2. Asymptotically Optimal Quantizers

Journal of Applied Mathematics Volume 2010, Article ID 378519,32pages doi:10.1155/2010/378519

Research Article

A Constructive Sharp Approach to Functional Quantization of Stochastic Processes

Stefan Junglen and Harald Luschgy

FB4-Department of Mathematics, University of Trier, 54286 Trier, Germany

Correspondence should be addressed to Harald Luschgy,luschgy@uni-trier.de Received 1 June 2010; Accepted 21 September 2010

Academic Editor: Peter Spreij

Copyrightq2010 S. Junglen and H. Luschgy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present a constructive approach to the functional quantization problem of stochastic processes, with an emphasis on Gaussian processes. The approach is constructive, since we reduce the infinite-dimensional functional quantization problem to a finite-dimensional quantization problem that can be solved numerically. Our approach achieves the sharp rate of the minimal quantization error and can be used to quantize the path space for Gaussian processes and also, for example, L´evy processes.

1. Introduction

We consider a separable Banach spaceE, · and a Borel random variableX:Ω,F, → E,BEwith finiterth momentX^rfor somer∈1,∞.

For a given natural numbern ∈ , the quantization problem consists in finding a set α⊂Ethat minimizes

erX,E,·, α erX, E, α:

min

a∈αX−a^r _1/r

1.1

over all subsetsα⊂Ewith cardα≤n. Such setsαare calledn-codebooks orn-quantizers. The corresponding infimum

en,rX,E,· en,rX, E: inf

α⊂E,cardα≤nerX, E, α 1.2

is called thenthL^r-quantization error ofXinE, and anyn-quantizerαfulfilling

erX, E, α en,rX, E 1.3 is calledr-optimal n-quantizer. For a given n-quantizer αone defines the nearest neighbor projection

πα:E−→α, x−→

a∈α

aχCaαx, 1.4

where the Voronoi partition{Caα, a∈α}is defined as a Borel partition ofEsatisfying

Caα⊂

x∈ :x−amin

b∈αx−b

. 1.5

The random variableπαXis called α-quantization of X. One can easily verify thatπαX is the best quantization ofX inα ⊂ E, which means that for every random variableY with values inαwe have

e_rX, E, α

X−π_αX^r_1/r

≤

X−Y^r_1/r

. 1.6

Applications of quantization go back to the 1940s, where quantization was used for the finite-dimensional settingE ^d, called optimal vector quantization, in signal compression and information processing see, e.g., 1, 2. Since the beginning of the 21st century, quantization has been applied for example in finance, especially for pricing path-dependent and American style options. Here, vector quantization3as well as functional quantization 4,5is useful. The terminology functional quantization is used when the Banach spaceEis a function space, such asE Lp0,1, · _porEC0,1, · _∞. In this case, the realizations ofXcan be seen as the paths of a stochastic process.

A question of theoretical as well as practical interest is the issue of high-resolution quantization which concerns the behavior of en,rX, E when n tends to infinity. By separability of E, · , we can choose a dense subset{ci, i ∈ } and we can deduce in view of

0≤ lim

n→ ∞ min

1≤i≤nX−c_i^r lim

n→ ∞min

1≤i≤nX−c_i^r0 1.7

thaten,rX, Etends to zero asntends to infinity.

A natural question is then if it is possible to describe the asymptotic behavior of e_n,rX, E. It will be convenient to write a_n ∼ b_n for sequences an_n∈Æ and bn_n∈Æ if an/bn −−−−−→n→ ∞ 1, an ^º bn if lim sup_{n→ ∞}an/bn ≤ 1 and an ≈ bn if 0 < lim infn→ ∞an/bn ≤ lim sup_{n→ ∞}an/bn<∞.

In the finite-dimensional setting^d, · this behavior can satisfactory be described by the Zador Theoremsee6for nonsingular distributions ^X. In the infinite dimensional case, no such global result holds so far, without some additional restrictions. To describe one

of the most famous results in this field, we call a measurable functionρ:s,∞ → 0,∞for ans≥0 regularly varying at infinity with indexb∈if for everyc >0

xlim→ ∞

ρcx

ρx c^b. 1.8

Theorem 1.1see7. LetXbe a centered Gaussian random variable with values in the separable Hilbert spaceH,·,·andλn, n∈the decreasing eigenvalues of the covariance operatorCX:H → H,u → u, XX (which is a symmetric trace class operator). Assume thatλn ∼ ρnfor some regularly varying functionρwith index−b <−1. Then, the asymptotics of the quantization error is given by

en,2X, H∼ b 2

_b−1 b b−1

_1/2 ω

logn_−1/2

, n−→ ∞, 1.9

whereωx:1/xρx.

