A Constructive Sharp Approach to Functional Quantization of Stochastic Processes

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.

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 number 𝑛∈ℕ, the quantization problem consists in finding a set 𝛼⊂𝐸 that minimizes 𝑒_𝑟(𝑋,(𝐸,‖⋅‖),𝛼)=𝑒_𝑟(𝑋,𝐸,𝛼)∶=𝔼min_𝑎∈𝛼‖𝑋−𝑎‖^𝑟1/𝑟 (1.1) over all subsets 𝛼⊂𝐸 with card (𝛼)≤𝑛. Such sets 𝛼 are called 𝑛-codebooks or 𝑛-quantizers. The corresponding infimum 𝑒_𝑛,𝑟(𝑋,(𝐸,‖⋅‖))=𝑒_𝑛,𝑟(𝑋,𝐸)∶=inf_{𝛼⊂𝐸,card(𝛼)≤𝑛}𝑒_𝑟(𝑋,𝐸,𝛼) (1.2) is called the 𝑛th 𝐿^𝑟-quantization error of 𝑋 in 𝐸, and any 𝑛-quantizer 𝛼 fulfilling 𝑒_𝑟(𝑋,𝐸,𝛼)=𝑒_𝑛,𝑟(𝑋,𝐸) (1.3) is called 𝑟-optimal 𝑛-quantizer. For a given 𝑛-quantizer 𝛼 one defines the nearest neighbor projection 𝜋_𝛼∶𝐸⟶𝛼,𝑥⟶_𝑎∈𝛼𝑎𝜒_{𝐶𝑎(𝛼)}(𝑥), (1.4) where the Voronoi partition {𝐶_𝑎(𝛼),𝑎∈𝛼} is defined as a Borel partition of 𝐸 satisfying 𝐶_𝑎(𝛼)⊂𝑥∈𝔼∶‖𝑥−𝑎‖=min_𝑏∈𝛼‖𝑥−𝑏‖. (1.5) The random variable 𝜋_𝛼(𝑋) is called 𝛼-quantization of 𝑋. One can easily verify that 𝜋_𝛼(𝑋) is the best quantization of 𝑋 in 𝛼⊂𝐸, which means that for every random variable 𝑌 with values in 𝛼 we have𝑒_𝑟(𝑋,𝐸,𝛼)=𝔼𝑋−𝜋_𝛼(𝑋)^𝑟1/𝑟≤𝔼‖𝑋−𝑌‖^𝑟1/𝑟. (1.6)

Applications of quantization go back to the 1940s, where quantization was used for the finite-dimensional setting 𝐸=ℝ^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 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=(𝐿_𝑝[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 of 𝑒_𝑛,𝑟(𝑋,𝐸) when 𝑛 tends to infinity. By separability of (𝐸,‖⋅‖), we can choose a dense subset {𝑐_𝑖,𝑖∈ℕ} and we can deduce in view of0≤lim_𝑛→∞𝔼min_{1≤𝑖≤𝑛}𝑋−𝑐_𝑖^𝑟=𝔼lim_𝑛→∞min_{1≤𝑖≤𝑛}𝑋−𝑐_𝑖^𝑟=0 (1.7) that 𝑒_𝑛,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 of 𝑒_𝑛,𝑟(𝑋,𝐸). It will be convenient to write 𝑎_𝑛∼𝑏_𝑛 for sequences (𝑎_𝑛)_𝑛∈ℕ and (𝑏_𝑛)_𝑛∈ℕ if 𝑎_𝑛/𝑏−−−−−_𝑛^𝑛→∞→1, 𝑎_𝑛≲𝑏_𝑛 if limsup_𝑛→∞𝑎_𝑛/𝑏_𝑛≤1 and 𝑎_𝑛≈𝑏_𝑛 if 0<liminf_𝑛→∞𝑎_𝑛/𝑏_𝑛≤limsup_𝑛→∞𝑎_𝑛/𝑏_𝑛<∞.

In the finite-dimensional setting (ℝ^𝑑,‖⋅‖) this behavior can satisfactory be described by the Zador Theorem (see [6]) for nonsingular distributions ℙ^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 function 𝜌∶(𝑠,∞)→(0,∞) for an 𝑠≥0  regularly varying at infinity with index 𝑏∈ℝ if for every 𝑐>0lim_𝑥→∞𝜌(𝑐𝑥)𝜌(𝑥)=𝑐^𝑏. (1.8)

Theorem 1.1 (see [7]). Let 𝑋 be a centered Gaussian random variable with values in the separable Hilbert space (𝐻,⟨⋅,⋅⟩) and 𝜆_𝑛,𝑛∈ℕ the decreasing eigenvalues of the covariance operator 𝐶_𝑋:𝐻→𝐻, 𝑢→𝔼⟨𝑢,𝑋⟩𝑋 (which is a symmetric trace class operator). Assume that 𝜆_𝑛∼𝜌(𝑛) for some regularly varying function 𝜌 with index −𝑏<−1. Then, the asymptotics of the quantization error is given by 𝑒_𝑛,2()∼𝑏𝑋,𝐻2^𝑏−1𝑏𝑏−1^1/2𝜔log(𝑛)^−1/2,𝑛⟶∞, (1.9) where 𝜔(𝑥)∶=1/𝑥𝜌(𝑥).

Note that any change of ~ in the assumption that 𝜆_𝑛∼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 Theorem 1.1 shows us how to construct asymptotically 𝑟-optimal 𝑛-quantizers for these processes, which means that sequences of 𝑛-quantizers 𝛼_𝑛,𝑛∈ℕ satisfy𝑒_𝑟𝑋,𝐸,𝛼_𝑛∼𝑒_𝑛,𝑟(𝑋,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 operator 𝐶_𝑋 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 rate𝑒_𝑟𝑋,𝐸,𝛼_𝑛≈𝑒_𝑛,𝑟(𝑋,𝐸),𝑛⟶∞. (1.11)

In this paper, we develop a constructive approach to calculate sequences of asymptotically 𝑟-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 ℝ^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 unknown (Sections 3.1 and 3.2), or for quantization problems in different Banach spaces (Sections 4 and 5).

