Functional quantization of Gaussian processes

Functional quantization of Gaussian

processes

Harald Luschgy

and Gilles Page`s

*

a_{FB IV, Mathematik, Universta¨t Trier, D-54286 Trier, BR Deutschland} b_{Labo. Probabilite´s et Mode`les ale´atoires, Universite´ Paris 6, Case 188, 4, Pl. Jussieu,}

UMR 7599, F-75252 Paris, Cedex 05, France

Received 10 January 2002; received in revised form 2 April 2002; accepted 8 April 2002

Abstract

Quantization consists in studying theLr_{-error induced by the approximation of a} random vectorXby a vector (quantized version) taking a finite numbernof values. ForRm-valued random vectors the theory and practice is quite well established and in particular, theasymptotics asn-Nof the resulting minimal quantization error

for nonsingular distributions is well known: it behaves like cðX;r;mÞn1=m_{: This} paper is a transposition of this problem to random vectors in an infinite dimensional Hilbert space and in particular, to stochastic processes ðXtÞ_tA½0;1 viewed as L2_{ð½0;1;dtÞ-valued random vectors. For Gaussian vectors and the L}2_{-error we}

present detailed results for stationary and optimal quantizers. We further establish a precise link between the rate problem and Shannon–Kolmogorov’s entropy ofX: This allows us to compute the exact rate of convergence to zero of the minimal

L2_{-quantization error under rather general conditions on the eigenvalues of the}

covariance operator. Typical rates are OððlognÞaÞ; a>0: They are obtained, for instance, for the fractional Brownian motion and the fractional Ornstein–Uhlenbeck process. The exponentais closely related with theL2_{-regularity of the process.}

r2002 Elsevier Science (USA). All rights reserved.

MSC:60E99; 60G15; 94A24; 94A34

Keywords:Quantization of probability distribution; Gaussian process; Shannon–Kolmogorov entropy; Fractional Brownian motion; Stationary processes

*Corresponding author.

E-mail addresses:[email protected] (H. Luschgy), [email protected] (G. Page`s).

0022-1236/02/$ - see front matterr2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 1 2 3 6 ( 0 2 ) 0 0 0 1 0 - 1

1. Introduction

LetX be a random vector in a real separable Hilbert spaceHwith scalar

product /;S and norm jj jj: For nAN and 0oroN; the n-level

Lr_{-quantization problem for X consists in minimizing}

E min

aAa

jjXajjr

over all setsaCH withjajpn:Theminimal nth quantization error is then

defined by en;rðXÞ ¼inf E min aAa jjXajj r 1=r :aCH; 1pjajpn ð1:1Þ under the integrability condition

EjjXjjroN: ð1:2Þ

In fact, the infimum in (1.1) holds as a (finite) minimum under (1.2), see Proposition 2.1.

Let aCH bea finitesubset with jajpn: One easily shows that the best

approximation ofX by ana-valued random vector is achieved by applying

the rule of the nearest neighbour which corresponds to the geometric object called Voronoi partition. So, if

f ¼X

aAa

a1Aa; ð1:3Þ

wherefAa:aAagis a Borel measurable partition ofH such that, for every

aAa;A_a is contained in the (closed and convex) Voronoi region

WðajaÞ ¼ xAH :jjxajj ¼min

bAa jjxbjj ; then EjjXfðXÞjjr_{¼E min} aAa jjXajjr_:

Functionf is called thenearest neighbour n-quantizerofa:Thus onearrives

at the representation en;rðXÞ ¼inf f ðEjjXfðXÞjj r_Þ1=r_¼inf Y ðEjjXYjj r_Þ1=r_{; ð1:4Þ}

where the first infimum is taken over all n-quantizing rules f; i.e., Borel

measurable mapsf:H-H withjfðHÞjpnand the second infimum is taken

over allH-valued random vectorsYwithjsuppðPY_Þjp_{ndefined on the same}

probability space O as X: Notethat thequantizing rulef is a purely

geometric object whereasen;rðXÞonly depends upon the distribution ofX:

Quantization of probability distributions on H ¼Rm _{is a very old story}

which starts in the early 1950s. The idea was to use a finite number ofn

codes (or quantizers) to transmit efficiently a continuous stationary signal (see [11]) for a recent overview of applications. Then it was essential to

evaluate the resulting error and to optimize the quantizers. It is easy to show (see [12] or [17]) that the error or distortionðEminaAajjXajjrÞ1=rreaches a

minimum at some n-optimal quantizer and that en;rðXÞ goes to zero as

n-N:The main result concerning the minimal quantization error in the

finite dimensional setting is the Zador Theorem from 1963 that rules the

exact rate of convergence of en;rðXÞ to zero. (The general version given

below was stated later by Bucklew and Wise in [10] and the complete proof can befound in [12].)

Theorem 1.1 (Zador, see Graf and Luschgy [12]). Assume that H ¼Rm _is

equipped with the Euclidean l2-norm and that EjjXjjrþdoNfor some d>0:

Then if h denotes the Lebesgue-density of the absolutely continuous part of P¼PX _(possibly,h¼0), lim n-Nn 1=m_e n;rðXÞ ¼qrðmÞ Z hðxÞm=ðmþrÞdx ðmþrÞ=mr ;

where qrðmÞ is a strictly positive finite constant depending only on r and the

dimension m:

TheconstantqrðmÞ corresponds to the case of uniform distributions on

sets whose Lebesgue measure is 1 (e.g.½0;1d_{). Except in dimensionm¼1 or}

m¼2;its truevalueis unknown (actually

qrð1Þ ¼ 1 2ðrþ1Þ1=r; q1ð2Þ ¼ 2þ3 logðpffiffiffi3Þ 37=4pffiffiffi₂ ; q2ð2Þ ¼ 5 18pffiffiffi3 !1=2

and in general, qrð2Þ is the rth root of thenormalizedrth moment of the

regular hexagon). However, some upper bounds can be obtained, using random quantization or latticequantization (see[7,12]).

IfPis singular, Theorem 1.1 shows thaten;rðXÞ ¼oðn1=mÞ:There is some

recent progress on the rate problem for such probabilities (see [12–14]). The main result (at the moment) concerns self-similar probabilities. In order to formulate rates, it is convenient to use the symbolsBandE;whereanBbn

meansan=bn-1 and anEbn meansan¼OðbnÞ andan¼OðbnÞ:

Theorem 1.2 (Graf and Luschgy [13]). Assume H¼Rm_:LetðS1;y;S_NÞbe

an iterated function system consisting of contractive similitudes Si:Rm-Rm

with contraction numbers siAð0;1Þwhich satisfies the usual open set condition

(or Moran’s condition):

(OCRm; open set; such that

S

1pipN SiðOÞCO;

8iaj; SiðOÞ-SjðOÞ ¼|: (

Letðp1;y_;pNÞ_{be a probability vector with pi>0for all i:If P}¼PX _denotes

the self-similar probability corresponding to(S1;y;SN;p1;y;pNÞ;then

en;rðXÞEn1=Dr as n-N;

where Dris the unique number inð0;msatisfyingPiN¼1 ðpisriÞ

Dr=ðrþDrÞ_¼1:

The idea of quantization is enlightened by the following result ([17] or [12]) which shows how an optimal quantizer asymptotically approximates

theoriginal distributionP¼PX_:

LetðanÞnX1be a sequence ofn-optimal sets of orderrX1 forX:Then the

weighted empirical measure P_aAan PðAaÞda weakly converges toward P;

where fAa:aAa_ng is any Voronoi partition of Rm with respect to a_n:

Furthermore, for every Lipschitz continuous functionF:Rm-R_;

X aAan PðAaÞFðaÞ Z Rm F dP p½F1en;rðXÞ:

where ½F₁ denotes the Lipschitz constant of F: Theaboveerror bound

holds for r-Ho¨lder functions with ½F_ren;rðXÞ as a left-hand term when

0oro1:Furthermore, if F is continuously differentiable with a Lipschitz

continuous derivative, it holds for a sequence of n-optimal quantizers of

orderr¼2 with½F0

1en;2ðXÞ2:

For a general introduction to quantization for probability measures on

Rm_{;one may consult the recent monograph by Graf and Luschgy [12] and}

the references therein. Beyond the classical applications to Signal Processing and Information Theory (see [8,9]), quantization seems to be a promising tool in some recent developments in Numerical Probability (see [1,2,17] or [3]).

The first basic properties of the quantization problem on H ¼Rm _can

straightforwardly be extended to infinite dimensional spacesH:This remark

yields a natural clue to define a notion of functional quantization for stochastic processes. The idea is simply to consider a bi-measurable (real) processðXtÞtA½0;1d with samplepaths in H¼L2ð½0;1d;dtÞa.s. asH-valued random vector.

This leads us to initiate in Section 2 some first elements of an abstract quantization theory for probability measures on a Hilbert space. With only

a few exceptions we concentrate throughout on the quadratic caser¼2:We

provide basic facts about the existence of optimal quantizers, stationarity and smoothness properties and the reduction of the quantization problem to

finite dimensional subspaces ofH:

Sections 3 and 4 are devoted to Gaussian random vectors. In Section 3 we

characterize the linear subspaces ofH spanned by stationary and optimal

quantizers extending results of Tarpey et al. [20] to an infinite dimensional setting. This is an important infinite dimensional issue and only of limited

interest in finite dimensions. In Section 4 we investigate the rate of convergence ofen;2ðXÞto zero asn-N:Here the asymptotic behaviour of

en;2 is more complex than in finite dimensions. One point of this paper is to

link thebehaviour of en;2ðXÞ to Shannon–Kolmogorov’s E-entropy of the

random vectorX:This connection is rather simple but it links two delicate

topics in a useful way. That is, entropy results regardingX will yield lower

bounds on therateofen;2ðXÞ:Combining with a ‘‘product quantizer’’ upper

bound, this allows to computetheexact rateas en;2ðXÞEcðlognÞ1=2 as n-N

in case the eigenvalues of the covariance operator of X areregularly

varying, wherecis an increasing, regularly varying function related to the

eigenvalues, see Theorem 4.12. The same arguments also yield the true rateofen;2ðXÞin special cases where the eigenvalues are rapidly decreasing,

see Corollary 4.13(c).