Note that any change of∼in the assumption thatλn ∼ ρnto either^º,≈or^²leads to the same change in1.9.Theorem 1.1can also be extended to an indexb 1 see7.

Furthermore, a generalization to an arbitrary momentrsee8as well as similar results for special Gaussian random variables and diffusions in non-Hilbertian function spacessee, e.g., 9–11have been developed. Moreover, several authors established a precise link between the quantization error and the behavior of the small ball function of a Gaussian measuresee, e.g.,12,13which can be used to derive asymptotics of quantization errors. More recently, for several types of L`evy processessharp optimal rates have been developed by several authorssee, e.g.,14–17.

Coming back to the practical use of quantizers as a good approximation for a stochastic process, one is strongly interested in a constructive approach that allows to implement the coding strategy and to computeat least numericallygood codebooks.

Considering again Gaussian random variables in a Hilbert space setting, the proof of Theorem 1.1 shows us how to construct asymptotically r-optimal n-quantizers for these processes, which means that sequences ofn-quantizersα_n, n∈ satisfy

erX, E, αn∼en,rX, E, n−→ ∞. 1.10 These quantizers can be constructed by reducing the quantization problem to a quantization problem of a finite-dimensional normal distributed random variable. Even if there are almost no explicit formulas known for optimal codebooks in finite dimensions, the existence is guaranteed see 6, Theorem 4.12 and there exist a lot of deterministic and stochastic numerical algorithms to compute optimal codebooks see e.g., 18, 19 or 20. Unfortunately, one needs to know explicitly the eigenvalues and eigenvectors of the covariance operatorCXto pursue this approach.

If we consider other non-Hilbertian function spacesE, · or non-Gaussian random variables in an infinite-dimensional Hilbert space, there is much less known on how to construct asymptotically optimal quantizers. Most approaches to calculate the asymptotics of the quantization error are either non-constructivee.g.,12,13or tailored to one specific

process typee.g.,9–11or the constructed quantizers do not achieve the sharp rate in the sense of1.10 e.g.,17or20but just the weak rate

e_rX, E, αn≈e_n,rX, E, n−→ ∞. 1.11

In this paper, we develop a constructive approach to calculate sequences of asymptoticallyr-optimal n-quantizersin the sense of1.10 for a broad class of random variables in infinite dimensional Banach spacesSection 2. Constructive means in this case that we reduce the quantization problem to the quantization problem of a^d-valued random variable, that can be solved numerically. This approach can either be used in Hilbert spaces in case the eigenvalues and eigenvectors of the covariance operator of a Gaussian random variable are unknownSections3.1and3.2, or for quantization problems in different Banach spacesSections4and5.

In Section 4, we discuss Gaussian random variables in C0,1, · _∞. This part is related to the PhD thesis of Wilbertz20. More precisely, we sharpen his constructive results by showing that the quantizers constructed in the thesis also achieve the sharp rate for the asymptotic quantization errorin the sense of1.10. Moreover, we can show that the dimensions of the subspaces wherein these quantizers are contained can be lessened without loosing the sharp asymptotics property.

InSection 5, we use some ideas of Luschgy and Pag`es17and develop for Gaussian random variables and for a broad class of L´evy processes asymptotically optimal quantizers in the Banach spaceLp0,1, · _p.

It is worth mentioning that all these quantizers can be constructed without knowing the true rate of the quantization error. This means precisely that we know aroughlower bound for the quantization error, that is,e_n,rX, E^²C₁logn^−b1and the true but unknown rate is en,rX, E ∼ C2logn^−b2 for constants C1, C2, b1, b2 ∈ 0,∞, then, we are able to construct a sequence ofn-quantizersα_n,n∈ that satisfies

erX, E, αn∼en,rX, E∼C2logn^−b2, n−→ ∞ 1.12

for the optimal but still unknown constantsC2, b2.

