# A Constructive Sharp Approach to Functional Quantization of Stochastic Processes

Authors: Stefan Junglen; Harald Luschgy
Journal: Journal of Applied Mathematics
 (2010)

---

## Abstract

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, Lévy processes.


---

## Body

## 1. Introduction

We consider a separable Banach space(E,∥·∥) and a Borel random variable X: (Ω,ℱ,ℙ)→(E,ℬ(E)) with finite rth moment 𝔼∥X∥r for some r∈[1,∞).For a given natural numbern∈ℕ, the quantization problem consists in finding a set α⊂E that minimizes (1.1)er(X,(E,‖⋅‖),α)=er(X,E,α):=(Emin⁡a∈α‖X-a‖r)1/r
over all subsets α⊂E with card (α)≤n. Such sets α are called n-codebooks or n-quantizers. The corresponding infimum (1.2)en,r(X,(E,‖⋅‖))=en,r(X,E):=inf⁡α⊂E,card(α)≤ner(X,E,α)
is called the nth Lr-quantization error of X in E, and any n-quantizer α fulfilling (1.3)er(X,E,α)=en,r(X,E)
is called r-optimaln-quantizer. For a given n-quantizer α one defines the nearest neighbor projection (1.4)πα:E⟶α,x⟶∑a∈αaχCa(α)(x),
where the Voronoi partition {Ca(α),a∈α} is defined as a Borel partition of E satisfying (1.5)Ca(α)⊂{x∈E:‖x-a‖=min⁡b∈α‖x-b‖}.
The random variable πα(X) is called α-quantization of X. One can easily verify that πα(X) is the best quantization of X in α⊂E, which means that for every random variable Y with values in α we have(1.6)er(X,E,α)=(E‖X-πα(X)‖r)1/r≤(E‖X-Y‖r)1/r.Applications of quantization go back to the 1940s, where quantization was used for the finite-dimensional settingE=ℝd, 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 quantization [3] as well as functional quantization [4, 5] is useful. The terminology functional quantization is used when the Banach space E is a function space, such as E=(Lp[0,1],∥·∥p) or E=C([0,1],∥·∥∞). In this case, the realizations of X can 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 ofen,r(X,E) when n tends to infinity. By separability of (E,∥·∥), we can choose a dense subset {ci,i∈ℕ} and we can deduce in view of(1.7)0≤lim⁡n→∞Emin⁡1≤i≤n‖X-ci‖r=Elim⁡n→∞min⁡1≤i≤n‖X-ci‖r=0
that en,r(X,E) tends to zero as n tends to infinity.A natural question is then if it is possible to describe the asymptotic behavior ofen,r(X,E). It will be convenient to write an~bn for sequences (an)n∈ℕ and (bn)n∈ℕ if an/bn→n→∞1, an≲bn if limsupn→∞an/bn≤1 and an≈bn if 0<liminfn→∞an/bn≤limsupn→∞an/bn<∞.In the finite-dimensional setting(ℝd,∥·∥) this behavior can satisfactory be described by the Zador Theorem (see [6]) for nonsingular distributions ℙX. 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 function ρ:(s,∞)→(0,∞) for an s≥0  regularly varying at infinity with index b∈ℝ if for every c>0(1.8)lim⁡x→∞ρ(cx)ρ(x)=cb.Theorem 1.1 (see [7]).
LetX be a centered Gaussian random variable with values in the separable Hilbert space (H,〈·,·〉) and λn,n∈ℕ the decreasing eigenvalues of the covariance operator CX:H→H, u→𝔼〈u,X〉X (which is a symmetric trace class operator). Assume that λn~ρ(n) for some regularly varying function ρ with index -b<-1. Then, the asymptotics of the quantization error is given by
(1.9)en,2(X,H)~((b2)b-1bb-1)1/2ω(log⁡(n))-1/2,n⟶∞,
where ω(x):=1/xρ(x).Note that any change of ~ in the assumption thatλn~ρ(n) to either ≲, ≈ or ≳ leads to the same change in (1.9). Theorem 1.1 can also be extended to an index b=1 (see [7]). Furthermore, a generalization to an arbitrary moment r (see [8]) as well as similar results for special Gaussian random variables and diffusions in non-Hilbertian function spaces (see, e.g., [9–11]) have been developed. Moreover, several authors established a precise link between the quantization error and the behavior of the small ball function of a Gaussian measure (see, e.g., [12, 13]) which can be used to derive asymptotics of quantization errors. More recently, for several types of Lèvy processes (sharp) optimal rates have been developed by several authors (see, 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 compute (at least numerically) good codebooks.Considering again Gaussian random variables in a Hilbert space setting, the proof of Theorem1.1 shows us how to construct asymptoticallyr-optimaln-quantizers for these processes, which means that sequences of n-quantizers αn,n∈ℕ satisfy(1.10)er(X,E,αn)~en,r(X,E),n⟶∞.
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 operator CX to pursue this approach.If we consider other non-Hilbertian function spaces(E,∥·∥) 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-constructive (e.g., [12, 13]) or tailored to one specific process type (e.g., [9–11]) or the constructed quantizers do not achieve the sharp rate in the sense of (1.10) (e.g., [17] or [20]) but just the weak rate(1.11)er(X,E,αn)≈en,r(X,E),n⟶∞.In this paper, we develop a constructive approach to calculate sequences of asymptoticallyr-optimal n-quantizers (in the sense of (1.10)) for a broad class of random variables in infinite dimensional Banach spaces (Section 2). Constructive means in this case that we reduce the quantization problem to the quantization problem of a ℝd-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 unknown (Sections 3.1 and 3.2), or for quantization problems in different Banach spaces (Sections 4 and 5).In Section4, we discuss Gaussian random variables in (C(0,1),∥·∥∞). This part is related to the PhD thesis of Wilbertz [20]. 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 error (in the sense of (1.10)). Moreover, we can show that the dimensions of the subspaces wherein these quantizers are contained can be lessened without loosing the sharp asymptotics property.In Section5, we use some ideas of Luschgy and Pagès [17] and develop for Gaussian random variables and for a broad class of Lévy processes asymptotically optimal quantizers in the Banach space (Lp([0,1]),∥·∥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 a (rough) lower bound for the quantization error, that is,en,r(X,E)≳C1log(n)-b1 and the true but unknown rate is en,r(X,E)~C2log(n)-b2 for constants C1,C2,b1,b2∈(0,∞), then, we are able to construct a sequence of n-quantizers αn, n∈ℕ that satisfies(1.12)er(X,E,αn)~en,r(X,E)~C2log⁡(n)-b2,n⟶∞
for the optimal but still unknown constants C2,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 Lévy 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 Lévy processes in Section6, and finally illustrate some of our results in Section 7.Notations and Definitions
If not explicitly differently defined, the following notations hold throughout the paper.(i)
We denote byX a Borel random variable in the separable Banach space (E,∥·∥) with card(supp(ℙX))=∞.(ii)
∥·∥ will always denote the norm in E whereas ∥·∥Lr(ℙ) will denote the norm in Lr(Ω,ℱ,ℙ).(iii)
The scalar product in a Hilbert spaceH will be denoted by 〈·,·〉.(iv)
The smallest integer above a given real numberx will be denoted by ⌈x⌉.(v)
A sequence(gj)j∈ℕ∈Eℕ is called admissible for a centered Gaussian random variable X in E if and only if for any sequence (ξi)i∈ℕ of independent N(0,1)-distributed random variables it holds that ∑i=1∞ξigi converges a.s. in (E,∥·∥) and X=d∑i=1∞ξigi. An admissible sequence (gj)j∈ℕ∈Eℕ is called rate optimal for X in E if and only if
(1.13)E‖∑i=m∞ξigi‖2≈inf⁡{E‖∑i=m∞ξifi‖2:(fi)i∈NadmissibleforX},
as m→∞. A precise characterization of admissible sequences can be found in [21].(vi)
An orthonormal system (ONS)(hi)i∈ℕ is called rate optimal for X in the Hilbert space H if and only if
(1.14)E‖∑i=m∞hi〈hi,X〉‖2≈inf⁡{E‖∑i=m∞fi〈fi,X〉‖2:(fi)i∈NONSinH},
as m→∞.