In Section 5 we apply these results to functional quantization for

Gaussian processes. For the fractional Brownian motion Br _{with Hurst}

exponentrAð0;1Þ;weshow that

en;2ðBrÞ ¼OððlognÞrÞ:

Similar upper bounds are obtained for the fractional integrated Brownian motion and a wide class of Gaussian stationary processes. For the fractional Brownian motions, this rateis shown to bethetrueone. Exact rates are

also derived for the once-integrated Brownian motion, Brownian

bridge, Brownian sheet and the fractional stationary Ornstein–Uhlenbeck process.

2. Quantization for measures on a Hilbert space

Let X bea H-valued random vector with distribution P satisfying the integrability condition (1.2). Then,

lim

n-Nen;r

ðXÞ ¼0: ð2:1Þ

As a matter of fact, the Hilbert spaceH being separable there exists a

sequenceðynÞnX1 everywhere dense in H:It is clear that

0per_n;rðXÞ_pE min

1pipn jjXyijj

r_{-0 as n-N}

by the Lebesgue dominated convergence theorem. On the other hand, the existence of optimal quantizers, i.e. the fact thaten;rðXÞactually stands as a

2.1. Optimal and stationary quantizers 2.1.1. Existence of optimal quantizers

A se taCH with 1pjajpnis calledn-optimal set of centersforX(of order

r) if

en;rðXÞr¼E min

aAa jjXajj

r :

The first results of existence for optimal quantizers are due to Cuesta– Albertos and Matra´n [8] and Parna¨ [18] in thelate1980s. Dueto the importance of these objects for our purpose, we provide here a short and self-contained proof.

Proposition 2.1. Assume that (1.2) holds. For every r>0 letCn;rðXÞdenote

the set of all n-optimal sets of centers.

(a)For every nAN;the setC_n;rðXÞis not empty.

(b) If jsuppðPÞjXn; then, for every aAC_n;rðXÞ; jaj ¼n;e_n;rðXÞoe_n1;rðXÞ

and for every aAa;PðW3ðajaÞÞ>0 (3is for interior).IfjsuppðPÞjis finite,then

for every nXjsuppðPÞj;e_n;rðXÞ ¼0 andsuppðPÞAC_n;rðXÞ:

Proof. Thekey of theproof is that functionFndefined onHn by

Fnða1;y;a_nÞ ¼E min

1pipn jjXaijj r

is weakly sequentially lower semi-continuous (Fn is theso-called distorsion

function).

LetaðkÞ:¼ ðað₁kÞ;y;a_nðkÞÞ,x:¼ ða1;y;a_nÞinHnwhere,is for (product)

weak convergence on Hn_{: For every iAf1;y;ng; jja}

iXjjrp

lim infn jjaðikÞXjjr:Hence

min 1pipn jjaiXjj r_{p min} 1pipnjja ðkÞ i Xjjr¼lim inf k 1minpipnjja ðkÞ i Xjjr:

Finally, taking the expectation and calling upon Fatou’s Lemma yields

FnðxÞpE lim inf k 1minpipnjja ðkÞ i Xjjrplim inf k Fkða ðkÞ_Þ:

(a) One proceeds by induction on n: If n¼1; let c>0 such that theset

fF1pcg is not empty. One checks that F1ðhÞX₂rjjhjjrEjjXjjr_:

Conse-quently,fF1pcgis a weakly compact set on whichFnachieves its minimum.

Now, assumethat argminFna| and let aðnÞ

AargminFn: Either

suppðPÞCfaðnÞ

i ;1pipng and the nþ1-tuple ða

ðnÞ

1 ;y;aðnnÞ;aðnnÞÞAargminFn

(among infinitely many others); or there existsanþ1AsuppðPÞ\faðnÞ

1 ;y;aðnnÞg:

Set aðnþ1Þ_{:¼ ða}ðnÞ

1 ;y;a

ðnÞ

n ;anþ1Þ: Since Wða˚ nþ1jaðnþ1ÞÞ is a nonempty open

the event fXAW 3 ðanþ1jaðnþ1ÞÞg; min 1pipnþ1 jja ðnþ1Þ i Xjjr¼ jjanþ1Xjjro min 1pipn jja ðnÞ i Xjjr;

whereas min1pipnþ1jjaðinþ1ÞXjjrpmin1pipnjjaiðnÞXjjr everywhere.

Subse-quently,Fnþ1ðaðnþ1ÞÞoFnðaðnÞÞ ¼minFn:

It follows that thesetfFnþ1ominFngis not empty. Hence, there exists a

real numbercominFnsuch thatFnþ1:¼ fFnþ1pcgis a nonempty (weakly)

closed set. Furthermore, it is obvious that any nþ1-tuple a in Fnþ1 has

pairwisedistinct components (if notFnþ1ðaÞXminFn).

Next step is to prove that Fnþ1 is bounded in Hnþ1: Otherwise, let

ak_AF

nþ1;kX0;be a sequence such that max_1pipnjak_ij ¼ þN:Up to at most

nþ1 extractions of subsequences, there is some subset ICf1;y;nþ1g;

jIjX1;such that

ak_i,aN

i ; iAI and lim k ja

k

ij ¼ þN; ieI:

The weak lower semi-continuity of the norm and Fatou’s Lemma imply that cXlim inf k Fnþ1ða k_{ÞXE lim inf} k 1minpipn jja k i Xjj r XE min iAI jja N i Xjjr XminFn_þ1jIj>c

hence the contradiction. Consequently,Fnþ1is weakly compact.Fnþ1being

weakly lower semi-continuous, so it reaches its minimum onFnþ1:This is

clearly the absolute minimum ofFnþ1 on Hnþ1:

(b) LetaAC_n;rðXÞand aAa such that PðW3ðajaÞÞ ¼0:Now cW3ðajaÞ ¼

S

bAa\fagWðbjaÞso that FnðaÞXFn1ða\fagÞXminFn1 (with obvious

nota-tions sincefunctionFn is permutation symmetric). This is impossible since

jsuppðPÞjXn:Other claims are by-products of (a). &

Remark.

* _{Of course this result embodies the classical finite dimensional case. Then}

thedistortion is simply continuous on Hn_{: On theother hand, the}

extension of the above proposition to reflexive Banach spaces is straightforward.

* _{Thel.s.c. property of thedistortion functionF}

nadmits a kind of converse

whose easy proof is left to the reader as a curiosity: let ðxk_Þ

kX0 bea

sequence ofHn-valuedn-tuples.

ðxk-xN

* _{Let aAC}

n;rðXÞ: Any nearest neighbour n-quantizer f ¼PaAa a1Aa as

defined by (1.3) provides ann-optimal quantizer, i.e.,

en;rðXÞ ¼ ðEjjXfðXÞjjrÞ1=r:

* _{If r¼2 and n¼1; theonly 1-optimal centreis fEXg and e}

1;2ðXÞ ¼

ðEjjXEXjj2Þ1=2:

It is now time to justify why and how an element ofCn;rðXÞquantizes the

distributionP:

Corollary 2.2. LetðanÞnX1be a sequence of setsanCH withja_njpn such that

EminaAanjjXajjr-0 as n-Nand let fAa:aAa_ngbe a Voronoi partition

of H with respect toan:Then

(a)

Pn:¼ X aAan

PðAaÞda-P weakly: ð2:2Þ

(b) Furthermore, if rAð0;1 and sA½r;þNÞ_{; for every r-Ho¨lder continuous}

functional F:H-R_; X aAan PðAaÞFðaÞ Z H FdP p½FrE min aAan jjXajj r p½F_r E min aAan jjXajjs r s :

The proof is as simple as in the finite dimensional setting and is reproduced for the reader’s convenience.

Proof. (a) follows from (b). Let us prove (b). Froman weconstruct

then-quantizerfnðXÞ ¼PaAan a1AaðXÞforX:Then X aAan PðAaÞFðaÞ Z H F dP ¼ jEFðXÞ EF3fnðXÞj p½F_rEjjXfnðXÞjjr ¼ ½F_rE min aAan jjXajjr:

The second inequality follows from the monotonicity oft/jjfjj_Lt_ðPÞ: &

Similar error bounds involving en;rðXÞ for rX1 areavailablefor locally

Lipschitz functionals satisfying

jFðuÞ FðvÞj_p½F_rjuvjð1þ jujr1_{þ jvj}r1_Þ:

Item (b) shows how the quantization error rules the rate of convergence of

the weighted empirical measurePn toward theoriginal distributionP:This

achieve the best rate of convergence in (2.2). For the same reason it suggests to investigate what is this optimal rate of convergence.

It is useful to observe the following equivariance properties.

Lemma 2.3. Let H1 and H2 be Hilbert spaces and let X be a H1-valued

random vector satisfying EjjXjjr_{oN: If T:H}

1-H2 is a bounded linear

operator,then

en;rðTðXÞÞpjjTjjen;rðXÞ:

If T:H1-H2 is a bijective isometry and c>0;then

en;rðcTðXÞÞ ¼cen;rðXÞ and Cn;rðcTðXÞÞ ¼cTCn;rðXÞ: Proof. Let us prove e.g. the first assertion. LetaAC_n;rðXÞ:Then

en;rðTðXÞÞp E min aAa jjTðXÞ Tajjr 1=r pjjTjj E min aAa jjXajjr 1=r ¼ jjTjjen;rðXÞ: &

2.1.2. The quadratic case (r¼2)

From now on, we will deal with the quadratic quantization error, i.e. the

caser¼2 (squareroot of thesquareerror). So, for thesakeof simplicity, we

will denoteenðXÞforen;2ðXÞandCnðXÞforCn;2ðXÞ:

Next we provide necessary conditions forn-optimality of quantizers. The

proof, similar to the finite dimensional setting (see [12, Theorem 4.1]), is partially reproduced for the reader’s convenience.