The crucial factors for the numerical implementation are the dimensions of the subspaces, wherein the asymptotically optimal quantizers are contained. We will calculate the dimensions of the subspaces obtained through our approach, and we will see that for all analyzed Gaussian processes, and also for many L´evy processes we are very close to the known asymptotics of the optimal dimension in the case of Gaussian processes in infinite- dimensional Hilbert spaces.

We will give some important examples of Gaussian and L´evy processes inSection 6, and finally illustrate some of our results inSection 7.

Notations and Definitions

If not explicitly differently defined, the following notations hold throughout the paper.

iWe denote byX a Borel random variable in the separable Banach spaceE, · with cardsupp ^X ∞.

ii · will always denote the norm inEwhereas · _Lr^Èwill denote the norm in LrΩ,F, .

iiiThe scalar product in a Hilbert spaceHwill be denoted by·,·.

ivThe smallest integer above a given real numberxwill be denoted byx.

vA sequence gj_j∈

Æ

∈ E^Æ is called admissible for a centered Gaussian random variable X in E if and only if for any sequence ξi_i∈Æ of independent N0,1- distributed random variables it holds that_∞

i1ξ_ig_i convergesa.s. in E, · and X ^d _∞

i1ξigi. An admissible sequencegj_j∈

Æ

∈E^Æis called rate optimal forXinE if and only if

∞ im

ξ_ig_i

2

≈inf

⎧⎨

⎩

∞ im

ξ_if_i

2

: f_i

i∈^Æ admissible forX

⎫⎬

⎭, 1.13

asm → ∞. A precise characterization of admissible sequences can be found in21.

viAn orthonormal system ONS hi_i∈Æ is called rate optimal forX in the Hilbert spaceHif and only if

∞ im

hihi, X

2

≈inf

⎧⎨

⎩

∞ im

fi

fi, X

2

: fi

i∈^ÆONS inH

⎫⎬

⎭, 1.14

asm → ∞.

2. Asymptotically Optimal Quantizers

The main idea is contained in the subsequent abstract result. The proof is based on the following elementary but very useful properties of quantization errors.

Lemma 2.1see22. LetE,Fbe separable Banach spaces,Xa random variable in E, andT :E → F.

1IfT is Lipschitz continuous with Lipschitz constantL, then

e_n,rTX, F≤Le_n,rX, E, 2.1

and for everyn-quantizerαforXit holds that

erTX, F, Tα≤LerX, E, α. 2.2

2LetT :E → Fbe linear, surjective, and isometric. Then, forc≥0 andf∈F en,r

cTX f, F

cen,rX, E, 2.3

and for everyn-quantizerαforXit holds that

er

cTX f, F, Tα

cTerX, E, α f. 2.4

To formulate the main result, we need for an infinite subsetJ⊂ the following.

Condition 1. There exist linear operatorsVm :E → Fm ⊂ Eform ∈J withVm_op ≤ 1, for finite dimensional subspacesFmwith dimFm m, where the norm · _opis defined as

Vm_op: sup

x∈E,x≤1Vmx. 2.5

Condition 2. There exist linear isometric and surjective operatorsφ_m:Fm, · → ^m,| · |_m with suitable norms| · |_min^mfor allm∈J.

Condition 3. There exist random variablesZmform∈J inEwithZm d

X, such that for the approximation errorX−VmZm_LrÈit holds that

X−V_mZm_LrÈ−→0, 2.6

asm → ∞alongJ.

Remark 2.2. The crucial point in Condition1is the norm one restriction for the operatorsV_m. Condition2becomes Important when constructing the quantizers in^m equipped with, in the best case, some well-known norm. As we will see in the proof of the subsequent theorem, to show asymptotic optimality of a constructed sequence of quantizers one needs to know only a rough lower bound for the asymptotic quantization error. In fact, this lower bound allows us in combination with Condition3to choose explicitly a sequencemn ∈J,n ∈ such that

X−V_mn

Z_mn

Lr^Èoen,rX, E, n−→ ∞. 2.7 Theorem 2.3. Assume that Conditions 1–3 hold for some infinite subset J ⊂ . One chooses a sequencemn_n∈Æ∈J^Æsuch that2.7is satisfied. Forn∈, letαnbe anr-optimaln-quantizer forξn :φ_mnV_mnZ_mnin^mn,| · |_mn.