In Section 4, 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 Section 5, 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 (𝐿_𝑝([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, 𝑒_𝑛,𝑟(𝑋,𝐸)≳𝐶₁log(𝑛)^−𝑏1 and the true but unknown rate is 𝑒_𝑛,𝑟(𝑋,𝐸)∼𝐶₂log(𝑛)^−𝑏2 for constants 𝐶₁,𝐶₂,𝑏₁,𝑏₂∈(0,∞), then, we are able to construct a sequence of 𝑛-quantizers 𝛼_𝑛, 𝑛∈ℕ that satisfies𝑒_𝑟𝑋,𝐸,𝛼_𝑛∼𝑒_𝑛,𝑟(𝑋,𝐸)∼𝐶₂log(𝑛)^−𝑏2,𝑛⟶∞ (1.12) 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 Section 6, 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 by 𝑋 a Borel random variable in the separable Banach space (𝐸,‖⋅‖) with card(supp(ℙ^𝑋))=∞. (ii)‖⋅‖ will always denote the norm in 𝐸 whereas ‖⋅‖_{𝐿𝑟(ℙ)} will denote the norm in 𝐿_𝑟(Ω,ℱ,ℙ). (iii)The scalar product in a Hilbert space 𝐻 will be denoted by ⟨⋅,⋅⟩. (iv)The smallest integer above a given real number 𝑥 will be denoted by ⌈𝑥⌉. (v)A sequence (𝑔_𝑗)_𝑗∈ℕ∈𝐸^ℕ is called admissible for a centered Gaussian random variable 𝑋 in 𝐸 if and only if for any sequence (𝜉_𝑖)_𝑖∈ℕ of independent 𝑁(0,1)-distributed random variables it holds that ∑^∞_𝑖=1𝜉_𝑖𝑔_𝑖 converges 𝑎.𝑠. in (𝐸,‖⋅‖) and 𝑋=^𝑑∑^∞_𝑖=1𝜉_𝑖𝑔_𝑖. An admissible sequence (𝑔_𝑗)_𝑗∈ℕ∈𝐸^ℕ is called rate optimal for 𝑋 in 𝐸 if and only if 𝔼^∞_𝑖=𝑚𝜉_𝑖𝑔_𝑖²≈inf𝔼^∞_𝑖=𝑚𝜉_𝑖𝑓_𝑖²∶𝑓_{𝑖𝑖∈ℕ}admissiblefor𝑋, (1.13) as 𝑚→∞. A precise characterization of admissible sequences can be found in [21]. (vi)An orthonormal system (ONS) (ℎ_𝑖)_𝑖∈ℕ is called rate optimal for 𝑋 in the Hilbert space 𝐻 if and only if 𝔼^∞_𝑖=𝑚ℎ_𝑖ℎ_𝑖,𝑋²≈inf𝔼^∞_𝑖=𝑚𝑓_𝑖𝑓_𝑖,𝑋²∶𝑓_{𝑖𝑖∈ℕ}ONSin𝐻, (1.14) 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]). Let 𝐸, 𝐹 be separable Banach spaces, 𝑋 a random variable in E, and 𝑇∶𝐸→𝐹. (1)If 𝑇 is Lipschitz continuous with Lipschitz constant 𝐿, then 𝑒_𝑛,𝑟(𝑇(𝑋),𝐹)≤𝐿𝑒_𝑛,𝑟(𝑋,𝐸), (2.1) and for every 𝑛-quantizer 𝛼 for 𝑋 it holds that 𝑒_𝑟(𝑇(𝑋),𝐹,𝑇(𝛼))≤𝐿𝑒_𝑟(𝑋,𝐸,𝛼). (2.2)(2)Let 𝑇∶𝐸→𝐹 be linear, surjective, and isometric. Then, for 𝑐≥0 and 𝑓∈𝐹𝑒_𝑛,𝑟𝑐𝑇(𝑋)+𝑓,𝐹=𝑐𝑒_𝑛,𝑟(𝑋,𝐸), (2.3) and for every 𝑛-quantizer 𝛼 for 𝑋 it holds that 𝑒_𝑟𝑐𝑇(𝑋)+𝑓,𝐹,𝑇(𝛼)=𝑐𝑇𝑒_𝑟(𝑋,𝐸,𝛼)+𝑓. (2.4)

To formulate the main result, we need for an infinite subset 𝐽⊂ℕ the following.

Condition 1. There exist linear operators 𝑉_𝑚∶𝐸→𝐹_𝑚⊂𝐸 for 𝑚∈𝐽 with ‖𝑉_𝑚‖_op≤1, for finite dimensional subspaces 𝐹_𝑚 with dim(𝐹_𝑚)=𝑚, where the norm ‖⋅‖_op is defined as 𝑉_𝑚op∶=sup_{‖𝑥‖𝑥∈𝐸,≤1}𝑉_𝑚(𝑥). (2.5)

Condition 2. There exist linear isometric and surjective operators 𝜙_𝑚∶(𝐹_𝑚,‖⋅‖)→(ℝ^𝑚,|⋅|_𝑚) with suitable norms |⋅|_𝑚 in ℝ^𝑚 for all 𝑚∈𝐽.

Condition 3. There exist random variables 𝑍_𝑚 for 𝑚∈𝐽 in 𝐸 with 𝑍=_𝑚^𝑑𝑋, such that for the approximation error ‖‖𝑋−𝑉_𝑚(𝑍_𝑚)‖‖_{𝐿𝑟(ℙ)}it holds that𝑋−𝑉_𝑚𝑍_{𝑚𝐿𝑟(ℙ)}⟶0, (2.6) as 𝑚→∞ along 𝐽.

Remark 2.2. The crucial point in Condition 1 is the norm one restriction for the operators 𝑉_𝑚. Condition 2 becomes Important when constructing the quantizers in ℝ^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 Condition 3 to choose explicitly a sequence 𝑚(𝑛)∈𝐽, 𝑛∈ℕ such that 𝑋−𝑉_𝑚(𝑛)𝑍_{𝑚(𝑛)𝐿𝑟(ℙ)}=𝑜𝑒_𝑛,𝑟(𝑋,𝐸),𝑛⟶∞. (2.7)

Theorem 2.3. Assume that Conditions 1–3 hold for some infinite subset 𝐽⊂ℕ. One chooses a sequence (𝑚(𝑛))_𝑛∈ℕ∈𝐽^ℕ such that (2.7) is satisfied. For 𝑛∈ℕ, let 𝛼_𝑛 be an 𝑟-optimal 𝑛-quantizer for 𝜉_𝑛∶=𝜙_𝑚(𝑛)(𝑉_𝑚(𝑛)(𝑍_𝑚(𝑛))) in (ℝ^𝑚(𝑛),|⋅|_𝑚(𝑛)).
Then, (𝜙⁻¹_𝑚(𝑛)(𝛼_𝑛))_𝑛∈ℕ is an asymptotically 𝑟-optimal sequence of 𝑛-quantizers for 𝑋 in 𝐸 and 𝑒_𝑛,𝑟(𝑋,𝐸)∼𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟∼𝑒_𝑟𝑋,𝐸,𝜙_𝑚(𝑛)⁻¹𝛼_𝑛, (2.8) as 𝑛→∞.

Remark 2.4. Note, that for 𝑛∈ℕ there always exist 𝑟-optimal 𝑛-quantizers for 𝜉_𝑛 ([6, Theorem 4.12]).