## 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 2.1 (see [22]).
LetE, F be separable Banach spaces, X a random variable in E, and T:E→F. (1)
IfT is Lipschitz continuous with Lipschitz constant L, then
(2.1)en,r(T(X),F)≤Len,r(X,E),
and for every n-quantizer α for X it holds that
(2.2)er(T(X),F,T(α))≤Ler(X,E,α).(2)
LetT:E→F be linear, surjective, and isometric. Then, for c≥0 and f∈F(2.3)en,r(cT(X)+f,F)=cen,r(X,E),
and for every n-quantizer α for X it holds that
(2.4)er(cT(X)+f,F,T(α))=cT(er(X,E,α))+f.To formulate the main result, we need for an infinite subsetJ⊂ℕ the following.Condition 1.
There exist linear operatorsVm:E→Fm⊂E for m∈J with ∥Vm∥op≤1, for finite dimensional subspaces Fm with dim(Fm)=m, where the norm ∥·∥op is defined as
(2.5)‖Vm‖op:=sup⁡x∈E,‖x‖≤1‖Vm(x)‖.Condition 2.
There exist linear isometric and surjective operatorsϕm:(Fm,∥·∥)→(ℝm,|·|m) with suitable norms |·|m in ℝm for all m∈J.Condition 3.
There exist random variablesZm for m∈J in E with Zm=dX, such that for the approximation error ∥∥X-Vm(Zm)∥∥Lr(ℙ)it holds that(2.6)‖‖X-Vm(Zm)‖‖Lr(P)⟶0,
as m→∞ along J.Remark 2.2.
The crucial point in Condition1 is the norm one restriction for the operators Vm. Condition 2 becomes Important when constructing the quantizers in ℝm 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 Condition 3 to choose explicitly a sequence m(n)∈J, n∈ℕ such that
(2.7)‖‖X-Vm(n)(Zm(n))‖‖Lr(P)=o(en,r(X,E)),n⟶∞.Theorem 2.3.
Assume that Conditions1–3 hold for some infinite subset J⊂ℕ. One chooses a sequence (m(n))n∈ℕ∈Jℕ such that (2.7) is satisfied. For n∈ℕ, let αn be an r-optimal n-quantizer for ξn:=ϕm(n)(Vm(n)(Zm(n))) in (ℝm(n),|·|m(n)).
Then,(ϕm(n)-1(αn))n∈ℕ is an asymptotically r-optimal sequence of n-quantizers for X in E and
(2.8)en,r(X,E)~(E‖X-πϕm(n)-1(αn)(Vm(n)(Zm(n)))‖r)1/r~er(X,E,ϕm(n)-1(αn)),
as n→∞.Remark 2.4.
Note, that forn∈ℕ there always exist r-optimal n-quantizers for ξn ([6, Theorem 4.12]).Proof.
Using Condition3 and the fact that en,r(X,E)>0 for all n∈ℕ since card(supp(ℙX))=∞, we can choose a sequence (m(n))n∈ℕ∈ℕℕ fulfilling (2.7). Using Lemma 2.1 and Condition 2, we see that ϕm(n)-1(αn) is an r-optimal n-quantizer for Vm(n)(Zm(n)) in Fm(n). Then, by using Condition 1, (2.7), and Lemma 2.1 we get
(2.9)en,r(X,E)≤(E‖X-πϕm(n)-1(αn)(Vm(n)(Zm(n)))‖r)1/r≤(E‖X-Vm(n)(Zm(n))‖r)1/r+(E‖Vm(n)(Zm(n))-πϕm(n)-1(αn)(Vm(n)(Zm(n)))‖r)1/r=(E‖X-Vm(n)(Zm(n))‖r)1/r+en,r(Vm(n)(Zm(n)),(Fm(n),‖⋅‖))≤(E‖X-Vm(n)(Zm(n))‖r)1/r+en,r(Zm(n),E)=(E‖X-Vm(n)(Zm(n))‖r)1/r+en,r(X,E)~en,r(X,E),n⟶∞.
The last equivalence of the assertion follows from (1.6).Remark 2.5.
We will usually chooseZm=X for all m∈ℕ, with an exception in Section 3 and J=ℕ.Remark 2.6.
The crucial factor for the numerical implementation of the procedure is the dimensions(m(n))n∈ℕ of the subspaces (Fm(n))n∈ℕ. For the well-known case of the Brownian motion in the Hilbert space H=L2([0,1]) it is known that this dimension sequence can be chosen as m(n)≈log(n), n→∞. In the following examples we will see that we can often obtain similar orders like log(n)c for constants c just slightly higher than one.We point out that there is a nonasymptotic version of Theorem2.3 for nearly optimal n-quantizers, that is, for n-quantizers, which are optimal up to ϵ>0. Its proof is analogous to the proof of Theorem 2.3.Proposition 2.7.
Assume that Conditions1–3 hold. Let m(ϵ):=inf{m∈ℕ:∥X-Vm(Zm)∥Lr(ℙ)<ϵ}, and for n∈ℕ one sets ξn:=ϕm(ϵ)(Vm(ϵ)(Zm(ϵ))). Then, it holds for every n∈ℕ and for every r-optimal n-quantizer αn for ξn in (ℝm(ϵ),|·|m(ϵ)) that
(2.10)er(X,E,πϕm(ϵ)-1(αn)(Vm(ϵ)(Zm(ϵ))))≤en,r(X,E)+ϵ.


## 3. Gaussian Processes with Hilbertian Path Space

In this chapter, letX be 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 sequence (ξi)i∈ℕ of independent N(0,1)-distributed random variables(3.1)X=d∑i=1∞λifiξi,
where λi denote the eigenvalues and fi denote the corresponding orthonormal eigenvectors of the covariance operator CX of X (Karhunen-Loève expansion). If these parameters are known, we can choose a sequence (dn)n∈ℕ such that a sequence of optimal quantizer αn for Xn=∑i=1d(n)λifiξi is asymptotically optimal for X in E.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(ℙ) or in H.Before we will use these representations forX to find suitable triples (Vm,Fm,ϕm) as in Theorem 2.3, note that for Gaussian random variables in H fulfilling suitable assumptions we know that(1)
Let(hi)i∈ℕ be an orthonormal basis of H. Then
(3.2)X=∑i=1∞hi〈hi,X〉a.s..
Compared to (3.1) we see that 〈hi,X〉 are still Gaussian but generally not independent.(2)
Let(gj)j∈ℕ be an admissible sequence for X in H such that
(3.3)X=d∑i=1∞ξigi.
Compared to (3.1) the sequence (gi)i∈ℕ is generally not orthogonal.(3.4)en,2(X,H)≈en,s(X,H),n⟶∞
for all s≥1; see [13]. Thus, we will focus on the case s=2 to search for lower bounds for the quantization errors.