Then,φ_mn⁻¹ αn_n∈

Æ

is an asymptoticallyr-optimal sequence ofn-quantizers forXinEand

en,rX, E∼

X−π_φ⁻¹

mnαn

V_mn

Z_mn^r1/r

∼er

X, E, φ⁻¹_mnαn

, 2.8

asn → ∞.

Remark 2.4. Note, that forn∈ there always existr-optimaln-quantizers forξn6, Theorem 4.12.

Proof. Using Condition3and the fact thaten,rX, E>0 for alln∈ since cardsupp ^X

∞, we can choose a sequence mn_n∈Æ ∈ ^Æ fulfilling 2.7. Using Lemma 2.1 and Condition2, we see thatφ_mn⁻¹ αnis anr-optimaln-quantizer forV_mnZmninF_mn. Then, by using Condition1,2.7, andLemma 2.1we get

en,rX, E≤

X−π_φ⁻¹

mnαn V_mn

Z_mn^r1/r

≤

X−V_mn

Z_mn^r1/r

Vmn Z_mn

−π_φ⁻¹

mnαn V_mn

Z_mn^r1/r

X−V_mn

Z_mn^r1/r

en,r

V_mn Z_mn

,

F_mn,·

≤

X−V_mn

Z_mn^r1/r

en,r

Z_mn, E

X−V_mn

Z_mn^r1/r

e_n,rX, E∼e_n,rX, E, n−→ ∞.

2.9

The last equivalence of the assertion follows from1.6.

Remark 2.5. We will usually chooseZmXfor allm∈, with an exception inSection 3and J.

Remark 2.6. The crucial factor for the numerical implementation of the procedure is the dimensionsmn_n∈Æof the subspacesF_mnn∈Æ. For the well-known case of the Brownian motion in the Hilbert spaceH L₂0,1it is known that this dimension sequence can be chosen asmn ≈ logn,n → ∞. In the following examples we will see that we can often obtain similar orders like logn^cfor constantscjust slightly higher than one.

We point out that there is a nonasymptotic version ofTheorem 2.3for nearly optimal n-quantizers, that is, forn-quantizers, which are optimal up to >0. Its proof is analogous to the proof ofTheorem 2.3.

Proposition 2.7. Assume that Conditions1–3hold. Letm:inf{m∈ :X−V_mZm_LrÈ<

}, and forn ∈ one setsξn :φ_mVmZm. Then, it holds for everyn ∈ and for every r-optimaln-quantizerαnforξnin^m,| · |_mthat

er

X, E, π_φ⁻¹

mαn

V_m

Z_m

≤en,rX, E . 2.10

3. Gaussian Processes with Hilbertian Path Space

In this chapter, letXbe a centered Gaussian random variable in the separable Hilbert space H,·,·. Following the approach used in the proof of Theorem 1.1, we have for every sequenceξi_i∈Æof independentN0,1-distributed random variables

X^{d ∞}

i1

λifiξi, 3.1

whereλidenote the eigenvalues andfidenote the corresponding orthonormal eigenvectors of the covariance operator CX of X Karhunen-Lo`eve expansion. If these parameters are known, we can choose a sequencedn_n∈Æsuch that a sequence of optimal quantizerα_n for Xn _dn

i1

λifiξiis asymptotically optimal forXinE.

In order to construct asymptotically optimal quantizers for Gaussian random variables with unknown eigenvalues or eigenvectors of the covariance operator, we start with more general expansions. In fact, we just need one of the two orthogonalities, either inL2 or inH.

Before we will use these representations forX to find suitable triplesVm, Fm, φmas inTheorem 2.3, note that for Gaussian random variables inHfulfilling suitable assumptions we know that

1Lethi_i∈^Æbe an orthonormal basis ofH. Then

X^∞

i1

hihi, X a.s.. 3.2

Compared to 3.1 we see that hi, X are still Gaussian but generally not independent.

2Letgj_j∈Æbe an admissible sequence forXinHsuch that

X^{d ∞}

i1

ξigi. 3.3

Compared to3.1the sequencegi_i∈Æis generally not orthogonal.

en,2X, H≈en,sX, H, n−→ ∞ 3.4 for alls≥1; see13. Thus, we will focus on the cases2 to search for lower bounds for the quantization errors.