Proof. Using Condition 3 and the fact that 𝑒_𝑛,𝑟(𝑋,𝐸)>0 for all 𝑛∈ℕ since card(supp(ℙ^𝑋))=∞, we can choose a sequence (𝑚(𝑛))_𝑛∈ℕ∈ℕ^ℕ fulfilling (2.7). Using Lemma 2.1 and Condition 2, we see that 𝜙⁻¹_𝑚(𝑛)(𝛼_𝑛) is an 𝑟-optimal 𝑛-quantizer for 𝑉_𝑚(𝑛)(𝑍_𝑚(𝑛)) in 𝐹_𝑚(𝑛). Then, by using Condition 1, (2.7), and Lemma 2.1 we get 𝑒_𝑛,𝑟(𝑋,𝐸)≤𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟≤𝔼𝑋−𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟+𝔼𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟=𝔼𝑋−𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟+𝑒_𝑛,𝑟𝑉_𝑚(𝑛)𝑍_𝑚(𝑛),𝐹_𝑚(𝑛),‖⋅‖≤𝔼𝑋−𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟+𝑒_𝑛,𝑟𝑍_𝑚(𝑛)=𝔼,𝐸𝑋−𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟+𝑒_𝑛,𝑟(𝑋,𝐸)∼𝑒_𝑛,𝑟(𝑋,𝐸),𝑛⟶∞. (2.9) The last equivalence of the assertion follows from (1.6).

Remark 2.5. We will usually choose 𝑍_𝑚=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 (𝐹_𝑚(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 Theorem 2.3 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 Conditions 1–3 hold. Let 𝑚(𝜖)∶=inf{𝑚∈ℕ∶‖𝑋−𝑉_𝑚(𝑍_𝑚)‖_{𝐿𝑟(ℙ)}<𝜖}, and for 𝑛∈ℕ one sets 𝜉_𝑛∶=𝜙_𝑚(𝜖)(𝑉_𝑚(𝜖)(𝑍_𝑚(𝜖))). Then, it holds for every 𝑛∈ℕ and for every 𝑟-optimal 𝑛-quantizer 𝛼_𝑛 for 𝜉_𝑛 in (ℝ^𝑚(𝜖),|⋅|_𝑚(𝜖)) that 𝑒_𝑟𝑋,𝐸,𝜋_𝜙⁻¹_{𝑚(𝜖)(𝛼𝑛)}𝑉_𝑚(𝜖)𝑍_𝑚(𝜖)≤𝑒_𝑛,𝑟(𝑋,𝐸)+𝜖. (2.10)

3. Gaussian Processes with Hilbertian Path Space

In this chapter, let 𝑋 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 (𝜉_𝑖)_𝑖∈ℕ of independent 𝑁(0,1)-distributed random variables𝑋=^𝑑∞_𝑖=1𝜆_𝑖𝑓_𝑖𝜉_𝑖, (3.1) where 𝜆_𝑖 denote the eigenvalues and 𝑓_𝑖 denote the corresponding orthonormal eigenvectors of the covariance operator 𝐶_𝑋 of 𝑋 (Karhunen-Loève expansion). If these parameters are known, we can choose a sequence (𝑑_𝑛)_𝑛∈ℕ such that a sequence of optimal quantizer 𝛼_𝑛 for 𝑋_𝑛=∑^𝑑(𝑛)_𝑖=1𝜆_𝑖𝑓_𝑖𝜉_𝑖 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 in 𝐿₂(ℙ) or in 𝐻.

Before we will use these representations for 𝑋 to find suitable triples (𝑉_𝑚,𝐹_𝑚,𝜙_𝑚) as in Theorem 2.3, note that for Gaussian random variables in 𝐻 fulfilling suitable assumptions we know that(1)Let (ℎ_𝑖)_𝑖∈ℕ be an orthonormal basis of 𝐻. Then 𝑋=^∞_𝑖=1ℎ_𝑖ℎ_𝑖,𝑋a.s.. (3.2) Compared to (3.1) we see that ⟨ℎ_𝑖,𝑋⟩ are still Gaussian but generally not independent. (2)Let (𝑔_𝑗)_𝑗∈ℕ be an admissible sequence for 𝑋 in 𝐻 such that 𝑋=^𝑑∞_𝑖=1𝜉_𝑖𝑔_𝑖. (3.3) Compared to (3.1) the sequence (𝑔_𝑖)_𝑖∈ℕ is generally not orthogonal. 𝑒_𝑛,2(𝑋,𝐻)≈𝑒_𝑛,𝑠(𝑋,𝐻),𝑛⟶∞ (3.4) 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 (ℎ_𝑚)_𝑚∈ℕ be an orthonormal basis of 𝐻. For the subsequent subsection we use the following notations. (1)We set 𝐹_𝑚=span{ℎ₁,…,ℎ_𝑚}.(2) We set 𝑉_𝑚∶=𝑝𝑟_𝐹𝑚∶𝐸→𝐹_𝑚, the orthogonal projection on 𝐹_𝑚. It is well known that ‖𝑉_𝑚‖_op=1. (3)Define the linear, surjective, and isometric operators 𝜙_𝑚 by 𝜙_𝑚∶𝐹_𝑚,‖⋅‖⟶ℝ^𝑚,‖⋅‖₂,ℎ_𝑖⟶𝑒_𝑖, (3.5) where 𝑒_𝑖 denotes the 𝑖th unit vector in ℝ^𝑚, 1≤𝑖≤𝑚.

Theorem 3.1. Assume that the eigenvalue sequence (𝜆_𝑗)_𝑗∈ℕ of the covariance operator 𝐶_𝑋 satisfies 𝜆_𝑗≈𝑗^−𝑏 for −𝑏<−1, and let 𝜖>0 be arbitrary. Assume further that (ℎ_𝑗)_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 (𝛼_𝑛)_𝑛∈ℕ of 𝑟-optimal 𝑛-quantizers for 𝜙_𝑚(𝑛)(𝑉_𝑚(𝑛)(𝑋)) in (ℝ^𝑚(𝑛),‖⋅‖₂) the asymptotics 𝑒_𝑛,𝑟(𝑋,𝐻)∼𝑒_𝑟𝑋,𝐻,𝜙_𝑚(𝑛)⁻¹𝛼_𝑛∼𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}𝑉_𝑚(𝑛)(𝑋)^𝑟1/𝑟, (3.6) as 𝑛→∞.

Proof. Let (𝑓_𝑗)_𝑗∈ℕ be the corresponding orthonormal eigenvector sequence of 𝐶_𝑋. Classic eigenvalue theory yields for every 𝑚∈ℕ𝔼^∞_𝑖=𝑚𝑓_𝑖𝑓_𝑖,𝑋²=^∞_𝑖=𝑚𝜆_𝑖≤^∞_𝑖=𝑚𝔼ℎ_𝑖,𝑋²=𝔼^∞_𝑖=𝑚ℎ_𝑖ℎ_𝑖,𝑋². (3.7) Combining this with rate optimality of the ONS (ℎ_𝑗)_𝑗∈ℕ for 𝑋, we get 𝔼𝑋−𝑉_𝑚(𝑛)(𝑋)²=𝔼^∞_{(𝑛)𝑖=𝑚+1}ℎ_𝑖ℎ_𝑖,𝑋²=^∞_{(𝑛)𝑖=𝑚+1}𝔼ℎ_𝑖,𝑋²≈^∞_{𝑖=𝑚(𝑛)+1}𝜆_𝑗≈𝑚(𝑛)^{−(𝑏−1)},𝑛⟶∞. (3.8) Using the equivalence of the 𝑟-norms of Gaussian random variables ([23, Corollary 3.2]), and since 𝑋−𝑉_𝑚(𝑛)(𝑋) is Gaussian, we get for all 𝑟≥1𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑟(ℙ)}≈𝑚(𝑛)^{−(1/2)(𝑏−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,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 𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑟(ℙ)}≈log(𝑛)^{−(1/2)(𝑏−1)(1+𝜖)}=𝑜𝑒_𝑛,𝑟(𝑋,𝐻),𝑛⟶∞, (3.10) and the assertion follows from Theorem 2.3.