### 3.1. Orthonormal Basis

Let(hm)m∈ℕ be an orthonormal basis of H. For the subsequent subsection we use the following notations. (1)
We setFm=span{h1,…,hm}.(2)
We setVm:=prFm:E→Fm, the orthogonal projection on Fm. It is well known that ∥Vm∥op=1.(3)
Define the linear, surjective, and isometric operatorsϕm by
(3.5)ϕm:(Fm,‖⋅‖)⟶(Rm,‖⋅‖2),hi⟶ei,
where ei denotes the ith unit vector in ℝm, 1≤i≤m.Theorem 3.1.
Assume that the eigenvalue sequence(λj)j∈ℕ of the covariance operator CX satisfies λj≈j-b for -b<-1, and let ϵ>0 be arbitrary. Assume further that (hj)j∈ℕ is a rate optimal ONS for X in H. One sets m(n)=⌈log(n)1+ϵ⌉ for n∈ℕ. Then, one gets for every sequence (αn)n∈ℕ of r-optimal n-quantizers for ϕm(n)(Vm(n)(X)) in (ℝm(n),∥·∥2) the asymptotics
(3.6)en,r(X,H)~er(X,H,ϕm(n)-1(αn))~(E‖X-πϕm(n)-1(αn)(Vm(n)(X))‖r)1/r,
as n→∞.Proof.
Let(fj)j∈ℕ be the corresponding orthonormal eigenvector sequence of CX. Classic eigenvalue theory yields for every m∈ℕ(3.7)E‖∑i=m∞fi〈fi,X〉‖2=∑i=m∞λi≤∑i=m∞E〈hi,X〉2=E‖∑i=m∞hi〈hi,X〉‖2.
Combining this with rate optimality of the ONS (hj)j∈ℕ for X, we get
(3.8)E‖X-Vm(n)(X)‖2=E‖∑i=m(n)+1∞hi〈hi,X〉‖2=∑i=m(n)+1∞E〈hi,X〉2≈∑i=m(n)+1∞λj≈m(n)-(b-1),n⟶∞.
Using the equivalence of the r-norms of Gaussian random variables ([23, Corollary 3.2]), and since X-Vm(n)(X) is Gaussian, we get for all r≥1(3.9)‖‖X-Vm(n)(X)‖‖Lr(P)≈m(n)-(1/2)(b-1),n⟶∞.
With ω as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics en,r(X,H)≈(ω(log(n)))-1/2≈(log(n))-(1/2)(b-1),n→∞. Therefore, the sequence (m(n))n∈ℕ satisfies (2.7) since
(3.10)‖‖X-Vm(n)(X)‖‖Lr(P)≈(log⁡(n))-(1/2)(b-1)(1+ϵ)=o(en,r(X,H)),n⟶∞,
and the assertion follows from Theorem 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 operator CX of a Gaussian random variable is symmetric and compact (in 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.
LetX be a centered Gaussian random variable with values in the Hilbert space H and X=X1+X2, where X1 and X2 are independent centered Gaussians. Then
(3.11)CX=CX1+CX2.
Let λi,λi(1),λi(2), i∈ℕ be the positive monotone decreasing eigenvalues of CX,CX1, and CX2. Then, for i∈ℕ it holds that
(3.12)λi(1),λi(2)≤λi.Proof.
SinceX1,X2 are independent centered Gaussians, we have 𝔼〈X1,u〉X2=𝔼〈X1,u〉𝔼X2=0 for all u∈H. This easily leads to
(3.13)CX(u)=E〈X1+X2,u〉(X1+X2)=E〈X1,u〉X1+E〈X2,u〉X2=CX1(u)+CX2(u).
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 operators (see, e.g., [24, page 213]), we get inequalities (3.12).Let(gi)i∈ℕ be an admissible sequence for X, and assume that ∑i=1∞ξigi=X a.s. In this subsection, we use the following notations.(1)
We setFm:=span{g1,…,gm}.(2)
We defineVm:H→Fm⊂H by
(3.14)Vm(fj):=fj(m)λj(m)λj,
for j≤m and Vm(fj):=0 for j>m, where λj and fj denote the eigenvalues and the corresponding eigenvectors of CX and λj(m) and fj(m) the eigenvalues and the corresponding eigenvectors of CXm, with Xm defined as
(3.15)Xm:=∑i=1mgiξi.
Note that Vm maps H onto Fm since
(3.16)span⁡{g1,…,gm}=span⁡{f1(m),…,fm(m)}.
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∈ℕ a random variable Zm=dX such that Vm(Zm)=∑i=1mξigi (see the proof of Theorem 3.3), which is explicitly known and sufficient to know for the construction.(3)
Define the linear, surjective, and isometric operatorsϕm by
(3.17)ϕm:(Fm,‖⋅‖)⟶(Rm,‖⋅‖2),fi(m)⟶ei,
where ei denotes the ith unit vector of ℝm for 1≤i≤m.Theorem 3.3.
Assume that the eigenvalue sequence(λj)j∈ℕ of the covariance operator CX satisfies λj≈j-b for -b<-1, and let ϵ>0 arbitrary. Assume that (gj)j∈ℕ is a rate optimal admissible sequence for X in H. One sets m(n)=⌈log(n)1+ϵ⌉ for n∈ℕ. Then, there exist random variables Zm,m∈ℕ, with Zm=dX such that for every sequence (αn)n∈ℕ of r-optimal n-quantizers for ϕm(n)(Vm(n)(Zm(n))) in (ℝm(n),∥·∥2)(3.18)en,r(X,H)~er(X,H,ϕm(n)-1(αn))~(E‖X-πϕm(n)-1(αn)(Vm(n)(Zm(n)))‖r)1/r,
as n→∞.Proof.
Linearity of(Vm)m∈ℕ follows from the orthogonality of the eigenvectors. In view of the inequalities for the eigenvalues in Lemma 3.2 and the orthonormality of the family (fi)i∈ℕ, we have for every h=∑i=1∞fiai∈H with ∥h∥2=∑i=1∞ai2≤1(3.19)‖Vm(h)‖2=‖Vm(∑i=1∞aifi)‖2=∑i=1mai2λi(m)λi≤∑i=1∞ai2≤1,
such that ∥Vm∥op≤1.
Note next that for everym∈ℕ there exist independent N(0,1)-distributed random variables (ζi(m))1≤i≤m satisfying
(3.20)∑i=1mξigi=∑i=1mλi(m)fi(m)ζi(m)a.s.
Then, we choose random variables (ζi(m))m+1≤i<∞ such that (ζi(m))1≤i<∞ is a sequence of independent N(0,1)-distributed random variables. We set
(3.21)Zm:=∑i=1∞ζi(m)λifi
and get by using rate optimality of the admissible sequences (gj)j∈ℕ and (λjfj)j∈ℕ(3.22)E‖X-Vm(Zm)‖2=E‖∑i=1∞giξi-Vm(∑i=1∞λifiζi(m))‖2=E‖∑i=1∞giξi-∑i=1mλi(m)fi(m)ζi(m)‖2=E‖∑i=m+1∞giξi‖2≈E‖∑i=m+1∞λifiξi‖2=∑i=m+1∞λi≈m-(b-1),m⟶∞,
where rate optimality of (λjfj)j∈ℕ is a consequence of
(3.23)‖‖X‖‖L2(P)2-E‖∑i=m+1∞giξi‖2=E‖∑i=1mgiξi‖2=∑i=1mλi(m)≤∑i=1mλi.
Using the equivalence of the r-norms of Gaussian random variables ([23, Corollary 3.2]), and since X-Vm(n)(X) is Gaussian, we get for all r≥1(3.24)‖‖X-Vm(n)(X)‖‖Lr(P)≈m(n)-(1/2)(b-1),n⟶∞.
With ω as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics en,r(X,H)≈(ω(log(n)))-1/2≈(log(n))-(1/2)(b-1), n→∞. Therefore, the sequence (m(n))n∈ℕ satisfies (2.7) since
(3.25)‖‖X-Vm(n)(X)‖‖Lr(P)≈(log⁡(n))-(1/2)(b-1)(1+ϵ)=o(en,r(X,H)),n⟶∞,
and the assertion follows from Theorem 2.3.


### 3.3. Comparison of the Different Schemes

At least in the caser=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 indexation i=1,2 for (Vm(i),ϕm(i),αn(i)) and m,n∈ℕ, where αn(i), for i=1,2, are defined as in Theorems 3.1 and 3.3. Note that for this purpose the size of the codebook n and the size of the subspaces dim(Fm)=m can be chosen arbitrarily (i.e., m does not depend on n). The ONS (hi)i∈ℕ is chosen as the ONS derived with the Gram-Schmidt procedure from the admissible sequence (gj)j∈ℕ for the Gaussian random variable X in the Hilbert space H, such that the definition of Fm coincides in the two subsections.Proposition 3.4.
It holds form,n∈ℕ that
(3.26)E‖X-π(ϕm(2))-1(αn(2))(Vm(2)(Zm))‖2≥E‖X-π(ϕm(1)(ϕm(1))-1(αn(1))(Vm(1)(X))‖2.Proof.
Consider forX the decomposition X=prFm⊥(X)+prFm(X). The key is the orthogonality of prFm⊥(X) to prFm(X), π(ϕm(2))-1(αn(2))(Vm(2)(Zm)), and π(ϕm(1))-1(αn(1))(Vm(1)(X)), which gives the two equalities in the following calculation:
(3.27)E‖X-π(ϕm(2))-1(αn(2))(Vm(2)(Zm))‖2=E‖prFm(X)-π(ϕm(2))-1(αn(2))(Vm(2)(Zm(2)))‖2+E‖prFm⊥(X)‖2≥(*)E‖prFm(X)-π(ϕm(1))-1(αn(1))(Vm(1)(X))‖2+E‖prFm⊥(X)‖2=E‖X-π(ϕm(1))-1(αn(1))(Vm(1)(X))‖2.
The inequality (*) follows from the optimality of the codebook (ϕm(1))-1(αn(1)) for prFm(X)=Vm(1)(X).