Proposition 2.4. If aAC_nðXÞ; and jsuppðPÞjXn; then jaj ¼n;

minaAaPðW 3_ðaj aÞÞ>0 and EðXjfðXÞÞ ¼fðXÞ a:s: where f ¼X aAa a1WðajaÞ: ð2:3Þ

In particular,for every aAa;

a¼EðXjXAWðajaÞÞ: ð2:4Þ

Furthermore,for every a;bAa; aab;

PðWðajaÞ-WðbjaÞÞ ¼0: ð2:5Þ

Proof. Let fAa; aAag bea Voronoi partition of H with respect toa:Let

j:¼P_aAa a1Aa and B:¼sðjðXÞÞ ¼sðfXAAag; aAaÞ: Using that

aAC_nðXÞand thatjaj ¼nyields

EjjjðXÞ Xjj2¼minfEjjZXjj2; jZðOÞjpng

pminfEjjZXjj2; Z B-measurableg ¼EjjEðXjBÞ Xjj2_:

Hence, jðXÞ ¼EðXjBÞ ¼EðXjjðXÞÞ: In particular, a¼EðXjXAA_aÞ for

everyaAa:LetaAa;(2.4) follows by choosing a Voronoi partition such that

Aa ¼WðajaÞ:

Concerning (2.5), one may choose another Voronoi partitionfA0_c; cAag

with respect to a such that A0_a¼WðajaÞ\WðbjaÞ: Notethat Aa\A0a¼

WðajaÞ_-WðbjaÞ: Then, it follows from the equality a¼EðXjXAA0_aÞ ¼

EðXjXAA_aÞand thestandard Bayes formula that

EðXjXAA_aÞ ¼EðXjXAA0 aÞ PðA0 aÞ PðAaÞ þEðX1fXAAa\A0agÞ PðAa\A0aÞ PðAaÞ :

Theonly way for this convex combination to hold is that

EðXjXAWðajaÞ-WðbjaÞÞ ¼EðXjXAA0

aÞ ¼a: A symmetric argument

shows that EðXjXAWðajaÞ-WðbjaÞÞ ¼b: Hence the contradiction since

aab:Finally (2.3) follows. &

A se t aCH satisfying jaj ¼n; min_aAaPðW

3_ðaj

aÞÞ>0; (2.4) and (2.5) is

called an-stationary setof means forX:Next corollary is obvious.

Corollary 2.5. Leta be a n-stationary set for X:We have

aCcl convðsuppðPÞÞ whereconv is for convex hull and cl is for closure;

ð2:6Þ EX ¼EðEðXjfðXÞÞÞ ¼E fðXÞ ¼X

aAa

aPðWðajaÞÞ:

2.1.3. First applications to functional quantization

The main interest of Proposition 2.4 for our purpose is that a stationary set necessarily lies in a very specific subspace ofH:Namely, ifEX ¼0;it lies in the reproducing kernel Hilbert space (or Cameron–Martin space) of the

covarianceoperator of X: This operator CX:H-H of X is defined by

CXy¼E/y;XSX: CX is a symmetric positive trace class operator. The

reproducing kernel Hilbert spaceKX is a subspaceofH that can be defined

as follows:

KX :¼ fEðZ XÞ:ZAcl_L2_ðPÞf/y;XS:yAHgg

¼ fEðgðXÞXÞ:gAcl_L2_ðPÞf/y; :S:yAHgg:

ThesetKX is equipped with the inner product

/_k1;k2S

X :¼EðZ1Z2Þ if ki¼EðZiXÞ; i¼1;2

so that ðKX;/:SÞ is a Hilbert space, isometric with the Hilbert space

clf/y;XS:yAHg: It is then straightforward that K_X is spanned as a

does not enlargeKX so that

KX¼ fEðgðXÞXÞ:gAL2ðPÞg: ð2:7Þ

Furthermore, we haveKX ¼CX1=2ðHÞ:

For everyy;zAH;one has using the Fubini Theorem

/_Eð/_y;XS_XÞ;Eð/_z;XS_XÞS

X¼Eð/y;XS/z;XSÞ

¼/Eð/y;XSXÞ;zS which in turn yields the so-called reproducing property:

/_k;CXyS

X ¼/k;yS; kAKX; yAH: ð2:8Þ

For these subjects see [5,21].

Proposition 2.6. If EX¼0 and aCH is a n-stationary set for X; then

aCKX:

Proof. By definition,

a¼EðXjXAWðajaÞÞ ¼EgðXÞX;

where g¼1WðajaÞ=PðWðajaÞÞAL2ðPÞ; aAa: Theassertion follows from

(2.7). &

The above proposition indicates that in a stochastic process setting the components of a stationary quantizer have certain smoothness properties. In particular, they have at least the same regularity as that of the processX in L1_{ðPÞ:In fact, consider the Hilbert space H¼L}2_{ðI;dtÞ with I¼ ½0;1}d

and a bi-measurable centered L2_{ðPÞ-process X ¼ ðX}

tÞtAI with paths in

L2_{ðI;dtÞ a.s. and covariancefunction GXðs;tÞ:¼EX}

sXt satisfying R

I GXðs;sÞdsoN:ThenX can be seen as aH-valued random vector with

EjjXjj2_oN;

CXy¼ Z

I

yðsÞGXðs;Þds; yAL2ðI;dtÞ;

and anyyAK_X admits a version (namelyt/EgðXÞX_t ify¼EgðXÞXÞthat

satisfies

jyðsÞ yðtÞjpjjyjj_XðEjXsXtj2Þ1=2 for alls;tAI: ð2:9Þ

Since the components of any stationary seta have a representation with a

bounded functiongAL2ðPÞ(cf. (2.7)), everyaAa admits a version (namely

t/EðXtjXAWðajaÞÞthat satisfies

jaðsÞ aðtÞjpPðWðajaÞÞ1EjXsXtj for all s;tAI: ð2:10Þ

The following facts about the reproducing space in that framework will be

functionsGXðs; :Þ; sA½0;1;lieinK_X:Furthermore,

KX¼cl spanfGXðs; :Þ:tA½0;1g ð2:11Þ

since, for everyfAfGXðs; :Þ; sA½0;1g>KX;the reproducing property implies

that

jjfjj_L2_ðI;dtÞ¼/f;C_XðfÞS_X ¼ Z 1

0

fðtÞ/f;GXðt; :ÞSXdt¼0:

2.2. Finite dimensional subproblems

Now wediscuss thereduction of thequantization problem to finite

dimensional subspaces ofH:For any finite dimensional linear subspaceU

ofH;letPUdenote the orthogonal projection fromHontoU:According to

(2.6) it makes no difference forenðPUðXÞÞwhetherPUðXÞis considered as

U-valued orH-valued random vector. Let us start by an easy proposition connecting both quadratic quantization errorsenðPUðXÞÞandenðXÞ: Proposition 2.7. Let U be a finite dimensional linear subspace of H:Then

enðPUðXÞÞ2penðXÞ2pinf E min aAa jjXajj2_:a CU;1pjajpn n o ¼EjjXPUðXÞjj2þenðPUðXÞÞ2:

Proof. LetbAC_nðXÞ:Then, the first inequality follows from

enðPUðXÞÞ2pE min bAb jjPUðXÞ PUðbÞjj2_{pE min} bAb jjXbjj2_¼e nðXÞ2:

The second inequality is obvious. LetaCU:Theequality follows from the

decomposition E min

aAa

jjXajj2_{¼EjjXPUðXÞjj}2_{þE min} aAa

jjPUðXÞ ajj2_{: &}

We see that the quadratic quantization error with respect toaCUconsists

of the projection error and the quantization error of the projected random vector.

Let us introduce the integral number

dnðXÞ ¼minfdim spanðaÞ:aAC_nðXÞg: ð2:12Þ

It represents the dimension of the levelnof thequantization problem forX:

Here spanðaÞ denotes the linear subspace spanned by a: It follows from

Proposition 2.7 that

e2_nðXÞ ¼minfEjjXPVðXÞjj2þe2_nðPVðXÞÞ:VCH

The following equivalence is a further immediate consequence of Proposition 2.7.

Corollary 2.8. Let U be a finite dimensional linear subspace of H andaCU:

The following statements are equivalent: (i) aAC_nðXÞ:

(ii) aAC_nðP_UðXÞÞand e_nðXÞ2¼EjjXP_UðXÞjj2þe_nðP_UðXÞÞ2:

The following remark contains an elementary fact aboutdnðXÞ:For the

asymptotic behaviour of dnðXÞ see remark (c) following Corollary 4.13

further on.

Remark. IfPis not concentrated on a finite dimensional linear subspace of H;then

sup

nX1

dnðXÞ ¼N:

As a matter of fact, assume dN:¼sup_nX1dnðXÞoN: Then by (2.13), for

everynAN

enðXÞ2XinffEjjXPVðXÞjj2:VCH linear subspace; dimV ¼dNg ¼EjjXPUðXÞjj2

for somesuitabledN-dimensional subspace U: It follows from (2.1) that

EjjXPUðXÞjj2_{¼0:This yieldsPðUÞ ¼1;a contradiction.}

2.3. Product quantizer upper bound

One natural question to investigate is the rate of convergence ofenðXÞto

zero. In finite dimension, the problem has been fully elucidated for nonsingular probability measures by Theorem 1.1 and for self-similar measures by Theorem 1.2.

We will use estimates in finite dimension and Proposition 2.7 to obtain some first estimates in infinite dimension based only on one-dimensional quantization problems. These bounds use optimal product quantizers or orthogonal grids (see [12,17]).

We need the following simple fact: let fu1;y;u_mg bean orthonormal

subset of H; U¼spanfu1;y;u_mg; Z¼ ð/u1;XS;y;/um;XSÞ and let

T:U-Rm _{be the bijective linear isometry given byTuj}¼bj;1pjpm;for thestandard basisfb1;y_;bmg _ofRm_;then

T3P_UðXÞ ¼X

m

j¼1

Hence by Lemma 2.3,

enðPUðXÞÞ ¼enðZÞ and TCnðPUðXÞÞ ¼CnðZÞ; ð2:14Þ

whereenðZÞdenotes the nth quantization error ofZwith respect to thel2

-norm onRm_:

Proposition 2.9. Assume (for simplicity)that EX ¼0:Let fuj:jX1g be an

orthonormal subset of H such that suppðPÞCcl spanfu_j :jX1g: Then, for

every n and every mAN;

enðXÞ2p X jXmþ1 Var/uj;XS þinf X m j¼1 enjð/uj;XSÞ 2_:n 1;y;nmAN; Ym j¼1 njpm ( ) :

Proof. Let U¼spanfu1;y;umg; Z¼ ð/u1;XS;y;/um;XSÞ:Using

Pro-position 2.7 and (2.14) yield, enðXÞ2p X jXmþ1 E/uj;XS2þenðPUðXÞÞ2 ¼ X jXmþ1 Var/uj;XSþenðZÞ2:

Now for njAN with Qm

j¼1njpn oneconsiders ajAC_njð/u_j;XSÞ and the

product quantizera¼#m j¼1aj:Oneobtains enðZÞ2pEmin aAa jjZajj 2_¼X m j¼1 E min bAaj j /_uj;XS_bj2 ¼X m j¼1 enjð/uj;XSÞ 2 : &

3. Quantization for Gaussian measures

In this section this section let X be a centred H-valued random vector

with Gaussian distribution P: Sincewewish to investigatetheinfinite

dimensional situation, we assume throughout that dimKX ¼N:Notethat

suppðPÞ ¼clðKXÞ:

In the Gaussian case Proposition 2.6 can be improved considerably.