3.2. Admissible Sequences

In order to show that linear operators 𝑉_𝑚 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 𝐶_𝑋 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. Let 𝑋 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 𝐶_𝑋=𝐶_𝑋1+𝐶_𝑋2. (3.11) Let 𝜆_𝑖,𝜆_𝑖⁽¹⁾,𝜆_𝑖⁽²⁾, 𝑖∈ℕ be the positive monotone decreasing eigenvalues of 𝐶_𝑋,𝐶_𝑋1, and 𝐶_𝑋2. Then, for 𝑖∈ℕ it holds that 𝜆_𝑖⁽¹⁾,𝜆_𝑖⁽²⁾≤𝜆_𝑖. (3.12)

Proof. Since 𝑋₁,𝑋₂ are independent centered Gaussians, we have 𝔼⟨𝑋₁,𝑢⟩𝑋₂=𝔼⟨𝑋₁,𝑢⟩𝔼𝑋₂=0 for all 𝑢∈𝐻. This easily leads to 𝐶_𝑋(𝑢)=𝔼𝑋₁+𝑋₂,𝑢𝑋₁+𝑋₂=𝔼𝑋₁,𝑢𝑋₁+𝔼𝑋₂,𝑢𝑋₂=𝐶_𝑋1(𝑢)+𝐶_𝑋2(𝑢). (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 operators (see, e.g., [24, page 213]), we get inequalities (3.12).

Let (𝑔_𝑖)_𝑖∈ℕ be an admissible sequence for 𝑋, and assume that ∑^∞_𝑖=1𝜉_𝑖𝑔_𝑖=𝑋 a.s. In this subsection, we use the following notations.(1)We set 𝐹_𝑚∶=span{𝑔₁,…,𝑔_𝑚}. (2) We define 𝑉_𝑚∶𝐻→𝐹_𝑚⊂𝐻 by 𝑉_𝑚𝑓_𝑗∶=𝑓_𝑗^(𝑚)𝜆_𝑗^(𝑚)𝜆_𝑗, (3.14) for 𝑗≤𝑚 and 𝑉_𝑚(𝑓_𝑗)∶=0 for 𝑗>𝑚, where 𝜆_𝑗 and 𝑓_𝑗 denote the eigenvalues and the corresponding eigenvectors of 𝐶_𝑋 and 𝜆_𝑗^(𝑚) and 𝑓_𝑗^(𝑚) the eigenvalues and the corresponding eigenvectors of 𝐶_𝑋𝑚, with 𝑋_𝑚 defined as 𝑋_𝑚∶=^𝑚_𝑖=1𝑔_𝑖𝜉_𝑖. (3.15) Note that 𝑉_𝑚 maps 𝐻 onto 𝐹_𝑚 since span𝑔₁,…,𝑔_𝑚=span𝑓₁^(𝑚),…,𝑓_𝑚^(𝑚). (3.16) Furthermore, it is important to mention that one does not need to know 𝜆_𝑗 and 𝑓_𝑗 explicitly to construct the subsequent quantizers, since we can find for any 𝑚∈ℕ a random variable 𝑍=_𝑚^𝑑𝑋 such that 𝑉_𝑚(𝑍_𝑚)=∑^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖 (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 operators 𝜙_𝑚 by 𝜙_𝑚∶𝐹_𝑚,‖⋅‖⟶ℝ^𝑚,‖⋅‖₂,𝑓_𝑖^(𝑚)⟶𝑒_𝑖, (3.17) where 𝑒_𝑖 denotes the 𝑖th unit vector of ℝ^𝑚 for 1≤𝑖≤𝑚.

Theorem 3.3. Assume that the eigenvalue sequence (𝜆_𝑗)_𝑗∈ℕ of the covariance operator 𝐶_𝑋 satisfies 𝜆_𝑗≈𝑗^−𝑏 for −𝑏<−1, and let 𝜖>0 arbitrary. Assume that (𝑔_𝑗)_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 𝑍_𝑚,𝑚∈ℕ, with 𝑍=_𝑚^𝑑𝑋 such that for every sequence (𝛼_𝑛)_𝑛∈ℕ of 𝑟-optimal 𝑛-quantizers for 𝜙_𝑚(𝑛)(𝑉_𝑚(𝑛)(𝑍_𝑚(𝑛))) in (ℝ^𝑚(𝑛),‖⋅‖₂)𝑒_𝑛,𝑟(𝑋,𝐻)∼𝑒_𝑟𝑋,𝐻,𝜙_𝑚(𝑛)⁻¹𝛼_𝑛∼𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}𝑉_𝑚(𝑛)𝑍_𝑚(𝑛)^𝑟1/𝑟, (3.18) as 𝑛→∞.