## 3.1. Orthonormal Basis

Let(hm)m∈ℕ be an orthonormal basis of H. For the subsequent subsection we use the following notations. (1)
We setFm=span{h1,…,hm}.(2)
We setVm:=prFm:E→Fm, the orthogonal projection on Fm. It is well known that ∥Vm∥op=1.(3)
Define the linear, surjective, and isometric operatorsϕm by
(3.5)ϕm:(Fm,‖⋅‖)⟶(Rm,‖⋅‖2),hi⟶ei,
where ei denotes the ith unit vector in ℝm, 1≤i≤m.Theorem 3.1.
Assume that the eigenvalue sequence(λj)j∈ℕ of the covariance operator CX satisfies λj≈j-b for -b<-1, and let ϵ>0 be arbitrary. Assume further that (hj)j∈ℕ is a rate optimal ONS for X in H. One sets m(n)=⌈log(n)1+ϵ⌉ for n∈ℕ. Then, one gets for every sequence (αn)n∈ℕ of r-optimal n-quantizers for ϕm(n)(Vm(n)(X)) in (ℝm(n),∥·∥2) the asymptotics
(3.6)en,r(X,H)~er(X,H,ϕm(n)-1(αn))~(E‖X-πϕm(n)-1(αn)(Vm(n)(X))‖r)1/r,
as n→∞.Proof.
Let(fj)j∈ℕ be the corresponding orthonormal eigenvector sequence of CX. Classic eigenvalue theory yields for every m∈ℕ(3.7)E‖∑i=m∞fi〈fi,X〉‖2=∑i=m∞λi≤∑i=m∞E〈hi,X〉2=E‖∑i=m∞hi〈hi,X〉‖2.
Combining this with rate optimality of the ONS (hj)j∈ℕ for X, we get
(3.8)E‖X-Vm(n)(X)‖2=E‖∑i=m(n)+1∞hi〈hi,X〉‖2=∑i=m(n)+1∞E〈hi,X〉2≈∑i=m(n)+1∞λj≈m(n)-(b-1),n⟶∞.
Using the equivalence of the r-norms of Gaussian random variables ([23, Corollary 3.2]), and since X-Vm(n)(X) is Gaussian, we get for all r≥1(3.9)‖‖X-Vm(n)(X)‖‖Lr(P)≈m(n)-(1/2)(b-1),n⟶∞.
With ω as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics en,r(X,H)≈(ω(log(n)))-1/2≈(log(n))-(1/2)(b-1),n→∞. Therefore, the sequence (m(n))n∈ℕ satisfies (2.7) since
(3.10)‖‖X-Vm(n)(X)‖‖Lr(P)≈(log⁡(n))-(1/2)(b-1)(1+ϵ)=o(en,r(X,H)),n⟶∞,
and the assertion follows from Theorem 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 operator CX of a Gaussian random variable is symmetric and compact (in 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.
LetX be a centered Gaussian random variable with values in the Hilbert space H and X=X1+X2, where X1 and X2 are independent centered Gaussians. Then
(3.11)CX=CX1+CX2.
Let λi,λi(1),λi(2), i∈ℕ be the positive monotone decreasing eigenvalues of CX,CX1, and CX2. Then, for i∈ℕ it holds that
(3.12)λi(1),λi(2)≤λi.Proof.
SinceX1,X2 are independent centered Gaussians, we have 𝔼〈X1,u〉X2=𝔼〈X1,u〉𝔼X2=0 for all u∈H. This easily leads to
(3.13)CX(u)=E〈X1+X2,u〉(X1+X2)=E〈X1,u〉X1+E〈X2,u〉X2=CX1(u)+CX2(u).
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 operators (see, e.g., [24, page 213]), we get inequalities (3.12).Let(gi)i∈ℕ be an admissible sequence for X, and assume that ∑i=1∞ξigi=X a.s. In this subsection, we use the following notations.(1)
We setFm:=span{g1,…,gm}.(2)
We defineVm:H→Fm⊂H by
(3.14)Vm(fj):=fj(m)λj(m)λj,
for j≤m and Vm(fj):=0 for j>m, where λj and fj denote the eigenvalues and the corresponding eigenvectors of CX and λj(m) and fj(m) the eigenvalues and the corresponding eigenvectors of CXm, with Xm defined as
(3.15)Xm:=∑i=1mgiξi.
Note that Vm maps H onto Fm since
(3.16)span⁡{g1,…,gm}=span⁡{f1(m),…,fm(m)}.
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∈ℕ a random variable Zm=dX such that Vm(Zm)=∑i=1mξigi (see the proof of Theorem 3.3), which is explicitly known and sufficient to know for the construction.(3)
Define the linear, surjective, and isometric operatorsϕm by
(3.17)ϕm:(Fm,‖⋅‖)⟶(Rm,‖⋅‖2),fi(m)⟶ei,
where ei denotes the ith unit vector of ℝm for 1≤i≤m.Theorem 3.3.
Assume that the eigenvalue sequence(λj)j∈ℕ of the covariance operator CX satisfies λj≈j-b for -b<-1, and let ϵ>0 arbitrary. Assume that (gj)j∈ℕ is a rate optimal admissible sequence for X in H. One sets m(n)=⌈log(n)1+ϵ⌉ for n∈ℕ. Then, there exist random variables Zm,m∈ℕ, with Zm=dX such that for every sequence (αn)n∈ℕ of r-optimal n-quantizers for ϕm(n)(Vm(n)(Zm(n))) in (ℝm(n),∥·∥2)(3.18)en,r(X,H)~er(X,H,ϕm(n)-1(αn))~(E‖X-πϕm(n)-1(αn)(Vm(n)(Zm(n)))‖r)1/r,
as n→∞.Proof.
Linearity of(Vm)m∈ℕ follows from the orthogonality of the eigenvectors. In view of the inequalities for the eigenvalues in Lemma 3.2 and the orthonormality of the family (fi)i∈ℕ, we have for every h=∑i=1∞fiai∈H with ∥h∥2=∑i=1∞ai2≤1(3.19)‖Vm(h)‖2=‖Vm(∑i=1∞aifi)‖2=∑i=1mai2λi(m)λi≤∑i=1∞ai2≤1,
such that ∥Vm∥op≤1.
Note next that for everym∈ℕ there exist independent N(0,1)-distributed random variables (ζi(m))1≤i≤m satisfying
(3.20)∑i=1mξigi=∑i=1mλi(m)fi(m)ζi(m)a.s.
Then, we choose random variables (ζi(m))m+1≤i<∞ such that (ζi(m))1≤i<∞ is a sequence of independent N(0,1)-distributed random variables. We set
(3.21)Zm:=∑i=1∞ζi(m)λifi
and get by using rate optimality of the admissible sequences (gj)j∈ℕ and (λjfj)j∈ℕ(3.22)E‖X-Vm(Zm)‖2=E‖∑i=1∞giξi-Vm(∑i=1∞λifiζi(m))‖2=E‖∑i=1∞giξi-∑i=1mλi(m)fi(m)ζi(m)‖2=E‖∑i=m+1∞giξi‖2≈E‖∑i=m+1∞λifiξi‖2=∑i=m+1∞λi≈m-(b-1),m⟶∞,
where rate optimality of (λjfj)j∈ℕ is a consequence of
(3.23)‖‖X‖‖L2(P)2-E‖∑i=m+1∞giξi‖2=E‖∑i=1mgiξi‖2=∑i=1mλi(m)≤∑i=1mλi.
Using the equivalence of the r-norms of Gaussian random variables ([23, Corollary 3.2]), and since X-Vm(n)(X) is Gaussian, we get for all r≥1(3.24)‖‖X-Vm(n)(X)‖‖Lr(P)≈m(n)-(1/2)(b-1),n⟶∞.
With ω as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics en,r(X,H)≈(ω(log(n)))-1/2≈(log(n))-(1/2)(b-1), n→∞. Therefore, the sequence (m(n))n∈ℕ satisfies (2.7) since
(3.25)‖‖X-Vm(n)(X)‖‖Lr(P)≈(log⁡(n))-(1/2)(b-1)(1+ϵ)=o(en,r(X,H)),n⟶∞,
and the assertion follows from Theorem 2.3.