3.1. Orthonormal Basis

Let hm_m∈Æ be an orthonormal basis of H. For the subsequent subsection we use the following notations.

1We setF_mspan{h1, . . . , h_m}.

2We setV_m:pr_Fm :E → F_m, the orthogonal projection onF_m. It is well known that Vm_op1.

3Define the linear, surjective, and isometric operatorsφmby

φm:Fm,·−→^m,·₂, hi−→ei, 3.5 whereeidenotes theith unit vector in^m, 1≤i≤m.

Theorem 3.1. Assume that the eigenvalue sequenceλj_j∈Æ

of the covariance operatorCX satisfies λ_j ≈ j^−b for−b <−1, and let >0 be arbitrary. Assume further thathj_j∈

Æ

is a rate optimal ONS forX inH. One setsmn logn¹ forn ∈. Then, one gets for every sequenceαn_n∈Æof r-optimaln-quantizers forφ_mnVmnXin^mn, · ₂the asymptotics

en,rX, H∼er

X, H, φ⁻¹_mnαn

∼

X−π_φ⁻¹

mnαn

V_mnX^r1/r

, 3.6

asn → ∞.

Proof. Let fj_j∈Æ be the corresponding orthonormal eigenvector sequence of C_X. Classic eigenvalue theory yields for everym∈

∞ im

f_i f_i, X

2

^∞

im

λ_i≤^∞

im

hi, X²

∞ im

h_ihi, X

2

. 3.7

Combining this with rate optimality of the ONShj_j∈ÆforX, we get

X−V_mnX²

∞ imn 1

hihi, X

2

^∞

imn 1

hi, X²

≈ ^∞

imn 1

λj≈mn^−b−1, n−→ ∞.

3.8

Using the equivalence of ther-norms of Gaussian random variables23, Corollary 3.2, and sinceX−V_mnXis Gaussian, we get for allr ≥1

X−V_mnX

Lr^È≈mn^−1/2b−1, n−→ ∞. 3.9 With ω as in Theorem 1.1, we get by using 3.4 and Theorem 1.1 the weak asymptotics e_n,rX, H≈ωlogn^−1/2≈logn^−1/2b−1, n → ∞. Therefore, the sequencemn_n∈Æ

satisfies2.7since X−V_mnX

Lr^È≈

logn_−1/2b−11

oen,rX, H, n−→ ∞, 3.10

and the assertion follows fromTheorem 2.3.

3.2. Admissible Sequences

In order to show that linear operatorsVm similar to those used in the subsection above are suitable for the requirements of Theorem 2.3, we need to do some preparations. Since the covariance operatorC_X of a Gaussian random variable is symmetric and compactin fact trace class, we will use a well-known result concerning these operators. This result can be used for quantization in the following way.

Lemma 3.2. LetXbe a centered Gaussian random variable with values in the Hilbert spaceHand XX1 X2, whereX1andX2are independent centered Gaussians. Then

CX CX1 CX2. 3.11

Letλi, λ¹_i , λ²_i ,i∈be the positive monotone decreasing eigenvalues ofCX, CX1, andCX2. Then, for i∈it holds that

λ¹_i , λ²_i ≤λ_i. 3.12

Proof. Since X₁, X₂ are independent centered Gaussians, we have X1, uX2

X1, uX2 0 for allu∈H. This easily leads to

C_Xu X1 X₂, uX1 X₂ X1, uX1

X2, uX2 CX1u CX2u. 3.13 The covariance operator of a centered Gaussian random variable is positive semidefinite.

Hence, by using a result on the relation of the eigenvalues of those operatorssee, e.g.,24, page 213, we get inequalities3.12.

Letgi_i∈Æbe an admissible sequence forX, and assume that_∞

i1ξ_ig_i Xa.s. In this subsection, we use the following notations.

1We setFm:span{g1, . . . , gm}.