Proof. Linearity of (𝑉_𝑚)_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 (𝑓_𝑖)_𝑖∈ℕ, we have for every ℎ=∑^∞_𝑖=1𝑓_𝑖𝑎_𝑖∈𝐻 with ‖ℎ‖²=∑^∞_𝑖=1𝑎²_𝑖≤1𝑉_𝑚(ℎ)²=𝑉_𝑚^∞_𝑖=1𝑎_𝑖𝑓_𝑖²=^𝑚_𝑖=1𝑎²_𝑖𝜆_𝑖^(𝑚)𝜆_𝑖≤^∞_𝑖=1𝑎²_𝑖≤1, (3.19) such that ‖𝑉_𝑚‖_𝑜𝑝≤1.
Note next that for every 𝑚∈ℕ there exist independent 𝑁(0,1)-distributed random variables (𝜁_𝑖^(𝑚))_{1≤𝑖≤𝑚} satisfying ^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖=^𝑚_𝑖=1𝜆_𝑖^(𝑚)𝑓_𝑖^(𝑚)𝜁_𝑖^(𝑚)a.s. (3.20) Then, we choose random variables (𝜁_𝑖^(𝑚))_{𝑚+1≤𝑖<∞} such that (𝜁_𝑖^(𝑚))_1≤𝑖<∞ is a sequence of independent 𝑁(0,1)-distributed random variables. We set 𝑍_𝑚∶=^∞_𝑖=1𝜁_𝑖^(𝑚)𝜆_𝑖𝑓_𝑖 (3.21) and get by using rate optimality of the admissible sequences (𝑔_𝑗)_𝑗∈ℕ and (𝜆_𝑗𝑓_𝑗)_𝑗∈ℕ𝔼𝑋−𝑉_𝑚𝑍_𝑚²=𝔼^∞_𝑖=1𝑔_𝑖𝜉_𝑖−𝑉_𝑚(^∞_𝑖=1𝜆_𝑖𝑓_𝑖𝜁_𝑖^(𝑚))²=𝔼^∞_𝑖=1𝑔_𝑖𝜉_𝑖−^𝑚_𝑖=1𝜆_𝑖^(𝑚)𝑓_𝑖^(𝑚)𝜁_𝑖^(𝑚)2=𝔼^∞_𝑖=𝑚+1𝑔_𝑖𝜉_𝑖²≈𝔼^∞_𝑖=𝑚+1𝜆_𝑖𝑓_𝑖𝜉_𝑖²=^∞_𝑖=𝑚+1𝜆_𝑖≈𝑚^{−(𝑏−1)},𝑚⟶∞, (3.22) where rate optimality of (𝜆_𝑗𝑓_𝑗)_𝑗∈ℕ is a consequence of ‖‖𝑋‖‖²_𝐿2(ℙ)−𝔼^∞_𝑖=𝑚+1𝑔_𝑖𝜉_𝑖²=𝔼^𝑚_𝑖=1𝑔_𝑖𝜉_𝑖²=^𝑚_𝑖=1𝜆_𝑖^(𝑚)≤^𝑚_𝑖=1𝜆_𝑖. (3.23) Using the equivalence of the 𝑟-norms of Gaussian random variables ([23, Corollary 3.2]), and since 𝑋−𝑉_𝑚(𝑛)(𝑋) is Gaussian, we get for all 𝑟≥1𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑟(ℙ)}≈𝑚(𝑛)^{−(1/2)(𝑏−1)},𝑛⟶∞. (3.24) With 𝜔 as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics 𝑒_𝑛,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 𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑟(ℙ)}≈log(𝑛)^{−(1/2)(𝑏−1)(1+𝜖)}=𝑜𝑒_𝑛,𝑟(𝑋,𝐻),𝑛⟶∞, (3.25) and the assertion follows from Theorem 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 indexation 𝑖=1,2 for (𝑉_𝑚^(𝑖),𝜙_𝑚^(𝑖),𝛼_𝑛^(𝑖)) and 𝑚,𝑛∈ℕ, where 𝛼_𝑛^(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(𝐹_𝑚)=𝑚 can be chosen arbitrarily (i.e., 𝑚 does not depend on 𝑛). The ONS (ℎ_𝑖)_𝑖∈ℕ is chosen as the ONS derived with the Gram-Schmidt procedure from the admissible sequence (𝑔_𝑗)_𝑗∈ℕ for the Gaussian random variable 𝑋 in the Hilbert space 𝐻, such that the definition of 𝐹_𝑚 coincides in the two subsections.

Proposition 3.4. It holds for 𝑚,𝑛∈ℕ that 𝔼𝑋−𝜋_{(𝜙𝑚(2))}⁻¹_{(𝛼𝑛(2))}𝑉_𝑚⁽²⁾𝑍_𝑚²≥𝔼𝑋−𝜋_{(𝜙𝑚(1)(𝜙𝑚(1))}⁻¹_{(𝛼𝑛(1))}(𝑉_𝑚⁽¹⁾(𝑋))². (3.26)

Proof. Consider for 𝑋 the decomposition 𝑋=𝑝𝑟_𝐹^⟂_𝑚(𝑋)+𝑝𝑟_𝐹𝑚(𝑋). The key is the orthogonality of 𝑝𝑟_𝐹^⟂_𝑚(𝑋) to 𝑝𝑟_𝐹𝑚(𝑋), 𝜋_{(𝜙𝑚(2))}⁻¹_{(𝛼𝑛(2))}(𝑉_𝑚⁽²⁾(𝑍_𝑚)), and 𝜋_{(𝜙𝑚(1))}⁻¹_{(𝛼𝑛(1))}(𝑉_𝑚⁽¹⁾(𝑋)), which gives the two equalities in the following calculation: 𝔼𝑋−𝜋_{(𝜙𝑚(2))}⁻¹_{(𝛼𝑛(2))}𝑉_𝑚⁽²⁾𝑍_𝑚²=𝔼𝑝𝑟_𝐹𝑚(𝑋)−𝜋_{(𝜙𝑚(2))}⁻¹_{(𝛼𝑛(2))}𝑉_𝑚⁽²⁾𝑍_𝑚⁽²⁾²+𝔼𝑝𝑟_𝐹^⟂_𝑚(𝑋)²_(∗)≥𝔼𝑝𝑟_𝐹𝑚(𝑋)−𝜋_{(𝜙𝑚(1))}⁻¹_{(𝛼𝑛(1))}𝑉_𝑚⁽¹⁾(𝑋)²+𝔼𝑝𝑟_𝐹^⟂_𝑚(𝑋)²=𝔼𝑋−𝜋_{(𝜙𝑚(1))−1(𝛼𝑛}⁽¹⁾₎(𝑉_𝑚⁽¹⁾(𝑋))². (3.27) The inequality (*) follows from the optimality of the codebook (𝜙_𝑚⁽¹⁾)⁻¹(𝛼_𝑛⁽¹⁾) for 𝑝𝑟_𝐹𝑚(𝑋)=𝑉_𝑚⁽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 Theorem 2.3. 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 𝑡_𝑖^(𝑚)∶=(𝑖−1)/(𝑚−1) and the hat functions 𝑓_𝑖^(𝑚)(𝑥)∶=𝜒_[𝑡𝑖^(𝑚)_,𝑡^(𝑚)_𝑖+1](𝑥)1−𝑥−𝑡_𝑖^(𝑚)(𝑚−1)+𝜒_[𝑡^(𝑚)_{𝑖−1,𝑡𝑖}^(𝑚)₎(𝑥)𝑥−𝑡^(𝑚)_𝑖−1(𝑚−1). (4.1) For the remainder of this subsection, we will use the following notations.(1)As subspaces 𝐹_𝑚 we set 𝐹_𝑚∶=span{𝑓_𝑗^(𝑚),1≤𝑗≤𝑚}.(2) As linear and continuous operators 𝑉_𝑚∶𝐶([0,1])→𝐹_𝑚 we set the quasiinterpolant 𝑉_𝑚𝑓∶=^𝑚_𝑖=1𝑓_𝑖^(𝑚)𝛽_𝑖^(𝑚)𝑓, (4.2) where 𝛽_𝑖^(𝑚)(𝑓)∶=𝑓(𝑡_𝑖^(𝑚)).(3) The linear and surjective isometric mappings 𝜙_𝑚 one defines as 𝜙_𝑚∶𝐹_𝑚,‖⋅‖_∞⟶𝑅^𝑚,‖⋅‖_∞,_𝑚𝑖=1𝑎_𝑖𝑓_𝑖^(𝑚)⟶𝑎₁,…,𝑎_𝑚. (4.3) It is easy to see that ‖∑^𝑚_𝑖=1𝑎_𝑖𝑓_𝑖^(𝑚)‖_∞=‖(𝑎₁,…,𝑎_𝑚)‖_∞ holds for every 𝑎∈ℝ^𝑚.