## 3.3. Comparison of the Different Schemes

At least in the caser=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 indexation i=1,2 for (Vm(i),ϕm(i),αn(i)) and m,n∈ℕ, where αn(i), for i=1,2, are defined as in Theorems 3.1 and 3.3. Note that for this purpose the size of the codebook n and the size of the subspaces dim(Fm)=m can be chosen arbitrarily (i.e., m does not depend on n). The ONS (hi)i∈ℕ is chosen as the ONS derived with the Gram-Schmidt procedure from the admissible sequence (gj)j∈ℕ for the Gaussian random variable X in the Hilbert space H, such that the definition of Fm coincides in the two subsections.Proposition 3.4.
It holds form,n∈ℕ that
(3.26)E‖X-π(ϕm(2))-1(αn(2))(Vm(2)(Zm))‖2≥E‖X-π(ϕm(1)(ϕm(1))-1(αn(1))(Vm(1)(X))‖2.Proof.
Consider forX the decomposition X=prFm⊥(X)+prFm(X). The key is the orthogonality of prFm⊥(X) to prFm(X), π(ϕm(2))-1(αn(2))(Vm(2)(Zm)), and π(ϕm(1))-1(αn(1))(Vm(1)(X)), which gives the two equalities in the following calculation:
(3.27)E‖X-π(ϕm(2))-1(αn(2))(Vm(2)(Zm))‖2=E‖prFm(X)-π(ϕm(2))-1(αn(2))(Vm(2)(Zm(2)))‖2+E‖prFm⊥(X)‖2≥(*)E‖prFm(X)-π(ϕm(1))-1(αn(1))(Vm(1)(X))‖2+E‖prFm⊥(X)‖2=E‖X-π(ϕm(1))-1(αn(1))(Vm(1)(X))‖2.
The inequality (*) follows from the optimality of the codebook (ϕm(1))-1(αn(1)) for prFm(X)=Vm(1)(X).


## 4. Gaussian Processes with Paths in(C([0,1]),∥·∥∞)

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 Theorem2.3. Since we now consider the non-Hilbertian separable Banach space (C([0,1]),∥·∥∞), we have to find different tools that are suitable to use Theorem 2.3.The tools used in [20] are B-splines of order s∈ℕ. In the case s=2, that we will consider in the sequel, these splines span the same subspace of C([0,1],∥·∥∞) as the classical Schauder basis. We set for x∈[0,1], m≥2, and 1≤i≤m the knots ti(m):=(i-1)/(m-1) and the hat functions (4.1)fi(m)(x):=χ[ti(m),ti+1(m)](x)(1-(x-ti(m))(m-1))+χ[ti-1(m),ti(m))(x)(x-ti-1(m))(m-1).
For the remainder of this subsection, we will use the following notations.(1)
As subspacesFm we set Fm:=span{fj(m),1≤j≤m}.(2)
As linear and continuous operatorsVm:C([0,1])→Fm we set the quasiinterpolant
(4.2)Vm(f):=∑i=1mfi(m)βi(m)(f),
where βi(m)(f):=f(ti(m)).(3)
The linear and surjective isometric mappingsϕm one defines as
(4.3)ϕm:(Fm,‖⋅‖∞)⟶(Rm,‖⋅‖∞),∑i=1maifi(m)⟶(a1,…,am).
It is easy to see that ∥∑i=1maifi(m)∥∞=∥(a1,…,am)∥∞ holds for every a∈ℝm.For the application of Theorem2.3, we need to know the error bounds for the approximation of X with the quasiinterpolant Vm(X). For Gaussian random variables, we can provide the following result based on the smoothness of an admissible sequence for X in E.Proposition 4.1.
Let(gj)j∈ℕ be admissible for the centered Gaussian random variable X in (C([0,1]),∥·∥∞). Assume that (1)
∥gj∥≤C1j-θ for every j≥1,θ>1/2, and C1<∞,(2)
gj∈C2([0,1]) with ∥gj′′∥≤C2j-θ+2 for every j≥1 and C2<∞. Then, for any ϵ>0 and some constant C<∞it holds that(4.4)‖‖X-Vm(X)‖‖Lr(P)≤Cm-0,8(θ-(1/2))+ϵ,
for every r≥1.Proof.
Using of [25, Theorem 1], we get
(4.5)‖‖∑i=k∞ξigi‖‖Lr(P)≤C3kθ-(1/2)-ϵ1
for an arbitrary ϵ1>0, some constant C3<∞, and every k∈ℕ. Thus, we have
(4.6)‖‖X-Vm(X)‖‖Lr(P)≤‖‖∑i=k∞ξigi‖‖Lr(P)+‖‖Vm(∑i=k∞ξigi)‖‖Lr(P)+‖‖∑i=1k-1ξigi-Vm(∑i=1k-1ξigi)‖‖Lr(P)≤2C3kθ-(1/2)-ϵ1+‖‖∑i=1k-1ξigi-Vm(∑i=1k-1ξigi)‖‖Lr(P).
Using of [26, Chapter 7, Theorem 7.3], we get for some constant C4<∞(4.7)‖∑i=1k-1ξigi-Wm(∑i=1k-1ξigi)‖≤C4ω(∑i=1k-1ξigi,1m-1),
where the module of smoothnessω(f,δ) is defined by
(4.8)ω(f,δ):=sup⁡0≤h<δ‖f(x)-2f(x+h)+f(x+2h)‖∞.
For an arbitrary f∈C2([0,1]) we have by using Taylor expansion
(4.9)‖f(x)-2f(x+h)+f(x+2h)‖∞h2≤2‖f′′‖∞.
Combining this, we get for an arbitrary ϵ2>0 and constants C5,C6,C7<∞, using again the equivalence of Gaussian moments,
(4.10)‖‖X-Vm(X)‖‖Lr(P)≤2C3kθ-(1/2)-ϵ1+1m2(E‖2∑i=1k-1ξigi′′‖∞r)1/r≤2C3kθ-(1/2)-ϵ1+1m2C5E‖2∑i=1k-1ξigi′′‖∞≤2C3kθ-(1/2)-ϵ1+1m2C6∑i=1k-1iθ+2+ϵ2E|ξi|≤2C3kθ-(1/2)-ϵ1+1m2C7k-θ+3+ϵ2.
To minimize over k, we choose k=k(m)=m0,8. Thus, we get for some constant C<∞ and an arbitrary ϵ>0(4.11)‖‖X-Vm(X)‖‖Lr(P)≤Cm-0,8(θ-(1/2))+ϵ.Now, we are able to prove the main result of this section.Theorem 4.2.
LetX be a centered Gaussian random variable and (gj)j∈ℕ an admissible sequence for X in C([0,1]) fulfilling the assumptions of Proposition 4.1 with θ=b/2, where the constant b>1 satisfies λj≳Kj-b with λj,j∈ℕ denoting the monotone decreasing eigenvalues of the covariance operator CX of X in H=L2([0,1]) and K>0. One sets m(n):=⌈log(n)(5/4)+ϵ⌉ for some ϵ>0. Then, for every sequence (αn)n∈ℕ of r-optimal n-quantizers for ϕm(n)(Vm(n)(X)) in (ℝm(n),∥·∥∞), it   holds that
(4.12)en,r(X,(C([0,1]),‖⋅‖∞))~er(X,C([0,1]),ϕm(n)-1(αn))~(E‖X-πϕm(n)-1(αn)(Vm(n)(X))‖∞r)1/r,
as n→∞.Proof.
For everyh∈C(([0,1]),∥·∥∞), with ∥h∥∞≤1it holds that(4.13)‖Vm(h)‖∞≤sup⁡x∈[0,1]∑i=1m|h(ti(m))|fi(m)(x)≤‖h‖∞sup⁡x∈[0,1]∑i=1mfi(m)(x)≤1,
since {fi(m),1≤i≤m} are partitions of the one for every m∈ℕ, so that ∥Vm∥op≤1.
We get a lower bound for the quantization erroren,r(X,C([0,1])) from the inequality
(4.14)‖f‖L2([0,1])≤‖f‖∞,
for all f∈C([0,1])⊂L2([0,1]). Consequently, we have
(4.15)en,r(X,C([0,1]))≥en,r(X,L2([0,1])).
From Theorem 1.1 and (3.4) we obtain
(4.16)(log⁡(n))-(1/2)(b-1)≈(ω(log⁡(n)))-1/2≲en,r(X,C([0,1])),n⟶∞,
where ω is given as in Theorem 1.1. Finally, we get by combining (4.16) and Proposition 4.1 for sufficiently small δ>0(4.17)‖‖X-Vm(n)(X)‖‖Lr(P)≤Cm(n)-0,8((1/2)(b-1))+δ=o((log⁡(n))-(1/2)(b-1))=o(en,r(X,C([0,1]))),n⟶∞,
and the assertion follows from Theorem 2.3.