2We defineV_m:H → F_m⊂Hby

V_m f_j

:f_j^m λ^m_j

λj

, 3.14

forj ≤ mandVmfj : 0 forj > m, where λj andfj denote the eigenvalues and the corresponding eigenvectors ofCX andλ^m_j andf_j^m the eigenvalues and the corresponding eigenvectors ofC_Xm, withX_mdefined as

Xm:^m

i1

giξi. 3.15

Note thatV_mmapsHontoF_msince span

g₁, . . . , g_m

span

f₁^m, . . . , f_m^m

. 3.16

Furthermore, it is important to mention that one does not need to know λj and fj explicitly to construct the subsequent quantizers, since we can find for any

m ∈ a random variableZm d X such thatVmZm _m

i1ξigi see the proof of Theorem 3.3, which is explicitly known and sufficient to know for the construction.

3Define the linear, surjective, and isometric operatorsφmby

φ_m:Fm,·−→^m,·₂, f_i^m−→e_i, 3.17

wheree_idenotes theith unit vector of^mfor 1≤i≤m.

Theorem 3.3. Assume that the eigenvalue sequenceλj_j∈

Æ

of the covariance operatorCX satisfies λj ≈j^−bfor−b <−1, and let >0 arbitrary. Assume thatgj_j∈

Æ

is a rate optimal admissible sequence forXinH. One setsmn logn¹ forn∈. Then, there exist random variablesZ_m, m∈, withZm d

Xsuch that for every sequenceαn_n∈^Æofr-optimaln-quantizers forφ_mnVmnZmn in^mn, · ₂

e_n,rX, H∼e_r

X, H, φ⁻¹_mnαn

∼

X−π_φ−1 mnαn

V_mn

Z_mn^r1/r

, 3.18

asn → ∞.

Proof. Linearity ofVm_m∈^Æfollows from the orthogonality of the eigenvectors. In view of the inequalities for the eigenvalues inLemma 3.2and the orthonormality of the familyfi_i∈Æ, we have for everyh_∞

i1f_ia_i∈Hwithh² _∞

i1a²_i ≤1

Vmh² Vm

∞ i1

aifi

2

^m

i1

a²_iλ^m_i λi ≤^∞

i1

a²_i ≤1, 3.19

such thatVm_op≤1.

Note next that for everym ∈ there exist independentN0,1-distributed random variablesζ^m_{i 1≤i≤m}satisfying

m i1

ξ_ig_i^m

i1

λ^m_i f_i^mζ^m_i a.s. 3.20

Then, we choose random variables ζ^m_{i m 1≤i<∞} such that ζ_i^m_1≤i<∞ is a sequence of independentN0,1-distributed random variables. We set

Zm:^∞

i1

ζ^m_i

λifi 3.21

and get by using rate optimality of the admissible sequencesgj_j∈Æand

λ_jf_jj∈Æ

X−V_mZm²

∞ i1

g_iξ_i−V_m^∞

i1

λ_if_iζ^m_i

2

∞ i1

giξi−^m

i1

λ^m_i f_i^mζ^m_i

2

∞ im 1

giξi

2

≈

∞ im 1

λifiξi

2

^∞

im 1

λi≈m^−b−1, m−→ ∞,

3.22

where rate optimality of λ_jf_j

j∈^Æis a consequence of

X²_L2È−

∞ im 1

giξi

2

m i1

giξi

2

^m

i1

λ^m_i ≤^m

i1

λi. 3.23

Using the equivalence of ther-norms of Gaussian random variables23, Corollary 3.2, and sinceX−V_mnXis Gaussian, we get for allr ≥1

X−V_mnX

Lr^È≈mn^−1/2b−1, n−→ ∞. 3.24 With ω as in Theorem 1.1, we get by using 3.4 and Theorem 1.1 the weak asymptotics en,rX, H≈ωlogn^−1/2≈logn^−1/2b−1,n → ∞. Therefore, the sequencemn_n∈Æ

satisfies2.7since X−V_mnX

Lr^È≈

logn_−1/2b−11

oen,rX, H, n−→ ∞, 3.25

and the assertion follows fromTheorem 2.3.

3.3. Comparison of the Different Schemes

At least in the case r 2, we have a strong preference for using the method as described in Section 3.1. We use the notations as in the above subsections including an additional indexationi 1,2 forV_mⁱ, φ_mⁱ, αⁱ_n and m, n ∈ , where αⁱ_n , for i 1,2, are defined as in Theorems3.1and3.3. Note that for this purpose the size of the codebooknand the size of the subspaces dimFm mcan be chosen arbitrarilyi.e.,mdoes not depend onn. The ONS hi_i∈^Æis chosen as the ONS derived with the Gram-Schmidt procedure from the admissible sequencegj_j∈

Æ

for the Gaussian random variableX in the Hilbert spaceH, such that the definition ofFmcoincides in the twosubsections.