For the application of Theorem 2.3, we need to know the error bounds for the approximation of 𝑋 with the quasiinterpolant 𝑉_𝑚(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 (𝑔_𝑗)_𝑗∈ℕ be admissible for the centered Gaussian random variable 𝑋 in (𝐶([0,1]),‖⋅‖_∞). Assume that (1)‖𝑔_𝑗‖≤𝐶₁𝑗^−𝜃 for every 𝑗≥1,𝜃>1/2, and 𝐶₁<∞, (2)𝑔_𝑗∈𝐶²([0,1]) with ‖𝑔_𝑗^′′‖≤𝐶₂𝑗^−𝜃+2 for every 𝑗≥1 and 𝐶₂<∞. Then, for any 𝜖>0 and some constant 𝐶<∞it holds that𝑋−𝑉_𝑚(𝑋)_{𝐿𝑟(ℙ)}≤𝐶𝑚^{−0,8(𝜃−(1/2))+𝜖}, (4.4) for every 𝑟≥1.

Proof. Using of [25, Theorem 1], we get ^∞_𝑖=𝑘𝜉_𝑖𝑔_{𝑖𝐿𝑟(ℙ)}≤𝐶₃𝑘^{𝜃−(1/2)−𝜖1} (4.5) for an arbitrary 𝜖₁>0, some constant 𝐶₃<∞, and every 𝑘∈ℕ. Thus, we have 𝑋−𝑉_𝑚(𝑋)_{𝐿𝑟(ℙ)}≤^∞_𝑖=𝑘𝜉_𝑖𝑔_{𝑖𝐿𝑟(ℙ)}+𝑉_𝑚^∞_𝑖=𝑘𝜉_𝑖𝑔_{𝑖𝐿𝑟(ℙ)}+^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖−𝑉_𝑚^𝑘−1_𝑖=1𝜉_𝑖𝑔_{𝑖𝐿𝑟(ℙ)}≤2𝐶₃𝑘^{𝜃−(1/2)−𝜖1}+^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖−𝑉_𝑚^𝑘−1_𝑖=1𝜉_𝑖𝑔_{𝑖𝐿𝑟(ℙ)}. (4.6) Using of [26, Chapter 7, Theorem 7.3], we get for some constant 𝐶₄<∞^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖−𝑊_𝑚^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖≤𝐶₄𝜔^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖,1,𝑚−1 (4.7) where the module of smoothness 𝜔(𝑓,𝛿) is defined by 𝜔𝑓,𝛿∶=sup_0≤ℎ<𝛿𝑓(𝑥)−2𝑓(𝑥+ℎ)+𝑓(𝑥+2ℎ)_∞. (4.8) For an arbitrary 𝑓∈𝐶²([0,1]) we have by using Taylor expansion 𝑓(𝑥)−2𝑓(𝑥+ℎ)+𝑓(𝑥+2ℎ)_∞ℎ²≤2𝑓^′′_∞. (4.9) Combining this, we get for an arbitrary 𝜖₂>0 and constants 𝐶₅,𝐶₆,𝐶₇<∞, using again the equivalence of Gaussian moments, 𝑋−𝑉_𝑚(𝑋)_{𝐿𝑟(ℙ)}≤2𝐶₃𝑘^{𝜃−(1/2)−𝜖1}+1𝑚²𝔼2^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖^′′𝑟_∞^1/𝑟≤2𝐶₃𝑘^{𝜃−(1/2)−𝜖1}+1𝑚²𝐶₅𝔼2^𝑘−1_𝑖=1𝜉_𝑖𝑔_𝑖^′′_∞≤2𝐶₃𝑘^{𝜃−(1/2)−𝜖1}+1𝑚²𝐶₆^𝑘−1_𝑖=1𝑖^𝜃+2+𝜖2𝔼𝜉_𝑖≤2𝐶₃𝑘^{𝜃−(1/2)−𝜖1}+1𝑚²𝐶₇𝑘^{−𝜃+3+𝜖2}. (4.10) To minimize over k, we choose 𝑘=𝑘(𝑚)=𝑚^0,8. Thus, we get for some constant 𝐶<∞ and an arbitrary 𝜖>0𝑋−𝑉_𝑚(𝑋)_{𝐿𝑟(ℙ)}≤𝐶𝑚^{−0,8(𝜃−(1/2))+𝜖}. (4.11)

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

Theorem 4.2. Let 𝑋 be a centered Gaussian random variable and (𝑔_𝑗)_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 𝜆_𝑗≳𝐾𝑗^−𝑏 with 𝜆_𝑗,𝑗∈ℕ denoting the monotone decreasing eigenvalues of the covariance operator 𝐶_𝑋 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 (𝛼_𝑛)_𝑛∈ℕ of 𝑟-optimal 𝑛-quantizers for 𝜙_𝑚(𝑛)(𝑉_𝑚(𝑛)(𝑋)) in (ℝ^𝑚(𝑛),‖⋅‖_∞), it holds that 𝑒_𝑛,𝑟𝑋,𝐶([0,1]),‖⋅‖_∞∼𝑒_𝑟𝑋,𝐶([0,1]),𝜙_𝑚(𝑛)⁻¹𝛼_𝑛∼𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛)(𝛼𝑛)}(𝑉_𝑚(𝑛)(𝑋))^𝑟_∞^1/𝑟, (4.12) as 𝑛→∞.

Proof. For every ℎ∈𝐶(([0,1]),‖⋅‖_∞), with ‖ℎ‖_∞≤1it holds that𝑉_𝑚(ℎ)_∞≤sup_[]^𝑚_{𝑥∈0,1𝑖=1}ℎ𝑡_𝑖^(𝑚)𝑓_𝑖^(𝑚)(𝑥)≤‖ℎ‖_∞sup_[]^𝑚_{𝑥∈0,1𝑖=1}𝑓_𝑖^(𝑚)(𝑥)≤1, (4.13) since {𝑓_𝑖^(𝑚),1≤𝑖≤𝑚} are partitions of the one for every 𝑚∈ℕ, so that ‖𝑉_𝑚‖_op≤1.
We get a lower bound for the quantization error 𝑒_𝑛,𝑟(𝑋,𝐶([0,1])) from the inequality 𝑓_𝐿2([0,1])≤𝑓_∞, (4.14) for all 𝑓∈𝐶([0,1])⊂𝐿₂([0,1]). Consequently, we have 𝑒_𝑛,𝑟(𝑋,𝐶([0,1]))≥𝑒_𝑛,𝑟𝑋,𝐿₂([0,1]). (4.15) From Theorem 1.1 and (3.4) we obtain (log(𝑛))^{−(1/2)(𝑏−1)}≈𝜔log(𝑛)^−1/2≲𝑒_𝑛,𝑟(𝑋,𝐶([0,1])),𝑛⟶∞, (4.16) where 𝜔 is given as in Theorem 1.1. Finally, we get by combining (4.16) and Proposition 4.1 for sufficiently small 𝛿>0𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑟(ℙ)}≤𝐶𝑚(𝑛)^{−0,8((1/2)(𝑏−1))+𝛿}=𝑜log(𝑛)^{−(1/2)(𝑏−1)}=𝑜𝑒_𝑛,𝑟(𝑋,𝐶([0,1])),𝑛⟶∞, (4.17) and the assertion follows from Theorem 2.3.