## 5. Processes with Path SpaceLp([0,1],∥·∥p)

Another useful tool for our purposes is the Haar basis inLp([0,1]) for 1≤p<∞, which is defined by (5.1)e0:=χ[0,1]e1:=χ[0,1/2)-χ[1/2,1]e2n+k:=2n/2e1(2n⋅-k),n∈N,k∈{0,…,2n-1}.
This is an orthonormal basis of L2([0,1]) and a Schauder basis of Lp([0,1]) for p∈[1,∞), that is, 〈f,e0〉+∑n=0∞∑k=12n-1〈f,e2n+k〉e2n+k converges to f in Lp([0,1]) for every f∈Lp([0,1]); see [27].The Haar basis was used in [17] to construct rate optimal sequences of quantizers for mean regular processes. These processes are specified through the property that for all 0≤s≤t≤1(5.2)E|Xt-Xs|p≤(ρ(t-s))p,
where ρ:ℝ+→[0,∞) is regularly varying with index b>0 at 0, which means that (5.3)lim⁡x→0ρ(cx)ρ(x)=cb,
for all c>0. Condition (5.2) also guarantees that the paths t→Xt lie in Lp([0,1]).For our approach, it will be convenient to define form∈ℕ and 1≤i≤m+1 the knots ti(m):=(i-1)/m and for 1≤i≤m-1 the functions (5.4)fi(m)(x):=χ[ti(m),ti+1(m))(x)m,fm(m)(x):=χ[tm(m),1](x)m
and the operators (5.5)Vm(f):=∑i=1mfi(m)〈fi(m),f〉.
Note that for f∈L1([0,1]), m=2n+1, and n∈ℕ0(5.6)〈e0,f〉e0+∑i=0n∑k=02i-1〈e2i+k,f〉e2i+k=∑i=1mfi(m)〈fi(m),f〉.For the remainder of the subsection, we set the following.(1)
We set form∈ℕ the subspaces Fm:=span{f1(m),…,fm(m)}.(2)
Set the linear and continuous operatorVm to
(5.7)Vm:Lp([0,1])⟶Fmf⟶∑i=1m〈fi(m),f〉fi(m).(3)
Forp∈[1,∞) we set the isometric isomorphisms ϕm,p:(Fm,∥·∥Lp)→(ℝm,∥·∥p) as
(5.8)ϕm,p(∑i=1maifi(m)):=m(1/2-1/p)(a1,…,am).Theorem 5.1.
LetX be a random variable in the Banach space (E,∥·∥)=(Lp([0,1]),∥·∥p) for some p∈[1,∞) fulfilling the mean pathwise regularity property
(5.9)‖Xt-Xs‖Lr∨p≤C(t-s)a,
for constants C,a>0 and t>s∈[0,1]. Moreover, assume that Klog(n)-b≲en,r(X,E) for constants K,b>0. Then, for an arbitrary ϵ>0 and m(n):=⌈(log(n))(b/a)+ϵ⌉it holds that every sequence of r-optimal n-quantizers (αn)n∈ℕ for ϕm(n),p(Vm(n)(X)) in (ℝm(n),∥·∥p) satisfies
(5.10)en,r(X,Lp([0,1]))~er(X,Lp([0,1]),ϕm(n),p-1(αn))~(E‖X-πϕm(n),p-1(αn)(Vm(n)(X))‖Lpr)1/r,
as n→∞.Proof.
As in the above subsections, we check that the sequencesVm and ϕm,p satisfy Conditions 1–3. Since Vm(f)=𝔼λ(f∣ℱm), where ℱm is defined by
(5.11)Fm:=σ(f1(m),…,fm(m)),
we get for f∈Lp([0,1]), with ∥f∥p≤1 and p∈[1,∞) by using Jensen's inequality,
(5.12)‖Vm(f)‖Lpp=∫[0,1]|Eλ(f∣Fm)|pdλ≤‖f‖Lpp,
and thus ∥Vm∥op≤1. The operators ϕm,p satisfy Condition 2 of Theorem 2.3 since
(5.13)‖∑i=1maifi(m)‖Lpp=∑i=1m|ai|p∫[0,1](fi(m))p=m(p/2-1)∑i=1m|ai|p=‖m(1/2-1/p)(a1,…,am)‖pp.
For Condition 3, we note that for t∈[0,1](5.14)Xt=∑i=1mfi(m)(t)m∫[ti(m),ti+1(m)]Xtdλ(s),(Vm(X))t=∑i=1mfi(m)(t)m∫[ti(m),ti+1(m)]Xsdλ(s).
Using the inequalities
(5.15)‖f‖Lp′≤‖f‖Lp,‖X‖Lr′(P)≤‖X‖Lr(P),
for r≥r′, p≥p′, f∈Lp and X∈Lr(ℙ), we get
(5.16)‖‖X-Vm(X)‖Lp([0,1])‖Lr(P)p∨r≤‖‖X-Vm(X)‖Lp∨r([0,1])‖Lp∨r(P)p∨r=‖‖X-Vm(X)‖Lp∨r(P)‖Lp∨r([0,1])p∨r=∫[0,1]E|∑i=1mfi(m)(t)m∫[ti(m),ti+1(m)](Xt-Xs)dλ(s)|p∨rdPdλ(t)≤∫[0,1]|∑i=1mfi(m)(t)m∫[ti(m),ti+1(m)]‖Xt-Xs‖Lr∨p(P)dλ(s)︸≤C/ma+1|p∨rdλ(t)≤∫[0,1]|χ[0,1](t)Cma|p∨rdλ(t)=Cp∨rma(p∨r).
Therefore, we know that the sequence (m(n))n∈ℕ satisfies (2.7) since we get with (5.16)
(5.17)‖‖X-Vm(n)(X)‖Lp([0,1])‖Lr(P)≤Cp∨rm(n)a=o(log⁡(n)-b)=o(en,r(X,E)),
as n→∞, and the assertion follows from Theorem 2.3.