Theorem 3.1. Let aCH be a n-stationary set of means for X and let U¼

spanðaÞ:ThenPUðXÞ and XPUðXÞare independent so that CXðUÞ ¼U:

In particular,aCC_XðHÞCK_X:

The proof is given below. Theorem 3.1 shows that linear subspacesU of

H spanned by n-stationary sets correspond to principal components of X;

Observe that by Corollary 2.5

dnðXÞpd%nðXÞ :¼maxfdim spanðaÞ:aAC_nðXÞg

pmaxfdim spanðaÞ:an-stationary for Xg

pn1: ð3:1Þ

In order to deal withn-optimal sets of means, letl1Xl2X?>0 bethe

ordered nonzero eigenvalues of CX (each written as many times as is its

multiplicity) and notethatEjjXjj2¼PN

j¼1lj:

Theorem 3.2. LetaAC_nðXÞ;U¼spanðaÞand m¼dimU:Then C_XðUÞ ¼U

and

EjjXPUðXÞjj2_¼ X jXmþ1

lj:

The proof is given below. Observe that

X jXmþ1

lj¼inffEjjXPVðXÞjj2:VCH linear subspace;dimV ¼mg:

Theorem 3.2 shows that m-dimensional subspaces of H spanned by

n-optimal sets of means are spanned by eigenvectors ofCX which belong to

them largest eigenvalues. Thus these subspaces correspond to the first m

principal components ofX:For finite dimensional Hilbert spaces Theorems

3.1 and 3.2 were derived by Tarpey et al. [20]. However, the theorems obviously achieve their full strength only in the infinite dimensional setting.

Let us deduce the final representation ofenðXÞand thecharacterization of

CnðXÞ:It follows from Theorem 3.2 and (2.14) in view of Proposition 2.7

and Corollary 2.8 that enðXÞ2¼ X jXmþ1 ljþen # m j¼1 N ð0;ljÞ 2 for mXd_nðXÞ; enðXÞ2o X jXmþ1 ljþen # m j¼1 Nð0;ljÞ 2 for 1pmodnðXÞ: ð3:2Þ

Concerning CnðXÞ; let fuj:jANg bean orthonormal basis of clðK_XÞ

consisting of eigenvectors ofCX such thatCXuj¼ljuj; jAN:FornX2;set

s¼d%nðXÞ and let r¼rn¼minfjXs:lj>ljþ1g; U¼spanfu1;y;urg and

T:U-Rr _{the corresponding isometry. Then}

CnðXÞ ¼T1Cn # r j¼1 N ð0;ljÞ : ð3:3Þ

This follows again from Theorem 3.2, (2.12) and Corollary 2.8. Notice that

Example 3.3. (a) Letn¼2:Wehaved2ðXÞ ¼d%2ðXÞ ¼1 and

e2ðXÞ2¼ X jX2

ljþe2ðNð0;l1ÞÞ2:

Let r bethemultiplicity of l1: Since e2ðNð0;l1ÞÞ2¼l1ð12=pÞ and

C2ð#r1Nð0;l1ÞÞ ¼ ffb;bg:bARr;jjbjj ¼ ð2l₁=pÞ1=2g (cf. [12, Example 4.20]) weobtain e2ðXÞ2¼EjjXjj2 2l1 p ¼e1ðXÞ 22l1 p and

C2ðXÞ ¼ ffa;ag:aAspanfu₁;y;u_rg; jjajj ¼ ð2l₁=pÞ1=2g:

(b) LetX ¼ ðXtÞtA½0;1beBrownian motion andH¼L

2_{ð½0;1;dtÞ:Then} lj ¼ ðpðj1 2ÞÞ 2 ; ujðtÞ ¼ ffiffiffi 2 p sinðt=pffiffiffiffiljÞ; jX1:

Sincee1ðXÞ2¼EjjXjj2¼12;onederives from (a)

e2ðXÞ2¼ 1 2 8 p3¼0:2419y and jC2ðXÞj ¼1; C2ðXÞ ¼ ff7ð8=p3Þ1=2u1gg:

(c) LetX¼ ðXtÞtA½0;1beBrownian bridgeandH ¼L2ð½0;1;dtÞ:Then

lj ¼ ðpjÞ2; ujðtÞ ¼ ffiffiffi 2 p sinðpjtÞ; jX1 andEjjXjj2¼1 6which yields e2ðXÞ2¼ 1 6 2 p3¼0:1021y and jC2ðXÞj ¼1; C2ðXÞ ¼ ff7ð2=p3Þ1=2u1gg:

We come to the proofs of both theorems.

Proof of Theorem 3.1. Set V¼U>: ThecoupleðPUðXÞ;PVðXÞÞ has a

Gaussian joint distribution. Westill denotef :¼P_aAaa1WðajaÞthestationary

quantizer associated toa:

OnehasfðXÞ ¼EðXjfðXÞÞ ¼EðPUðXÞjfðXÞÞ þEðPVðXÞjfðXÞÞ:Hence EðPVðXÞjfðXÞÞ ¼fðXÞ EðPUðXÞjfðXÞÞAV-U¼ f0g;

i.e.EðPVðXÞjfðXÞÞ ¼0:

On the other hand, for every aAa; jjPUðXÞ ajj2¼ jjXajj2

jjPVðXÞjj2; hence PUðXÞAWðajaÞ-U if and only if XAWðajaÞ:

Therefore

Now, for every yAH; the conditional expectation of /y;P_VðXÞS given

PUðXÞcoincides with the linear regression, i.e. there existslyAH such that

Eð/y;PVðXÞSjPUðXÞÞ ¼/ly;PUðXÞS:

Now,fðPUðXÞÞisPUðXÞ-measurable andEðPUðXÞjfðPUðXÞÞÞ ¼fðPUðXÞÞ

so that

Eð/y;PVðXÞSjfðPUðXÞÞÞ ¼/ly;fðPUðXÞÞS and

Eð/y;PVðXÞSjfðPUðXÞÞÞ ¼Eð/y;PVðXÞSjfðXÞÞ ¼/_y;EðPVðXÞjfðXÞÞS¼0:

It follows that /y;fðXÞS¼/y;fðPUðXÞÞS¼0 a.s. This implies that

yAa>¼Usincemin_aAaPðWðajaÞÞ>0:Consequently, for every yAH

Eð/y;PVðXÞSjPUðXÞÞ ¼0

which in turn implies that PUðXÞ and PVðXÞ are independent since they

havea Gaussian joint distribution. HenceCXðUÞCUsince, for everyyAU

CXðyÞ ¼Eð/y;XSXÞ ¼Eð/y;PUðXÞSXÞ

¼Eð/y;PUðXÞSPUðXÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} AU þEð/y;PUðXÞSPVðXÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0 AU:

If CXðyÞ ¼0 for some yAU; then E/y;XS2¼/C_XðyÞ;yS¼0; i.e.

/_y;XS_{¼0 a.s. But then} /_y;fðXÞS_¼Eð/_y;XS_{jfðXÞÞ ¼0 a.s. which in}

turn implies yAa> sincemin_aAaPðWðajaÞÞ>0: Hence y¼0 which

completes the proof. &

Proof of Theorem 3.2. By Proposition 2.4 and Theorem 3.1, we have CXðUÞ ¼U: Therefore, there exists an orthonormal basis fuj:jANg of

clðKXÞconsisting of eigenvectors ofCX such thatU¼spanfu1;y;u_mg:Let

m_j; jANbe the corresponding (unordered) eigenvalues of C_X;i.e.,C_Xu_j¼

mjuj for alljAN:Then

EjjXPUðXÞjj2_¼ X jXmþ1

mj:

Setxj¼m

1=2

j /uj;XS; jAN: ThenðxjÞjX1 is an i.i.d. sequence ofNð0;

1Þ-distributed random variables. Consequently, X ¼X N j¼1 ffiffiffiffi m_j p _x juj a:s: and LpHðPÞ; pX1:

Let f ¼P_aAaa1WðajaÞ: By (2.3), fðXÞ ¼EðXjfðXÞÞ since a is n-stationary.

Consequently, fðXÞ ¼X N j¼1 ffiffiffiffi m_j p _Z juj;

where Zj ¼EðxjjfðXÞÞ ¼m

1=2

j /uj;fðXÞS: WehaveZj¼0 a.s. if jXmþ1

sincefðXÞis U-valued andP_ðZ

ja0Þ>0 ifjpm:Now letsbea permutation

of N_{; that is, a bijective function from} N _{onto itself, with jfj}AN:

sðjÞajgjoN:Set Xs_¼X N j¼1 ffiffiffiffi m_j p _x sðjÞuj and fðXÞs¼EðXsjfðXÞÞ: NotethatXs¼d X , fðXÞs¼X N j¼1 ffiffiffiffi mj p _Z sðjÞuj¼:g13fðXÞAg₁ðaÞ a:s: and X ¼X N j¼1 ffiffiffiffiffiffiffiffiffiffiffiffi m_j m_s1_ðjÞ s /_u s1_ðjÞ;XsSu_j¼:g₂ðXsÞ: HencefðXÞs_¼g 13f3g₂ðXsÞ:It follows that EjjXsfðXÞsjj2¼EjjXg13f3g2ðXÞjj2 Xe_nðXÞ2¼EjjXfðXÞjj2 which reads XN j¼1 mjEjxsðjÞZsðjÞj2X XN j¼1 mjEjxjZjj2:

Now, settingsðjÞ ¼k; sðkÞ ¼j andsðrÞ ¼rfor refj;kg;1pjpmandk>

myields m_jþm_kEjxjZjj2XmjEjxjZjj2þmk; that is, ðm_jm_kÞð1EjxjZjj2ÞX0: Therefore, m_jXm_k since Ejx_jZ_jj2¼1EZ2 jo1: Thus theproof is complete. &

4. Rates of decay for the quantization error

LetX be a centred Gaussian random vector with values inH such that

dimKX ¼N:In this section we investigate the rate of convergence to zero

ofenðXÞunder various conditions on the eigenvalues or more generally, on

thevariances Var/uj;XS coming from an orthonormal basis fujg: For

solution of theproblem. ½x denotes the integral part of a number x and throughout all logarithms arenatural logarithms.