5. Processes with Path Space 𝐿_𝑝([0,1],‖⋅‖_𝑝)

Another useful tool for our purposes is the Haar basis in 𝐿_𝑝([0,1]) for 1≤𝑝<∞, which is defined by 𝑒₀∶=𝜒_[0,1]𝑒₁∶=𝜒_[0,1/2)−𝜒_[1/2,1]𝑒₂^𝑛_+𝑘∶=2^𝑛/2𝑒₁2^𝑛⋅−𝑘,𝑛∈ℕ,𝑘∈0,…,2^𝑛−1. (5.1) This is an orthonormal basis of 𝐿₂([0,1]) and a Schauder basis of 𝐿_𝑝([0,1]) for 𝑝∈[1,∞), that is, ⟨𝑓,𝑒₀⟩+∑^∞_𝑛=0∑^2𝑛−1_𝑘=1⟨𝑓,𝑒₂^𝑛_+𝑘⟩𝑒₂^𝑛_+𝑘 converges to 𝑓 in 𝐿_𝑝([0,1]) for every 𝑓∈𝐿_𝑝([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≤1𝔼𝑋_𝑡−𝑋_𝑠^𝑝≤𝜌(𝑡−𝑠)^𝑝, (5.2) where 𝜌∶ℝ₊→[0,∞) is regularly varying with index 𝑏>0 at 0, which means that lim_𝑥→0𝜌(𝑐𝑥)𝜌(𝑥)=𝑐^𝑏, (5.3) for all 𝑐>0. Condition (5.2) also guarantees that the paths 𝑡→𝑋_𝑡 lie in 𝐿_𝑝([0,1]).

For our approach, it will be convenient to define for 𝑚∈ℕ and 1≤𝑖≤𝑚+1 the knots 𝑡_𝑖^(𝑚)∶=(𝑖−1)/𝑚 and for 1≤𝑖≤𝑚−1 the functions 𝑓_𝑖^(𝑚)(𝑥)∶=𝜒_[𝑡𝑖^(𝑚)_,𝑡^(𝑚)_𝑖+1)(𝑥)√𝑚,𝑓_𝑚^(𝑚)(𝑥)∶=𝜒_[𝑡𝑚^(𝑚)_,1](𝑥)√𝑚 (5.4) and the operators 𝑉_𝑚𝑓∶=^𝑚_𝑖=1𝑓_𝑖^(𝑚)𝑓_𝑖^(𝑚),𝑓. (5.5) Note that for 𝑓∈𝐿₁([0,1]), 𝑚=2^𝑛+1, and 𝑛∈ℕ₀𝑒₀,𝑓𝑒₀+^𝑛2_𝑖=0^𝑖−1_𝑘=0𝑒₂^𝑖_+𝑘,𝑓𝑒₂^𝑖_+𝑘=^𝑚_𝑖=1𝑓_𝑖^(𝑚)𝑓_𝑖^(𝑚),𝑓. (5.6)

For the remainder of the subsection, we set the following. (1)We set for 𝑚∈ℕ the subspaces 𝐹_𝑚∶=span{𝑓₁^(𝑚),…,𝑓_𝑚^(𝑚)}.(2) Set the linear and continuous operator 𝑉_𝑚 to 𝑉_𝑚∶𝐿_𝑝([0,1])⟶𝐹_𝑚𝑓⟶^𝑚_𝑖=1𝑓_𝑖^(𝑚),𝑓𝑓_𝑖^(𝑚). (5.7)(3)For 𝑝∈[1,∞) we set the isometric isomorphisms 𝜙_𝑚,𝑝∶(𝐹_𝑚,‖⋅‖_𝐿𝑝)→(ℝ^𝑚,‖⋅‖_𝑝) as 𝜙_𝑚,𝑝^𝑚_𝑖=1𝑎_𝑖𝑓_𝑖^(𝑚)∶=𝑚^{(1/2−1/𝑝)}𝑎₁,…,𝑎_𝑚. (5.8)

Theorem 5.1. Let 𝑋 be a random variable in the Banach space (𝐸,‖⋅‖)=(𝐿_𝑝([0,1]),‖⋅‖_𝑝) for some 𝑝∈[1,∞) fulfilling the mean pathwise regularity property 𝑋_𝑡−𝑋_{𝑠𝐿𝑟∨𝑝}≤𝐶(𝑡−𝑠)^𝑎, (5.9) for constants 𝐶,𝑎>0 and 𝑡>𝑠∈[0,1]. Moreover, assume that 𝐾log(𝑛)^−𝑏≲𝑒_𝑛,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 (𝛼_𝑛)_𝑛∈ℕ for 𝜙_{𝑚(𝑛),𝑝}(𝑉_𝑚(𝑛)(𝑋)) in (ℝ^𝑚(𝑛),‖⋅‖_𝑝) satisfies 𝑒_𝑛,𝑟𝑋,𝐿_𝑝([0,1])∼𝑒_𝑟𝑋,𝐿_𝑝([0,1]),𝜙_𝑚(𝑛)⁻¹_,𝑝𝛼_𝑛∼𝔼𝑋−𝜋_𝜙⁻¹_{𝑚(𝑛),𝑝(𝛼𝑛)}𝑉_𝑚(𝑛)(𝑋)^𝑟_𝐿𝑝^1/𝑟, (5.10) as 𝑛→∞.