## 6. Examples

In this section, we want to present some processes that fulfill the requirements of the Theorems3.1, 3.3, 4.2, and 5.1. Firstly, we give some examples for Gaussian processes that can be applied to all of the four Theorems, and secondly we describe how our approach can be applied to Lévy processes in view of Theorem 5.1.Examples 6.1.
Gaussian Processes and Brownian Diffusions(i) Brownian Motion and Fractional Brownian Motion
Let(Xt(H))t∈[0,1] be a fractional Brownian motion with Hurst parameter H∈(0,1) (in the case H=1/2 we have an ordinary Brownian motion). Its covariance function is given by
(6.1)EXs(H)Xt(H)=12(s2H+t2H-|s-t|2H).
Note that except for the case of an ordinary Brownian motion the eigenvalues and eigenvectors of the fractional Brownian motion are not known explicitly. Nevertheless, the sharp asymptotics of the eigenvalues has been determined (see, e.g., [7]).In [28] the authors constructed an admissible sequence (gj)j∈ℕ in C([0,1]) that satisfies the requirements of Proposition 4.1 with θ=1/2+H. Furthermore, the eigenvalues λj of CX(H) in L2([0,1]) satisfy λj≈j-(1+2H), see, for example, [7], such that the requirements for Theorem 4.2 are satisfied. Additionally, this sequence is a rate optimal admissible sequence for X(H) in L2([0,1]), such that the requirements for Theorem 3.3 are also met. Constructing recursively an orthonormal sequence (hj)j∈ℕ by applying Gram-Schmidt procedure on the sequence (gj)j∈ℕ yields a rate optimal ONS for X(H) in L2([0,1]) that can be used in the application of Theorem 3.1. In Section 7 we will illustrate the quantizers constructed for X(H) with this ONS for several Hurst parameters H. Note that there are several other admissible sequences for the fractional Brownian motion which can be applied similarly as described above; see, for example, [29] or [30]. Moreover, we have for s,t∈[0,1] the mean regularity property (6.2)E‖XtH-XsH‖p=CH,p|t-s|pH,
and the asymptotics of the quantization error is given as (6.3)en,r(XH,Lp([0,1]))≈en,2(XH,L2([0,1]))≈(log⁡(n))-H,n⟶∞
for all r,p≥1 (see [13]), such that the requirements of Theorem 5.1 are met with a=b=H. Note that in [11] the authors showed the existence of constants k(H,E) for E=C([0,1]) and E=Lp([0,1]) independent of r such that (6.4)en,r(XH,E)~k(H,E)(log⁡(n))-H,n⟶∞.
Therefore, the quantization errors of the sequences of quantizers constructed via Theorems 3.1, 3.3, 4.2, and 5.1 also fulfill this sharp asymptotics.(ii) Brownian Bridge
Let(Bt)t∈[0,1] be a Brownian bridge with covariance function
(6.5)EBsBt=min⁡(s,t)-st.
Since the eigenvalues and eigenvectors of the Brownian bridge are explicitly known, we do not have to search for any other admissible sequence or ONS for (Bt)t∈[0,1] to be applied in H=L2([0,1]). This (the eigenvalue-eigenvector) admissible sequence also satisfies the requirements of Theorem 4.2. The mean pathwise regularity for the Brownian bridge can be deduced by
(6.6)(E|Bt-Bs|p)1/p≤Cp,2(E|Bt-Bs|2)1/2=Cp,2(|t-s|-|t-s|2)1/2≤C|t-s|1/2,
for any p≥1. Combining [31, Theorem 3.7] and [13, Corollary 1.3] yields
(6.7)en,r(B,Lp([0,1]))≈(log⁡(n))-1/2,n⟶∞,
for all r,p≥1, such that Theorem 5.1 can be applied with a=b=1/2.(iii) Stationary Ornstein-Uhlenbeck Process
The stationary Ornstein-Uhlenbeck process(Xt)t∈[0,1] is a Gaussian process given through the covariance function
(6.8)EXsXt=σ22αexp⁡(-α|s-t|),
with parameters α,σ>0. An admissible sequence for the stationary Ornstein-Uhlenbeck process in C([0,1]) and L2([0,1]) can be found in [21]. This sequence that can be applied to Theorems 3.3 and 4.2 and also by applying Gram-Schmidt procedure to Theorem 3.1. According to [13] we have
(6.9)en,r(X,Lp([0,1]))≈(log⁡(n))-1/2,n⟶∞
for all r,p≥1. Furthermore, it holds that
(6.10)(E|Xt-Xs|p)1/p=Cp,2(E|Xt-Xs|2)1/2=Cp,2(σ2α(1-exp⁡(-α|s-t|)))1/2≤C|s-t|1/2,
and therefore we can choose a=b=1/2 to apply Theorem 5.1.(iv) Fractional Ornstein-Uhlenbeck Process
The fractional Ornstein-Uhlenbeck process(Xt(H))t∈[0,1] for H∈(0,2) is a continuous stationary centered Gaussian process with the covariance function
(6.11)EXs(H)Xt(H)=e-α|t-s|H,α>0.
In [22] the authors constructed an admissible sequence (gj)j∈ℕ(H) for H∈(0,1] that satisfies the requirements of Proposition 4.1 with θ=1/2+H/2. Since the eigenvalues λj of CX(H) in L2([0,1]) satisfy λj≈j-1+H, we get again that the assumptions of Theorem 4.2 are satisfied. Similarly, we can use this sequence in Theorems 3.3 and 3.1.(v) Brownian Diffusions
We consider a 1-dimensional Brownian diffusion(Xt)t∈[0,1] fulfilling the SDE
(6.12)Xt=∫0tb(s,Xs)ds+∫0tσ(s,Xs)dWs,
where the deterministic functions b,σ:[0,1]×ℝ→ℝ satisfy the growth assumption
(6.13)|b(t,x)|+|σ(t,x)|≤C(1+|x|).
Under some additional ellipticity assumption on σ, the asymptotics of the quantization error in (Lp([0,1],∥·∥p) is then given by
(6.14)en,r(X,Lp([0,1]))≈(log⁡(n))-1/2,
as n→∞ (see [10] and also [32]). Furthermore, one shows that for 0≤s≤t≤1(6.15)(E‖Xt-Xs‖p)1/p≤C(t-s)1/2
(see [17, Examples 3.1]) such that Theorem 5.1 can be applied with a=b=1/2.Examples 6.2 (Lévy processes).
Let(Xt)t∈[0,1] be a real Lévy process, that is, X is a càdlàg process with ℙ(X0=0)=1 and stationary and independent increments. The characteristic exponent ψ(u) given through the equation
(6.16)∫Rexp⁡(iux)PX1(dx)=exp⁡(-ψ(u)),u∈R,
is characterized by the Lévy-Khintchine formula(6.17)ψ(u)=iau+12σ2u2+∫R(1-eiux+iuxχ(|x|<1))Π(dx),
where the characteristic triple (a,σ,Π) contains constants a∈ℝ,σ≥0, and a measure Π on ℝ∖{0} satisfying ∫ℝ(1∧x2)Π(dx)<∞. By definition, we know that
(6.18)E|Xt-Xs|p=E|Xt-s|p,
and it is further known that the latter moment is finite if and only if
(6.19)∫(|x|≥1)|x|pΠ(dx)<∞.
Furthermore, by the Lévy-Ito decomposition, X can be written as the sum of independent Lévy processes
(6.20)X=X(1)+X(2)+X(3),
where X(3) is a Brownian motion with drift, X(2) is a Compound Poisson process, and X(1) is a Lévy process with bounded jumps and without Brownian component.Firstly, we  will analyze the mean pathwise regularity of these three types ofLévy processes to combine these results with lower bounds for the asymptotical quantization error. (1)
Mean Pathwise Regularity of the 3 Components of the Lévy-Ito Decomposition:
(i)
According to an extendedMillar's Lemma [17, Lemma 5], we have, for all Lévy processes with bounded jumps and without Brownian component, that there is for every p≥2 a constant C<∞ such that for every t∈[0,1](6.21)E|Xt|p≤Ct=C(t1/p)p.
Combining (6.18) and (6.21), we can choose ρ in (5.2) as ρ1,p(x)=x1/p. For p∈[1,2) we have by using (6.21) with p=2(6.22)E|Xt|p≤(E|Xt|2)p/2≤(Ct)p/2=(C1/2t1/2)p,
and thus we can choose ρ1,p(x)=Cx1/2. Combining these facts, we get ρ1,p(x)=Cx1/2∨p for p≥1.(ii)
We consider the Compound Poisson process(6.23)Xt=∑k=1KtZk,
where K denotes a standard Poisson process with intensity λ=1 and (Uk)k∈ℕ is an i.i.d sequence of random variables with ∥Z1∥Lp(ℙ)<∞. Then, one shows that
(6.24)E|∑k=1KtZk|p≤t‖Z1‖Lp(P)pexp⁡(-t)∑k=1∞tk-1kpk!≤C(t1/p)p,
so that (5.2) is satisfied with ϕ2,p(x)=x1/p.(iii)
We consider a Brownian motion with drift. Using Examples6.1 (i) and Lemma 2.1 we can choose ρ in (5.2) as ρ3,p(x)=ρ3(x)=x1/2 for all p≥1.(2)
Lévy Processes with Nonvanishing Brownian ComponentLet X be a Lévy process with non vanishing Brownian component, which  means that σ in the characteristic triple satisfies σ>0. in [17, Proposition 4] for r,p≥1, it holds that
(6.25)(log⁡(n))-1/2≈Cen,r(W,Lp)≲en,r(X,Lp),n⟶∞
for some constant C∈(0,∞), and W denotes a Brownian motion. We consider the Lévy-Ito decomposition X=X(1)+X(2)+X(3) and assume that for Xt(2)=∑k=1KtZkit holds that∥Z1∥Lp∨r(ℙ)<∞. Therefore, we receive the mean pathwise regularity for X, all p,r≥1, and some constant C<∞(6.26)ρp(x):=Cx1/2∨r∨p.
Thus, we can apply Theorem 5.1 with a=1/2∨p∨r and b=1/2.(3)
Compound Poisson ProcessesFor a Compound Poisson process X we know that the rate for the asymptotic quantization error under suitable assumptions is given by
(6.27)en,r(X,Lp)≈exp⁡(-κlog⁡(n)log⁡(log⁡(n))),n⟶∞;
see [16, Theorems 13, 14] and [17, Proposition 3] for a constant κ∈(0,∞). Thus, the sequence (m(n))n∈ℕ has to grow faster than in the examples above. To fulfill
(6.28)‖‖X-Vm(X)‖Lp([0,1])‖Lr(P)=o(exp⁡(-κlog⁡(n)log⁡(log⁡(n)))),
as n→∞ (see the proof of Theorem 5.1), we need to choose m(n)=⌈(p∨r)exp(κlog(n)log(log(n))(1+ϵ))⌉ for an arbitrary ϵ>0.(4)
α-stable Lévy Processes withα∈(0,2)These are Lévy processes satisfying the self-similarity property
(6.29)Xt=dt1/αX1,
and furthermore
(6.30)E|X1|α=∞,sup⁡r{E|X1|r<∞}=α.
Thus, we can choose ρ(x)=Cx1/α for any p≥1 and constants Cp<∞. The asymptotics of the quantization error for X is given by
(6.31)en,r(X,Lp)≈log⁡(n)-1/α,n⟶∞
for r,p≥1 [14], such that we meet the requirements of Theorem 5.1 by setting a=b=α.