Let fuj :jANg denote any orthonormal subset of H such that

KXCcl spanfu_j:jANgand let

mj¼Var/uj;XS¼/uj;CXujS and Sm¼ ð/uj;CXukSÞ0pj;kpm: ð4:1Þ

Observe that detP_m>0 provided fuj :jANgCclðK_XÞ:FornAN;set

gnðmÞ ¼enðNð0;SmÞÞ; mX1:

In the finite dimensional Gaussian setting, Theorem 1.1 takes the following form.

Proposition 4.1. Assumefuj:jANgCclðK_XÞ:Then

lim n-Nn 1=m_g nðmÞ ¼QðmÞ for every mX1; where QðmÞAð0;NÞand QðmÞBðmðdetSmÞ1=mÞ1=2 as m-N_: In particular, limm-NQðmÞ ¼0:

Proof. The limiting statement forgnðmÞholds with coefficientQðmÞgiven by

QðmÞ ¼qðmÞpffiffiffiffiffiffi2pðdetSmÞ1=2m

mþ2 m

ðmþ2Þ=4 ;

where the constantqðmÞAð0;NÞ_satisfies

qðmÞB m 2pe

1=2

as m-N

(cf. [12, Theorem 6.2, Corollary 9.4]). Usingðm!Þ1=mBm=eand 1

m

Xm j¼1

jm_j-0

which follows from Kronecker’s lemma, the assertion for QðmÞ finally

follows from mðdetSmÞ1=mpm Ym j¼1 m_j !1=m ¼ m ðm!Þ1=m Ym j¼1 jm_j !1=m p m ðm!Þ1=m Pm j¼1jmj m -0: &

Remark. (a) Since enðXÞXgnðmÞ (cf. Proposition 2.6), an immediate

consequence of the former proposition is that enðXÞ decreases slower to

zero than any powerna; a>0:Indeed, if 1=moa;then naenðXÞXna1=mn1=mg_nðmÞ-N as n-N:

(b) Proposition 4.1 suggests the conjecture that gnðmÞ2Bn2=mmðdetSmÞ1=m

for suitablechoices ofm¼mðnÞ-Nprovidedfu_j:jANgCclðK_XÞ:

4.1. Upper bounds

We rely on the product quantizer bounds of Proposition 2.9 to get upper bounds for thenth quantization error: letnAN;then for everymAN;

enðXÞ2p X jXmþ1 m_jþgnðmÞ2 p X jXmþ1 m_jþinf X m j¼1 enjð/uj;XSÞ 2_:n 1;y;nmAN; Ym j¼1 njpn ( ) :

In the Gaussian case one can derive a simpler form of the above inequality. As a matter of fact,

/_uj;XS_BNð0;m jÞ and enðNð0;mjÞÞ2¼mjenðNð0;1ÞÞ2 so that Xm j¼1 enjð/uj;XSÞ 2_¼X m j¼1 m_jn_j 2ðn_j2enjðNð0;1ÞÞ 2_{Þ: ð4:2Þ}

Theorem 1.1 says thatk2_e

kðNð0;1ÞÞ2 converges to some finite limit when

k-Nso that

c0:¼sup kX1

k2ekðNð0;1ÞÞ2oN:

Hence, for every (fixed)nAN;

enðXÞ2pc0 inf mAN X jXmþ1 m_j þinf X m j¼1 m_jn_j2:n1;y;nmAN; Ym j¼1 njpn ( )! : ð4:3Þ Note in connection with the solution of the minimization problem (4.3) that, for real numbersn1;y;nm>0;

inf X m j¼1 njy_j2:yj>0; Ym j¼1 yjpn ( ) ¼X m j¼1 njz_j 2¼n2=mm Y m j¼1 nj !1=m ; where zj ¼n1=mn1j=2ð Qm j¼1njÞ 1=2m

: This follows from the

arithmetic–geo-metric mean inequality. Combining this observation with remark (b) following Proposition 4.1 we can expect that bound (4.3) does not increase the order, provided the orthonormal setfuj:jANgis suitably chosen.

In the sequel we assume that

m_j ¼OðnjÞ as j-N ð4:4Þ

for some decreasing sequence ðnjÞ_jX1 of numbers nj>0 satisfying

PN

j¼1 njoN: By c;c1;y we shall denote finite numerical constants not

depending on the quantization leveln:Now we can present the basic lemma.

Lemma 4.2. For nAN;let

I¼IðnÞ ¼ mX1:n2=mn_m Y m j¼1 nj !1=m X1 8 < : 9 = ;: ð4:5Þ

Then I is a nonempty finite set, I¼ f1;y;mng where mn¼mnðnÞ ¼_maxI

and enðXÞ2pcinf X jXmþ1 njþn2=mm Ym j¼1 nj !1=m :mAI 8 < : 9 = ;: ð4:6Þ Moreover,mn ðnÞincreases to Nas n-Nand mn

ðnÞ ¼OðlognÞ if lim infn-N Qn

j¼1nj

1=n

=nn >1;

mn

ðnÞ ¼OðlognÞ if lim sup_n-N Qn j¼1 nj 1=n =nnoN: 8 > < > :

Notethat theabovelim inf-condition is satisfied as soon as ðnnnÞ is

decreasing since ðQn_j¼1njÞ1=n nn ¼ nðQn_j¼1jnjÞ1=n ðn!Þ1=nnnn Beð Qn j¼1jnjÞ1=n nnn Xe:

The two inequalities below (valid for arbitrary numbers nn>0) are

sometimes useful:

lim infðnn=nnþ1Þnplim inf Yn j¼1 nj

!1=n =nn;

lim supðnn=nnþ1ÞnXlim sup Yn j¼1 nj

!1=n

=nn: ð4:7Þ

Proof of Lemma 4.2. Setting

an ¼ 1 2log Yn j¼1 nj=nn_n ! ¼n 2log Yn j¼1 nj !1=n =nn 0 @ 1 A

weseethat

IðnÞ ¼ fmX1:a_mplogng:

Onechecks that an is increasing in nAN: Moreover a_n increases to N as

n-Nsince Qn j¼1 nj nn n Xn1 nn-N:

Consequently,I is finite, 1AI andI¼ f1;y_;mng_{:Furthermore,}

amnplognoa_mn_þ1

for all nX1; which implies both assertions about the order of

mn

¼mn

ðnÞ:Now choosea constantc1 such thatmjpc1nj for everyj:Then

by (4.3), enðXÞ2pc2 X jXmþ1 njþinf X m j¼1 njn_j2:n1;y;n_mAN; Ym j¼1 njpn ( )!

for everymAN:For everymAI;set for every jAf1;y;mg;

nj ¼njðnÞ:¼ n1=mn1j=2 Ym j¼1 nj !1=2m 2 4 3 5:

Then, for everyjAf1;y;mg;

njX1; Ym j¼1 njpn and n 1=2 j ðnjþ1ÞXn1=m Ym k¼1 nk !1=2m : Consequently Xm j¼1 njn_j2pX m j¼1 n2=m Y m k¼1 nk !1=m njþ1 nj 2 p4m n2=m Y m k¼1 nk !1=m :

Settingc¼4c2 completes the proof of (4.6). &

Let us deduce simpler bounds.

Lemma 4.3. Let I ¼IðnÞand mn

¼mn ðnÞbe as in Lemma4.2.Then enðXÞ2pcinf X jXmþ1 njþmnm:mAI ( ) ¼c X jXmn_þ1 njþmn nmn ! :

Proof. Using that for everymAI; n2=m_m Ym k¼1 nj !1=m ¼mnm n2=m nm Y m j¼1 nj !1=m 0 @ 1 A 1 pmnm

the inequality follows from Lemma 4.2. One easily checks thatP_jXmþ1njþ

mnm is decreasing inmAN:This yields the equality. &

Wefirst consider thecasem_j¼OðnjÞ where nj is a rapidly decreasing

sequence, i.e. lim infnj=njþ1>1:

Proposition 4.4 (Rapidly decreasing variances). Assume nj¼jðjÞ for all

jX1;wherej:ð0;NÞ-ð0;NÞis a decreasing and log-concave function with

nonvanishing right derivativej0_r:For nAN;let

J ¼JðnÞ ¼ mX1:n2=mn_m=j mþ1 2 X1 : Then enðXÞ2pcinf n2m=jj 0 rðmÞj þn 2=m_mj mþ1 2 :mAJ pc1inffmnm:mAJg:

Proof. Wewill apply Lemma 4.2. FormAN;wehave

Ym j¼1 nj !1=m ¼exp X m j¼1 logjðjÞ=m ! pexp logj mðmþ1Þ 2m ¼j mþ1 2 and XN jXmþ1 njp Z N m jðxÞdx¼ Z N m jðxÞ j0 rðxÞ ðj0_rðxÞÞdx: Nowj0

r=jo0 andj0r=jis decreasing due to log-concavity so thatj=j0r>

0 andj=j0

r is decreasing. Consequently, for everymAN;

X jXmþ1 njpjðmÞ j0 rðmÞ Z N m ðj0_rðxÞÞdx¼ jðmÞ j0 rðmÞ jðmÞ ¼ n 2 m jj0 rðmÞj : SinceformAN n2=mnm Ym j¼1 nj !1=m Xn2=mn_m=j mþ1 2 ;

wehaveJCI and thus Lemma 4.2 yields the first inequality. Since, for mAJ; n2=mmj mþ1 2 pmnm

andnm=jj0rðmÞj ¼ jjðmÞ=j0rðmÞjis decreasing inmX1;weget n2_m=jj0_rðmÞj þn2=mmj mþ1