Proposition 3.4. It holds form, n∈that

X−π

φ²m⁻¹α²n

V_m²Zm ²≥

X−π

φm¹φ¹m⁻¹α¹n Vm¹X

². 3.26 Proof. Consider forXthe decompositionX pr_F^⊥_mX prFmX. The key is the orthogonality ofpr_F^⊥_mXtoprFmX,π_φ2

m⁻¹α²n V_m²Zm, andπ_φ1

m⁻¹α¹n V_m¹X, which gives the two equalities in the following calculation:

X−π

φ²m⁻¹α²n

V_m²Zm ²

prFmX−π_φ2 m⁻¹α²n

Vm²

Z²m

² pr_F^⊥_mX²

∗≥

pr_FmX−π

φ¹m⁻¹α¹n

V_m¹X

² pr_F⊥

mX²

X−π_φ¹

m⁻¹α¹n Vm¹X².

3.27

The inequality∗follows from the optimality of the codebookφ¹m⁻¹α¹n forprFmX V_m¹X.

4. Gaussian Processes with Paths in C0, 1, ·

In the previous section, where we worked with Gaussian random variables in Hilbert spaces, we saw that special Hilbertian subspaces, projections, and other operators linked to the Gaussian random variable were good tools to develop asymptotically optimal quantizers based on Theorem 2.3. Since we now consider the non-Hilbertian separable Banach space C0,1, · _∞, we have to find different tools that are suitable to useTheorem 2.3.

The tools used in20are B-splines of order s ∈ . In the case s 2, that we will consider in the sequel, these splines span the same subspace ofC0,1, · _∞as the classical Schauder basis. We set forx∈0,1,m≥2, and 1≤i≤mthe knotst^m_i : i−1/m−1and the hat functions

f_i^mx:χ_t^m

i ,t^m_{i 1}x 1−

x−t^m_i

m−1 χ_t^m

i−1,t^m_i x x−t^m_i−1

m−1. 4.1

For the remainder of this subsection, we will use the following notations.

1As subspacesFmwe setFm:span{f_j^m,1≤j ≤m}.

2As linear and continuous operatorsVm:C0,1 → Fmwe set the quasiinterpolant

Vm

f :^m

i1

f_i^mβ^m_i f

, 4.2

whereβ^m_i f:ft^m_i .

3The linear and surjective isometric mappingsφmone defines as φ_m:Fm,·_∞−→R^m,·_∞,

m i1

a_if_i^m−→a1, . . . , a_m. 4.3

It is easy to see that_m

i1aif_i^m_∞a1, . . . , am_∞holds for everya∈^m. For the application of Theorem 2.3, we need to know the error bounds for the approximation ofX with the quasiinterpolantV_mX. For Gaussian random variables, we can provide the following result based on the smoothness of an admissible sequence forXin E.

Proposition 4.1. Let gj_j∈

Æ

be admissible for the centered Gaussian random variable X in C0,1, · _∞. Assume that

1gj ≤C₁j^−θfor everyj ≥1, θ >1/2, andC₁<∞,

2gj ∈C²0,1withg_j ≤C2j^{−θ 2}for everyj≥1 andC2 <∞.

Then, for any >0 and some constantC <∞it holds that

X−VmX_Lr^È≤Cm^{−0,8θ−1/2} , 4.4

for everyr ≥1.

Proof. Using of25, Theorem 1, we get

∞ ik

ξ_ig_i Lr^È

≤ C3

k^θ−1/2−1 4.5

for an arbitrary₁>0, some constantC₃<∞, and everyk∈. Thus, we have X−V_mX_LrÈ≤

∞ ik

ξ_ig_i Lr^È

V_m

∞ ik

ξ_ig_i Lr^È

k−1

i1

ξigi−Vm

k−1 i1

ξigi

Lr^È

≤ 2C3

k^θ−1/2−1

k−1

i1

ξ_ig_i−V_m

k−1

i1

ξ_ig_i Lr^È

.