## 7. Numerical Illustrations

In this section, we want to highlight the steps needed for a numerical implementation of our approach and also give some illustrating results. For this purpose, it is useful to regard ann-quantizer αn as an element of En (again denoted by αn) instead of being a subset of E. Then, r-optimality of an n-quantizer αn for the random variable X in the separable Banach space E reads (7.1)αn=argmin⁡α=(a1,…,an)∈En⁡Dn,rX(α),Dn,rX(α):=Emin⁡1≤i≤n‖X-ai‖r,
with Dn,rX(α) also called distortion function for X. The differentiability of the distortion function was treated in [6] for finite-dimensional Banach spaces (what is sufficient for our purpose) and later in [33] for the general case.Proposition 7.1 (see [6, Lemma 4.10]).
Assume that the norm∥·∥ of ℝd is smooth. Let r>1, and assume that any Voronoi diagram {Vai(α),ai∈α,1≤i≤n} with Vai(α):={x∈ℝd:∥x-ai∥=mina∈α∥x-a∥} satisfies ℙX(Vai(α)∩Vaj(α))=0 for i≠j. Then, the distortion function is differentiable at every admissible n-tuple α=(a1,…,an) (i.e., ai≠aj for i≠j) with
(7.2)∇Dn,rX(α)=rE(χCai(α)∖{ai}(X)‖X-ai‖r-1∇‖⋅‖(X-ai))∈(Rd)n,
where {Cai(α):1≤i≤n} denotes any Voronoi partition induced by α={a1,…,an}.Remark 7.2.
Whenr=1, the above result extends to admissible n-tuples with ℙX({a1,…,an})=0. Furthermore, if the norm is just smooth on a set A∈ℬ(E) with ℙX(A)=1, then the result still holds true. This is, for example, the case for (E,∥·∥)=(ℝd,∥·∥∞) and random variables X with ℙX(H)=0 for all hyperplanes H, which includes the case of normal distributed random variables.Classic optimization theories now yield that any local minimum is contained in the set of stationary points. So letn∈ℕ,m=m(n)∈ℕ,r≥1,X,Vm, and ϕm be given. The procedure looks as follows.Step 1.
Calculation of the Distribution of theℝm-Valued Random Variableζ:=ϕm(Vm(X)). This step strongly depends on the shape of the random variable X and the operators Vm.
Being in the setting of Section3.1 one starts with an orthonormal system (hi)i∈ℕ in H providing
(7.3)ϕm(Vm(X))=ϕm(Vm(∑i=1∞hi〈hi,X〉))=ϕm(∑i=1mhi〈hi,X〉))=∑i=1mei〈hi,X〉,
where (ei)1≤i≤m denote the unit vectors in ℝm. Thus, the covariance matrix of the random variable ζ admits the representation
(7.4)Eζζ⊥=(E〈hi,X〉〈hj,X〉)1≤i,j,≤m=〈CX(hi),hj〉1≤i,j,≤m,
with CX being the covariance operator of X.
Similarly, we get for Gaussian random variables in the framework of Section3.2(7.5)Eζζ⊥=(λi(m)λj(m))1≤i,j,≤m,
in the setting of Section 4(7.6)Eζζ⊥=(EXti(m)Xtj(m))1≤i,j,≤m=(δti(CX(δtj)))1≤i,j,≤m,
and in the setting of Section 5(7.7)Eζζ⊥=(m1-2/pfj(m)(CX(fi(m))))1≤i,j,≤m,
with fj(m) associated with ∫[0,1]·fj(m)(s)ds. If one considers in the latter framework a non-Brownian Lévy process, for example, and a compound Poisson process (we use the notations as in Examples 6.2 (1) (ii)), the simulation of the gradient leads to the problem of simulating
(7.8)∫titi-1∑j=1KtZkdt,
which is still possible.Step 2.
Use a (stochastic) optimization algorithm to solve the stationarity equation(7.9)∇Dn,rζ(α)=0∈Rm,
for ζ=ϕm(Vm(X)). For this purpose, the computability of the gradient (7.2) is of enormous importance. One may either apply a deterministic gradient-based optimization algorithm (e.g., BFGS) combined with a (Quasi) Monte-Carlo approximation for the gradient, such as the one used in [20], or use a stochastic gradient algorithm, which is in the Hilbert space setting also known as CLVQ (competitive learning vector quantization) algorithm (see, e.g., [19] for more details). In both cases, the random variable ℙζ needs to be simulated, which is the case for the above described examples.Step 3.
Reconstruct the quantizerβ=(b1,…,bn) for the random variable X by setting
(7.10)bi:=ϕm-1(ai)∈Fm⊂E,
for 1≤i≤n with α=(a1,…,an) being some solution of the stationarity (7.9).Illustration
For illustration purposes, we will concentrate on the case described in Section3.1 for r=2. Examples for quantizers as constructed in Section 4 can be found in [20]. The quantizers shown in the sequel were calculated numerically, by using the widely used CLVQ-algorithm as described in [19]. To achieve a better accuracy, we finally performed a few steps of a gradient algorithm by approximating the gradient with a Monte Carlo simulation.Let X(H) be a fractional Brownian motion with Hurst parameter H. We used the admissible sequence as described in [28]:
(7.11)Xt(H)=d∑n=1∞2cH|J1-H(xn)|sin⁡⁡(xnt)xn1+Hζn1+∑n=1∞2cH|J-H(yn)|1-cos⁡⁡(ynt)yn1+Hζn2,
where cH is given as
(7.12)cH2:=sin⁡(πH)Γ(1+2H)π,J1-H and J-H are Bessel functions with corresponding parameters, and xn and yn are the ordered roots of the Bessel functions with parameters -H and 1-H. After ordering the elements of the two parts of the expansion in an alternating manner and applying Gram-Schmidt's procedure for orthogonalization to construct a rate optimal ONS, we used the method as described in Section 3.1. We show the results we obtained for n=10, m=4 and the Hurst parameters H=0.3,0.5, and 0.7 (Figures 1, 2, and 3). To show the effects of changing parameters, we also present the quantizers obtained after increasing the size of the containing subspace (m=8) (Figures 4, 5, and 6) and in addition the effect of increasing the quantizer size (n=30) (Figures 7, 8, and 9). Since X(H) is for H=0.5 an ordinary Brownian motion, one can compare the results with the results obtained for the Brownian motion by using the Karhunen-Loève expansion (see, e.g., [18]).Figure 1
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.3 in a 4-dimensional subspace.Figure 2
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.5 in a 4-dimensional subspace.Figure 3
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.7 in a 4-dimensional subspace.Figure 4
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.3 in an 8-dimensional subspace.Figure 5
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.5 in an 8-dimensional subspace.Figure 6
The 10-quantizer for the fractional Brownian motion with Hurst parameterH=0.7 in an 8-dimensional subspace.Figure 7
The 30-quantizer for the fractional Brownian motion with Hurst parameterH=0.3 in an 8-dimensional subspace.Figure 8
The 30-quantizer for the fractional Brownian motion with Hurst parameterH=0.5 in an 8-dimensional subspace.Figure 9
The 30-quantizer for the fractional Brownian motion with Hurst parameterH=0.7 in an 8-dimensional subspace.



---

*Source: 378519-2010-12-09.xml*