2

pjjð1Þ=j0_rð1Þjnmþmnm

pðjjð1Þ=j0_rð1Þj þ1Þmnm for mAJ:

This yields the second inequality of the proposition. &

Now, we pass to regularly varying variances, i.e.m_j¼OðnjÞwithnj¼jðjÞ;

j regularly varying. A function j:R_þ-_ð0;NÞ is regularly varying at

infinity with indexbif

lim

x-N

jðtxÞ

jðxÞ ¼t

b _{for every t>0:}

Lemma 4.5. Let rX_{0and letc:}ðr_;NÞ-ð0_;NÞ_{be an increasing,unbounded,}

regularly varying function at infinity of indexX_0:Assume

(i) _ðQn j¼1njÞ 1=n_¼Oð nnÞ, (ii) P_jXnþ1 njþnnn¼Oð1=cðnÞÞ: Then enðXÞ ¼Oð1=cðlognÞ1=2Þ:

Notethat therestriction on theindex ofcis necessary: otherwisecðxÞ-0

asx-N:

Proof. Using (ii) and Lemma 4.3, we see that for sufficiently largen;

enðXÞ2pc1=cðm n

ðnÞÞ and by (i) and Lemma 4.2,

mn

ðnÞXc₂logn

forc2>0:Consequently

enðXÞ2pc1=cðc2lognÞ

and thus assertion follows from the fact thatcis regularly varying. &

The following theorem provides sharp upper bounds on the rate ofenðXÞ

for regularly varying sequences nj (sharpness will be a consequence of

Theorem 4.6 (Regularly varying variances). Let rX0:Assumen_j¼jðjÞ; j>

r; where j:ðr;NÞ-ð0;NÞ is a decreasing, regularly varying function at

infinity of indexb_p1:Set for every x>r;

cðxÞ:¼_RN 1 x jðyÞdy if b¼1 and cðxÞ:¼ 1 xjðxÞ if b>1: Then enðXÞ ¼Oð1=cðlognÞ1=2Þ: ð4:8Þ Moreover,we have (i) nn=nnþ1-1; (ii) ðQn_j¼1njÞ1=n_Beb_nn; (iii) P_jXnþ1 njþnnnBc=cðnÞ; where c¼1if b¼1and c¼b=ðb1Þif b>1:

Notethat theaboverestrictionbp1 on theindex ofjis natural since

otherwisexjðxÞ-Nasx-N:

Proof. Thefunction c is regularly varying at infinity of index b1:

Therefore, (4.8) follows from the properties (ii) and (iii) and Lemma 4.5. It

remains to prove (i)–(iii). Let jðxÞ ¼xb_{gðxÞ with g slowly varying at}

infinity.

(i) By the Uniform Convergence Theorem [4, Theorem 1.2.1],gðxÞ=gðxþ

1Þ-1 asx-N:This yields (i).

(ii) By Theorem 1.3.3 in [4] there exists a differentiable, slowly varying functiong0>0 of elasticity zero at infinity, i.e.xg00ðxÞ=g0ðxÞ-0 asx-N;

such thatgðxÞBg0ðxÞasx-N:Let j0ðxÞ ¼xbg0ðxÞ:Observe that

xj0 0ðxÞ j₀ðxÞ ¼ bþ xg0 0ðxÞ g0ðxÞ b; x-N and ðQn_j¼1njÞ1=n nn B ðQn_j¼1j₀ðjÞÞ1=n j₀ðnÞ :

In view of inequalities (4.7) it is sufficient to show that lim n-N j₀ðnÞ j₀ðnþ1Þ n ¼eb:

Now xðlogj₀ðxÞ logj₀ðxþ1ÞÞ ¼ xj 0 0ðxxÞ j₀ðxxÞ for some xoxxoxþ1 ¼xj 0 0ðxxÞ j₀ðxxÞ

for sufficiently large x

¼x xx x_x j 0 0ðxxÞ j₀ðxxÞ -b as x-N

which provides the assertion.

(iii) Using thefact thatjis decreasing we get

X jXnþ1 njþnnnB Z N n jðyÞdyþnjðnÞ:

In case b¼1; wehavexjðxÞ ¼gðxÞ and theslowly varying function g

satisfies gðxÞ RN x jðyÞdy -0 as x-N (cf. [4, Proposition 1.5.9b]). Consequently Z N n jðyÞdyþnjðnÞB Z N n jðyÞdy¼1=cðnÞ: In caseb>1;weget Z N x jðyÞdyBxjðxÞ b1 as x-N (cf. [4, Proposition 1.5.10]) and hence

Z N n jðyÞdyþnjðnÞBbnjðnÞ b1 ¼ b ðb1ÞcðnÞ: Thus the proof of (iii) is complete. &

The most prevalent form forjin Theorem 4.6 is

jðxÞ ¼xbðlogxÞa; x>ea=b and x>1;

whereb>1; aARorb¼1; a>1;and in Proposition 4.4

jðxÞ ¼ebx; x>0

forb>0:We state these special cases as a corollary. Parts (a) and (c) below comprise all applications to Gaussian processes we have in mind. Sharpness of thebound in part (c) follows from Corollary 4.13(c).

Corollary 4.7. (a)Ifm_j¼Oðjb_ðlogjÞa_{Þas j-Nfor b>1and a}

AR;then

(b)Ifmj¼Oðj1ðlogjÞaÞas j-Nfor a>1;then

enðXÞ ¼Oððlog lognÞða1Þ=2Þ:

(c)Ifmj¼OðebjÞas j-Nfor b>0;then

enðXÞ ¼OððlognÞ1=4eðblognÞ 1=2

Þ:

Proof. (a) Apply Theorem 4.6 with jðxÞ ¼xb_ðlogxÞa _{and cðxÞ ¼}

1=xjðxÞ ¼xb1_ðlogxÞa :

(b) Apply Theorem 4.6 withjðxÞ ¼x1_ðlogxÞa _and

cðxÞ ¼_RN 1

x jðyÞdy

¼ ða1ÞðlogxÞa1:

(c) Let us apply Proposition 4.4. Letnj¼ebjandjðxÞ ¼ebxforx>0:The

constraintmAJðnÞreads as 2 logn m bmþ bðmþ1Þ 2 X0; that is m2mp4 logn b :

Sincem/ebm_{;mX1=b is decreasing, the best choice ofmis then given by}

mðnÞ ¼ ½1 2þ ð14þ 4 logn b Þ 1=2_{31 but setting} m¼mðnÞ ¼ 2 logn b 1=2 " # 31

will beenough. Pluggingminto Proposition 4.4 yields

enðXÞ2pcmebmpc1ðlognÞ1=2e2ðblognÞ 1=2

: & 4.2. Lower bounds

We deduce lower bounds on the rate ofenðXÞfrom theentropy behaviour

of therandom vectorX:ForEX₀;_{Shannon–Kolmogorov’sE-entropy}R_Xð_EÞ

ofX [16] is defined by

RXðEÞ ¼inffIðQÞ:Q probability on HH with first marginal

Q1¼P and Z HH jjxyjj2_dQðx;yÞpE2 ;

wherePis thedistribution ofXand theaveragemutual informationIðQÞof

Qis equal to the Kullback–Leibler divergence

Z

log dQ

dQ1#Q2

ifQis absolutely continuous with respect to the product of the marginals

Q1#Q2 and equal to N otherwise. The function RX is also called rate

distortion function; it is decreasing and continuous on R_{þ and satisfies}

Rð0Þ ¼N:NotethatR_XðEÞis theminimum mutual information onehas to

transmit in order to reproduceXwithL2_{-error (orL}2_{-distortion) not greater}

thanE:The link betweenRXðEÞandenðXÞis as follows.

Lemma 4.8. Letc:ðr;NÞ-ð0;NÞbe an increasing,unbounded function for

some rX0 such that

cðRXðEÞÞ ¼OðE2Þ as E-0:

Then

enðXÞ ¼Oð1=cðlognÞ1=2Þ as n-N: Proof. For nAN; aAC_nðXÞ; f ¼P

aAaa1WðajaÞ is an n-optimal quantizing

ruleforX:LetQdenote the distribution ofðX;fðXÞÞ:Then, one checks

dQ dQ1#Q2 ðx;yÞ ¼X aAa 1WðajaÞðxÞ1fagðyÞ 1 PðWðajaÞÞ so that RXðenðXÞÞpIðQÞ ¼ X aAa logðPðWðajaÞÞÞPðWðajaÞÞ ¼entropy of fðXÞplogn: ð4:9Þ

Consequently,cðRXðenðXÞÞÞpcðlognÞwhich completes the proof. &

Next theorem shows that, for Gaussian vectors, there is an

explicit expression forRXðEÞ in terms of the eigenvalues of the covariance

operator.

Theorem (cf. Ihara [15, Theorem 6.9.1]). Letl1X_l2X?>0be the nonzero

eigenvalues of CX (each written as many times as its multiplicity). For

0oEoe1ðXÞ;set m:¼mðEÞ ¼max kX1: X jXkþ1 ljþklk>E2 ( ) ð4:10Þ and c:¼cðEÞA½l_mþ1;l_mÞ uniquely defined by X jXmþ1 ljþmc¼E2: ð4:11Þ Then, XN j¼1

minflj;cg ¼E2 and hence RXðEÞ ¼

1 2

Xm j¼1

NoticethatmðEÞ-Nas E-0:If wecombinethis formula with Lemma

4.8 we get the following result.

Proposition 4.9. Letc:ðr;NÞ-ð0;NÞbe an increasing unbounded function

for some rX0such that

c 1 2 Xn j¼1 logðlj=lnÞ ! ¼O X jXnþ1 ljþnlnþ1 !1 0 @ 1 A as n-N: ð4:13Þ Then enðXÞ ¼Oð1=cðlognÞ1=2Þ:

An easy consequence is as follows.