4.6

Using of26, Chapter 7, Theorem 7.3, we get for some constantC₄<∞

k−1

i1

ξ_ig_i−W_m k−1

i1

ξ_ig_i

≤C₄ω k−1

i1

ξ_ig_i, 1 m−1

, 4.7

where the module of smoothnessωf, δis defined by ω

f, δ : sup

0≤h<δ

fx−2fx h fx 2h

∞. 4.8

For an arbitraryf ∈C²0,1we have by using Taylor expansion fx−2fx h fx 2h_∞

h² ≤2f_∞. 4.9

Combining this, we get for an arbitrary₂ >0 and constantsC₅, C₆, C₇ <∞, using again the equivalence of Gaussian moments,

X−VmX_LrÈ≤ 2C3

k^θ−1/2−1 1 m²

2

k−1

i1

ξig_i

r

∞

1/r

≤ 2C₃ k^θ−1/2−1

1 m²C5

2

k−1

i1

ξig_i ∞

≤ 2C3

k^θ−1/2−1 1 m²C6

k−1 i1

i^{θ 2 2}|ξi|

≤ 2C3

k^θ−1/2−1 1

m²C7k^{−θ 3 2}.

4.10

To minimize over k, we choosek km m^0,8. Thus, we get for some constantC <∞and an arbitrary >0

X−VmX_LrÈ≤Cm^{−0,8θ−1/2} . 4.11

Now, we are able to prove the main result of this section.

Theorem 4.2. LetXbe a centered Gaussian random variable andgj_j∈Æ

an admissible sequence for XinC0,1fulfilling the assumptions ofProposition 4.1withθ b/2, where the constantb >1 satisfiesλj ^² Kj^−b with λj, j ∈ denoting the monotone decreasing eigenvalues of the covariance operatorCX ofXin H L20,1andK > 0. One setsmn : logn^5/4 for some > 0.

Then, for every sequenceαn_n∈Æofr-optimaln-quantizers forφ_mnV_mnXin^mn, · _∞, it holds that

en,rX,C0,1,·_∞∼er

X, C0,1, φ⁻¹_mnαn

∼

X−π_φ−1

mnαnVmnX^r

∞

1/r

,

4.12

asn → ∞.

Proof. For everyh∈C0,1, · _∞, withh_∞≤1it holds that Vmh_∞≤ sup

x∈0,1

m i1

h

t^m_i f_i^mx≤ h_∞sup

x∈0,1

m i1

f_i^mx≤1, 4.13

since{f_i^m,1≤i≤m}are partitions of the one for everym∈, so thatVm_op≤1.

We get a lower bound for the quantization errore_n,rX, C0,1from the inequality f

L20,1≤f

∞, 4.14

for allf∈C0,1⊂L₂0,1. Consequently, we have

en,rX, C0,1≥en,rX, L20,1. 4.15 FromTheorem 1.1and3.4we obtain

logn^−1/2b−1 ≈ ω

logn_−1/2

ºen,rX, C0,1, n−→ ∞, 4.16 whereωis given as inTheorem 1.1. Finally, we get by combining4.16andProposition 4.1 for sufficiently smallδ >0

X−V_mnX

Lr^È≤Cmn−0,81/2b−1 δ

o

logn_−1/2b−1

oen,rX, C0,1, n−→ ∞, 4.17

and the assertion follows fromTheorem 2.3.

5. Processes with Path Space L

0, 1, ·

Another useful tool for our purposes is the Haar basis inL_p0,1for 1 ≤p < ∞, which is defined by

e₀:χ_0,1 e₁:χ_0,1/2−χ_1/2,1

e₂ⁿ _k:2^n/2e₁2ⁿ· −k, n∈, k∈ {0, . . . ,2ⁿ−1}. 5.1 This is an orthonormal basis ofL₂0,1and a Schauder basis ofL_p0,1forp∈1,∞, that is,f, e0 _∞

n0₂ⁿ₋₁

k1f, e2ⁿ ke2ⁿ kconverges tofinLp0,1for everyf ∈Lp0,1; see 27.

The Haar basis was used in 17to construct rate optimal sequences of quantizers for mean regular processes. These processes are specified through the property that for all 0≤s≤t≤1

|Xt−Xs|^p≤

ρt−s_p

, 5.2