC([0,1];H).(4.2) n→∞ In the next section we shall apply this theorem witha nj satisfyinga nj (0) = 0, n≥1,j∈Z, in which special case Condition (C1) is an immediate consequence of (C2).

ClassicallyZ n converges weakly toZ Q inC([0,1],H)if and only if a) the “finite dimensional” distributions ofZ n converge to those ofZ Q ; b) the sequence(Z n ) n≥1 is tight inC([0,1];H). It is worth noticing here that the expression “finite dimensional” used to keep the analogy with the classical settingH=R, may be misleading. The meaning of a) is that the following convergence holds true for anyd≥1and any choice ofddifferent realt 1 , . . . , t d ∈[0,1]: (Z n (t 1 ), . . . , Z n (t d ))−− D −→(Z Q (t 1 ), . . . , Z Q (t d ))inH d , n→∞