Lemma 4.10. Let rX0 and let c:ðr;NÞ-ð0;NÞ be an

increasing, unbounded, regularly varying function at infinity of index X0:

Assume

(i) lim infn-Nð Qn

j¼1ljÞ1=n=ln >1;

(ii) P_jXnþ1 ljþnlnþ1 ¼Oð1=cðnÞÞ:

Then

enðXÞ ¼Oð1=cðlognÞ1=2Þ:

Proof. Let us apply Proposition 4.9. Using (i), we see that fornlarge, 1 2 Xn j¼1 logðlj=lnÞ ¼ n 2log Yn j¼1 lj !1=n =ln 0 @ 1 AXc1n 2 withc1>0:By (ii), X jXnþ1 ljþnlnþ1 !1 pc2cðnÞ

for 0oc2oN:This yields

c 1 2 Xn j¼1 logðlj=lnÞ ! Xcðc₁n=2ÞXcðc1n=2Þ c2cðnÞ X jXnþ1 ljþnlnþ1 !1

and using regular variation of the functionc;weseethat condition (4.13) is

fulfilled. &

The following result is a kind of comparison lemma for ratesenðXÞbased

Lemma 4.11. Let Y be a H-valued centred Gaussian random vector with dimKY ¼Nand nonzero eigenvaluesr1Xr2X?>0:IfljXcrj for all jX1

and c>0;then

enðXÞXc enðYÞ for all nX1:

In particular,if KX*KY as sets,then

enðXÞ ¼OðenðYÞÞ as n-N:

IfljEr_j as j-Nand,in particular,if K_X¼K_Y as sets,then

enðXÞEenðYÞ as n-N:

Proof. If ljXcrj;then it is an easy consequence of (3.2) and Lemma 2.3

thatenðXÞXcenðYÞ:IfKX*KY;then there is a constantc>0 such that

/_y;CXySXc/y;C_YyS for allyAH

(cf. [5, Theorem 3.3.4]). Using the representation of eigenvalues of symmetric positive trace class operators as values of minimax problems, this impliesljXcrj: &

Under the subsequent conditions on the eigenvalues the previous upper and lower bounds match.

Theorem 4.12. Assumelj¼OðjðjÞÞas j-N;wherej:ðr;NÞ-ð0;NÞis a

decreasing,regularly varying function at infinity of indexb_p1for some rX_0:Let cðxÞ ¼_RN 1 x jðyÞdy if b¼1 and cðxÞ ¼ 1 xjðxÞ if b>1; x>r: Then enðXÞ ¼Oð1=cðlognÞ1=2Þ: ð4:14Þ IfljEjðjÞ;then enðXÞEcðlognÞ1=2: ð4:15Þ IfljBjðjÞ;then lim n-NcðRXðenðXÞÞÞ 1=2_e nðXÞ ¼ ðcðb=2Þb1Þ1=2; ð4:16Þ where c¼1if b¼1and c¼b=ðb1Þif b>1:

Proof. LetY beaH-valued centred Gaussian random vector with nonzero eigenvalues jðjÞ; jAN: These eigenvalues satisfy conditions (i) and (ii) of

Lemma 4.10 (cf. Theorem 4.6). Consequently, enðYÞ ¼Oð1=cðlognÞ1=2Þ

which yields (4.14) in view of Lemma 4.11. Under the assumptionljEjðjÞ;

assertion (4.15) follows from (4.14) and the upper estimate of Theorem 4.6.

Now assumethatljBjðjÞasj-N:In order to prove (4.16) first observe

that ðQn_j¼1ljÞ1=n ln B ðQn_j¼1jðjÞÞ1=n jðnÞ and X jXnþ1 ljþnlnB X jXnþ1 jðjÞ þnjðnÞ:

Therefore, by the second part of Theorem 4.6, (i) ln=lnþ1-1;

(ii) _ðQn

j¼1 ljÞ1=nBebln;

(iii) P_jXnþ1 ljþnlnBc=cðnÞ;

wherec¼1 ifb¼1 andc¼b=ðb1Þifb>1:Letm¼mðEÞbeas in (4.10). Then by formula (4.12), (i) and (ii),

RXðEÞB

mb

2 as E-0:

Let cðxÞ ¼xb1_{gðxÞ with g slowly varying at infinity. Weobserveby}

applying the Uniform Convergence Theorem [4, Theorem 1.2.1] that gðRXðEÞÞBgðmÞ:

Hence

cðRXðEÞÞBcðmÞðb=2Þb1 as E-0:

By (4.11), (i) and (iii),

E2Bc=cðmÞ as E-0 and thus lim E-0 cðRXðEÞÞE 2_¼cðb=2Þb1 : & ð4:17Þ

If jðxÞ ¼c1xbðlogxÞa with b>1; aAR and 0oc₁oN; then (4.17)

yields RXðEÞB b 2 c1b b1 b1 2 a 1=ðb1Þ E2=ðb1Þlogð1=EÞa=ðb1Þ as E-0:

IfX is Brownian motion on [0,1] andH¼L2ð½0;1;dtÞ;then (4.17) reduces to theclassical fact that (see[16])

lim E-0 RXðEÞE

2_¼2=p2_:

Corollary 4.13. (a) IfljEjbðlogjÞa as j-Nfor b>1 and aAR;then

enðXÞEðlognÞðb1Þ=2ðlog lognÞa=2

and ifljBcjbðlogjÞa for0ocoN;then

lim n-NRXðenðXÞÞ ðb1Þ=2_ðlogR XðenðXÞÞÞa=2enðXÞ ¼ cbb ðb1Þ2b1 1=2 :

(b)IfljEj1ðlogjÞa as j-Nfor a>1;then

enðXÞEðlog lognÞða1Þ=2

and ifljBcj1_ðlogjÞa _{for0ocoN;then}

lim n-N ðlogRXðenðXÞÞÞða1Þ=2enðXÞ ¼ c a1 1=2 : (c)IfljEebj _{as j-Nfor b>0;then} enðXÞEðlognÞ1=4eðblognÞ 1=2 :

Proof. Conditions (a) and (b) are immediate consequences of Theorem 4.12.

(c) For the lower bound we will apply Proposition 4.9. In view of Lemma

4.11 we may assume without loss of generality thatlj¼ebj _{for allj:Then}

wehave 1 2 Xn j¼1 logðlj=lnÞ ¼1 4nðn1Þb and X jXnþ1 ljþnlnþ1¼ebðnþ1Þ nþ 1 1eb :

Therefore, condition (4.13) is satisfied for the function

cðxÞ ¼x1=2e2ðbxÞ1=2

; x>1=4b:

(Notethatc is not regularly varying but rapidly varying of index N:) It

follows from Proposition 4.9 that enðXÞ ¼OððlognÞ1=4eðblognÞ 1=2

Þ:

Remark. (a) It remains an open question whether in the situation of

Theorem 4.12 under the conditionljBjðjÞthe

lim

n-Nc

ðlognÞ1=2_e nðXÞ

exists inð0;NÞ;or, what is thesamein caseb>₁;_whether

RXðenðXÞÞBclogn

for somecAð0;1:(In finite dimensions the latter relation withc¼1 follows

from [15, Proposition 4.1, Theorem 5.8.1].)

(b) Without any condition imposed on the eigenvalues we have

X jXmn_þ1 ljþmn lmn_þ1pe_nðXÞ2pc X jXmn_þ1 ljþmn lmn ! ð4:18Þ for allnX1;wheremn¼mnðnÞis defined as in Lemma 4.2 withn_j replaced

bylj:Consequently, under the mild condition

lim inf n-N lnþ1=ln >0 weobtain enðXÞ2E X jXmn_þ1 ljþm n lmn as n-N: In fact, setting am :¼ m 2log Ym j¼1 lj !1=m =lm 0 @ 1 A; then mn ðnÞ ¼maxfmX1:a_mplogng:

On theother hand, by (4.12) forEoe1ðxÞ;

RXðEÞ>amðEÞ

so that by (4.9),

lognXR_Xðe_nðXÞÞ>a_mðenÞ:

Consequently,mn

ðnÞ þ1>mðenÞfor allnX1 and using (4.10) this yields the

lower estimate. The upper estimate is taken from Lemma 4.3.

(c) Theorem 4.12 allows to derive a lower bound on the dimension

dn¼dnðXÞ of the level n quantization problem. Let the eigenvalues be

as in Theorem 4.12 satisfy ljEjðjÞ: Combining this theorem and (3.2),

weget c1 cðdnÞp X jXdnþ1 ljpenðXÞ2p c2 cðlognÞ:

Using thefact thatcis regularly varying and increasing, this implies dnðXÞ ¼OðlognÞ: ð4:19Þ

It would beuseful to know if onehas dnðXÞElogn:

5. Application to Gaussian processes

In this section we use the results of Section 4 to get rates for the quantization error of some classes of Gaussian processes. We consider

centered L2_{ðPÞ-continuous Gaussian processes X¼ ðX}

tÞtAI on I¼ ½0;1 d_:

Then X has a bi-measurable version and thus can be seen as a centered

random vector with values in the Hilbert space H ¼L2_{ðI;dtÞ: The}

covariancefunctionGX ofX is continuous andKX consists of (equivalence

classes of) continuous functions; see (2.9). We will start our investigations by stationary processes because these results will be called upon to elucidate the case of other processes.

5.1. Stationary Gaussian processes, Ornstein–Uhlenbeck process and fractional Ornstein–Uhlenbeck process

In this example we deal with centered L2_{ðPÞ-continuous stationary}

Gaussian processesX ¼ ðXtÞtA½0;1:This means that

GXðs;tÞ ¼gðstÞ whereg:R-R is continuous;symmetric;

positivedefinite:

It is classical background (see e.g. [6]) on weakly stationary processes that

these assumptions imply the existence of a finite symmetric Borel measurem

onR_{such that} GXðs;tÞ ¼ Z R e2iplðtsÞdmðlÞ ¼ Z R cosð2plðtsÞÞdmðlÞ: ð5:1Þ

Themeasuremis called thespectral measureof theprocessX:

Thereproducing spaceKX can beeasily characterized by the

spectral measure. As a matter of fact, one derives from (5.1) and (2.11) that KX¼ t/R Z R e2iplt_{fðlÞmðdlÞ:fAL}2 CðmÞ : ð5:2Þ

Theproposition below shows that onemay also read on (the

quantization error. For most part of it, the result relies on a theorem by Rosenblatt [19].

Proposition 5.1. (a) Let a>1

2 and bA½0; 1

2Þ: Assume that mðdlÞ ¼jðlÞdl

where the spectral densityjsatisfies or everylAR;

jðlÞp c

ðjljb41Þð1þ jlj2aÞ: ð5:3Þ Then

enðXÞ ¼OððlognÞða1=2ÞÞ: ð5:4Þ

(b)If ap1;then the above bounds also holds if bA½1

2;1Þ:

Remark. The coefficienta is related to the regularity oft/Xt from ½0;1

into L2_{ðPÞ—i.e. a}1

2 (at least)—whereas b is a ‘‘long-rangememory’’

coefficient. So, according to the intuition, the quantization rate seems to strongly depend on the regularity of the trajectories, but not on the dependency properties of the process: the above distinction seems to be essentially technical.

Proof. (a) Let ðYtÞtA½0;1 be a centred stationary Gaussian process with

spectral density j₁ given by j₁ðlÞ ¼ c

ðjljb_41Þð1þjlj2a_Þ: Notethat

j₁AðL1-L2ÞðR;dlÞ since a>1

2 and bo12: Then it follows from a theorem

due to Rosenblatt [19, Theorem 3] and Widom [22], that the eigenvalues

r₁Xr₂X?>0 of thecovarianceoperatorC_Y ofY satisfy

r_jBc1j2a as j-N:

It follows from Corollary 4.13(a) thatenðYÞEðlognÞða 1

2Þ_{: Moreover, one}

checks that KXCK_Y as sets. The comparison Lemma (Lemma 4.11)

completes the proof.

(b) Whenjis no longer square integrable, Rosenblatt’s Theorem cannot

be applied and a direct approach is needed. If one wishes to quantize such a process and to estimate the quantization error, it seems natural to introduce the (real-valued) trigonometric orthonormal basisfuj :jX0gofL2ð½0;1;dtÞ

defined by u0 :¼1; u2jðtÞ:¼ ffiffiffi 2 p cosð2pjtÞ; u2j1ðtÞ ¼ ffiffiffi 2 p sinð2pjtÞ; jX1;

and to rely on the positive real coefficients

mj ¼Var Z 1 0 XtujðtÞdt ¼ Z 1 0 Z 1 0 gðstÞujðsÞujðtÞds dt:

One introduces for computational convenience the complex-valued basis ˜

so thatu0¼u˜0; u2j¼ ffiffiffi 2 p Sð˜ujÞand u2j1¼ ffiffiffi 2 p

ðu˜jÞfor jX1:Now, set for

jX0; * mj :¼ Z 1 0 Z 1 0 gðstÞu˜jðsÞu˜jðtÞds dt:

On theonehand,m*₀¼m₀pgð0Þandm*j¼2ðm2j1þm2jÞfor every jX1:On

the other hand, Fubini Theorem yields for everyjX1;

* mj ¼ Z R je2ipl_1j2 ð2pðlþjÞÞ2dmðlÞ: ð5:5Þ Themain step is to show, using (5.5), that

*

m_j ¼Oðj2a_{Þ as j-N ð5:6Þ}

Letcdenote a real constant that may vary from line to line. First, note that

forjX1; * m_jpc Z R je2ipl_1j2 l2 dl ð14jljjbÞð1þ jljj2aÞ: Sincel/je2ipl_1j2 =ðl2jl1jbÞAL1ðR;dlÞ;onegets Z 1 N je2ipl_1j2 l2 dl ð1₄jljjb_{Þð1þ jljj}2a_Þpsup lp1 1 ð1þ jljj2aÞ Z R je2ipl_1j2 jl1jb_l2 dl p c 1þ ðj1Þ2a

since both singularities in the above integral are false. Now forjX2;

Z j1 1 je2ipl_1j2 l2 dl ð14jljjbÞð1þ jljj2aÞ p22b Z j1 1 dl l2ðjlÞ2a p22bð2pÞb Z j1 1 dl l2ðjlÞ2a pc Z j1 1 1 l2ð1aÞ dl ðlðjlÞÞ2a pc j2a Z j1 1 1 lþ 1 jl 2a dl l2ð1aÞ

pc22a j2a Z j1 1 1 l2aþ 1 ðjlÞ2a dl l2ð1aÞ pc j2a Z þN 1 1 l2þ 1 l2a dl¼Oðj2a_Þ: Furthermore Z jþ1 j1 je2ipl_1j2 l2 dl ð14jljjbÞð1þ jljj2aÞpc Z jþ1 j1 dl jljjbl2 p c ðj1Þ2 Z 1 1 dl jljb ¼Oðj2_{Þ ¼Oðj}2a_{Þ: ð5:7Þ} Finally, Z þN jþ1 je2ipl_1j2 l2 dl ð14jljjbÞð1þ jljj2aÞp4 Z þN jþ1 dl l2ðljÞ2a p c j2a Z þN 1 1 l2þ 1 l2a dl ¼Oðj2aÞ:

Thus (5.6) holds and, in turn, it follows thatm_j¼Oðj2a_{Þasj-Nsincethe} m_j’s are nonnegative. Hence (5.4) follows from Corollary 4.7(a). &

Application to the fractional Ornstein–Uhlenbeck process: Thefractional

Ornstein–Uhlenbeck process with index rAð0;2Þ is a stationary centred

Gaussian processðX_trÞ_tA½0;1with covariancefunction

Grðs;tÞ ¼expðajstjrÞ; a>0:

The spectral measure of the process is a symmetricr-stabledistribution. Its

Lebesgue densityjis (symmetric) continuous and satisfies

jðlÞBcðrÞlð1þrÞ as l-N:

Therefore, it follows from Theorem 3 in [19] (or [22, Theorem 1]) that

eigenvalues of the covariance operator ofXr_satisfies

ljBc1jð1þrÞ as j-N: ð5:8Þ

Thus, by Corollary 4.13(a)

enðXrÞEðlognÞr=2: ð5:9Þ

Ifr¼1;one gets the standard stationary Ornstein–Uhlenbeck processX1

Application to a stationary process with smooth covariance: The1-periodic

Poisson kernel defined for every 0oao1 by

gðtÞ ¼ 1a

2

1þa2_2acosð2ptÞ

provides an example of a stationary centred Gaussian process X on ½0;1

with a very smooth covariance functionGXðs;tÞ ¼gðstÞ:Since

gðtÞ ¼ X N

j¼N

ajjje2pijt; tAR;

we deduce that the (real) trigonometric orthonormal basis consists of eigenfunctions ofCX and the eigenvalues are given byl0 ¼1; l2j¼l2j1¼

aj_{;jX1:Therefore, Corollary 4.13(c) (withb¼ logðaÞ=2Þyields}

enðXÞEðlognÞ1=4eðlogð1=aÞlognÞ 1=2

=pffiffi2_{: ð5:10Þ}

(Hence enðXÞ ¼OððlognÞrÞ for every r>0:) Moreover, by (3.3), any aAC_nðXÞsatisfies aCspanf1g if n¼2; aC spanf1; ffiffiffi 2 p sinð2pjtÞ;pffiffiffi2cosð2pjtÞ:j¼1;y;ðn2Þ=2g if nX3; n even; aCspanf1; ffiffiffi 2 p sinð2pjtÞ;pffiffiffi2cosð2pjtÞ:j¼1;y;ðn1Þ=2g if nX3; n odd:

5.2. Brownian motion and fractional Brownian motion

The fractional Brownian motion with Hurst exponentrAð0;1is a centred

continuous Gaussian process Br¼ ðBr_tÞ_tA½0;1 having

thecovariancefunc-tion,

Grðs;tÞ ¼1₂ðs2rþt2r jstj2rÞ:

Letfuj :jX0gbean orthonormal basis ofL2ð½0;1;dtÞwithu0¼1:Wewill

rely on the numbers

mj ¼Var Z 1 0 BrtujðtÞdt ¼ Z 1 0 Z 1 0 Grðs;tÞujðsÞujðtÞds dt

to estimate the quantization error. In fact one checks that

mj ¼ 1 2 Z 1 0 Z 1 0 jstj2r_u jðsÞujðtÞds dt; jX1:

Before dealing with its quantization error, let us mention that by (2.9), optimal sets of means for Br _{have r-Ho¨lder components since ðEjB}r

t

Br

sj2Þ1=2¼ jtsjr:

Proposition 5.2. For everyrAð0;1Þ;

enðBrÞEðlognÞr: ð5:11Þ Remark.

* _{One question is left open by such a result. Does ðlognÞ}r_e

nðBrÞ havea

finitenonzero limit asn-N;similarly to Theorem 1.1? This seems to be

a natural conjecture.

* _Ifr¼1

2; one obtains standard Brownian motion denoted simply by B:

Then enðBÞEðlognÞ1=2: * _{Furthermore, by (3.3), anyaAC} nðBÞ; nX2;satisfies (cf. Example 3.3) aCspanf ffiffiffi 2 p sinðpðj1=2ÞtÞ:j¼1;y;n1g:

Proof. One considers the celebrated Haar orthonormal basis defined by u0¼1; u1¼1½0;1=2Þ1½1=2;1; u2m_þkðtÞ ¼2m=2u₁ð2mtkÞ;

mAN; k¼0;y;2m1:

Using its wavelet character, a standard computation shows that

m₂m_þk¼

m₁

2mð1þ2rÞ:

Consequently, for everyjX1;

m₁

j1þ2rpmjp 21þ2r_m

1

j1þ2r : ð5:12Þ

Thus Corollary 4.7(a) yields, for everyrAð0;1;

enðBrÞ ¼OððlognÞrÞ: ð5:13Þ

This rate of convergence is the true one when 0oro1 (whenr¼1;Br_t ¼ tZ; ZBNð0;1Þ;so thatenðBrÞBcn1 by Theorem 1.1).

The main step is to show that the nonzero eigenvaluesl1Xl2X?>0 of

thecovarianceoperatorCBr satisfy

lj ¼Oðjð1þ2rÞÞ as j-N: ð5:14Þ

First, notethat thechangeof variabledl¼ jsjduyields

jsj2r 2 ¼R Z R ð1e2iplsÞ c jlj1þ2rdl