Proof. As in the above subsections, we check that the sequences 𝑉_𝑚 and 𝜙_𝑚,𝑝 satisfy Conditions 1–3. Since 𝑉_𝑚(𝑓)=𝔼_𝜆(𝑓∣ℱ_𝑚), where ℱ_𝑚 is defined by ℱ_𝑚∶=𝜎𝑓₁^(𝑚),…,𝑓_𝑚^(𝑚), (5.11) we get for 𝑓∈𝐿_𝑝([0,1]), with ‖𝑓‖_𝑝≤1 and 𝑝∈[1,∞) by using Jensen's inequality, 𝑉_𝑚𝑓^𝑝_𝐿𝑝=_[0,1]𝔼_𝜆(𝑓∣ℱ_𝑚)^𝑝𝑑𝜆≤𝑓^𝑝_𝐿𝑝, (5.12) and thus ‖𝑉_𝑚‖_op≤1. The operators 𝜙_𝑚,𝑝 satisfy Condition 2 of Theorem 2.3 since ^𝑚_𝑖=1𝑎_𝑖𝑓_𝑖^(𝑚)𝑝_𝐿𝑝=^𝑚_𝑖=1𝑎_𝑖^𝑝_[0,1]𝑓_𝑖^(𝑚)𝑝=𝑚^{𝑚(𝑝/2−1)}_𝑖=1𝑎_𝑖^𝑝=𝑚^{(1/2−1/𝑝)}𝑎₁,…,𝑎_𝑚^𝑝_𝑝. (5.13) For Condition 3, we note that for 𝑡∈[0,1]𝑋_𝑡=^𝑚_𝑖=1𝑓_𝑖^(𝑚)(𝑡)√𝑚_{[𝑡𝑖(𝑚),𝑡(𝑚)𝑖+1]}𝑋_𝑡(𝑠),𝑉𝑑𝜆_𝑚(𝑋)_𝑡=^𝑚_𝑖=1𝑓_𝑖^(𝑚)(𝑡)√𝑚_{[𝑡𝑖(𝑚),𝑡(𝑚)𝑖+1]}𝑋_𝑠𝑑𝜆(𝑠). (5.14) Using the inequalities 𝑓_𝐿𝑝′≤𝑓_𝐿𝑝,‖𝑋‖_{𝐿𝑟′(ℙ)}≤‖𝑋‖_{𝐿𝑟(ℙ)}, (5.15) for 𝑟≥𝑟^′, 𝑝≥𝑝^′, 𝑓∈𝐿_𝑝 and 𝑋∈𝐿_𝑟(ℙ), we get 𝑋−𝑉_𝑚(𝑋)_{𝐿𝑝([0,1])𝐿}^𝑝∨𝑟_𝑟(ℙ)≤𝑋−𝑉_𝑚(𝑋)_{𝐿𝑝∨𝑟([0,1])𝐿}^𝑝∨𝑟_{𝑝∨𝑟(ℙ)}=𝑋−𝑉_𝑚(𝑋)_{𝐿𝑝∨𝑟(ℙ)𝐿}^𝑝∨𝑟_{𝑝∨𝑟([0,1])}=_[0,1]𝔼^𝑚_𝑖=1𝑓_𝑖^(𝑚)(𝑡)√𝑚_{[𝑡𝑖(𝑚),𝑡(𝑚)𝑖+1]}𝑋_𝑡−𝑋_𝑠𝑑𝜆(𝑠)^𝑝∨𝑟(𝑡)≤𝑑ℙ𝑑𝜆_{[0,1]𝑚𝑖=1}𝑓_𝑖^(𝑚)(𝑡)√𝑚_{[𝑡𝑖(𝑚),𝑡(𝑚)𝑖+1]}𝑋_𝑡−𝑋_{𝑠𝐿𝑟∨𝑝(ℙ)}𝑑𝜆(𝑠)_{≤𝐶/𝑚𝑎+1}^𝑝∨𝑟(𝑡)≤𝑑𝜆_[0,1]𝜒_[0,1](𝑡)𝐶𝑚^{𝑎𝑝∨𝑟}(𝑡)=𝐶𝑑𝜆^𝑝∨𝑟𝑚^{𝑎(𝑝∨𝑟)}. (5.16) Therefore, we know that the sequence (𝑚(𝑛))_𝑛∈ℕ satisfies (2.7) since we get with (5.16) 𝑋−𝑉_𝑚(𝑛)(𝑋)_{𝐿𝑝([0,1])𝐿𝑟(ℙ)}≤𝐶^𝑝∨𝑟𝑚(𝑛)^𝑎=𝑜log(𝑛)^−𝑏=𝑜𝑒_𝑛,𝑟(𝑋,𝐸), (5.17) 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 Theorems 3.1, 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 (𝑋_𝑡^(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 𝔼𝑋_𝑠^(𝐻)𝑋_𝑡^(𝐻)=12𝑠^2𝐻+𝑡^2𝐻−|𝑠−𝑡|^2𝐻. (6.1) 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 (𝑔_𝑗)_𝑗∈ℕ in 𝐶([0,1]) that satisfies the requirements of Proposition 4.1 with 𝜃=1/2+𝐻. Furthermore, the eigenvalues 𝜆_𝑗 of 𝐶_𝑋^(𝐻) in 𝐿₂([0,1]) satisfy 𝜆_𝑗≈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 (ℎ_𝑗)_𝑗∈ℕ by applying Gram-Schmidt procedure on the sequence (𝑔_𝑗)_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 𝔼𝑋^𝐻_𝑡−𝑋^𝐻_𝑠^𝑝=𝐶_𝐻,𝑝|𝑡−𝑠|^𝑝𝐻, (6.2) and the asymptotics of the quantization error is given as 𝑒_𝑛,𝑟𝑋^𝐻,𝐿_𝑝([0,1])≈𝑒_𝑛,2𝑋^𝐻,𝐿₂([0,1])≈log(𝑛)^−𝐻,𝑛⟶∞ (6.3) 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=𝐿_𝑝([0,1]) independent of 𝑟 such that 𝑒_𝑛,𝑟𝑋^𝐻,𝐸∼𝑘(𝐻,𝐸)log(𝑛)^−𝐻,𝑛⟶∞. (6.4) 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 (𝐵_𝑡)_𝑡∈[0,1] be a Brownian bridge with covariance function 𝔼𝐵_𝑠𝐵_𝑡=min(𝑠,𝑡)−𝑠𝑡. (6.5) 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 (𝐵_𝑡)_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 𝔼𝐵_𝑡−𝐵_𝑠^𝑝1/𝑝≤𝐶_𝑝,2𝔼𝐵_𝑡−𝐵_𝑠^21/2=𝐶_𝑝,2|𝑡−𝑠|−|𝑡−𝑠|^21/2≤𝐶|𝑡−𝑠|^1/2, (6.6) for any 𝑝≥1. Combining [31, Theorem 3.7] and [13, Corollary 1.3] yields 𝑒_𝑛,𝑟𝐵,𝐿_𝑝([0,1])≈log(𝑛)^−1/2,𝑛⟶∞, (6.7) 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 (𝑋_𝑡)_𝑡∈[0,1] is a Gaussian process given through the covariance function 𝔼𝑋_𝑠𝑋_𝑡=𝜎²(|,2𝛼exp−𝛼𝑠−𝑡|) (6.8) 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 𝑒_𝑛,𝑟𝑋,𝐿_𝑝([0,1])≈log(𝑛)^−1/2,𝑛⟶∞ (6.9) for all 𝑟,𝑝≥1. Furthermore, it holds that 𝔼𝑋_𝑡−𝑋_𝑠^𝑝1/𝑝=𝐶_𝑝,2𝔼𝑋_𝑡−𝑋_𝑠^21/2=𝐶_𝑝,2𝜎²𝛼1−exp(−𝛼|𝑠−𝑡|)^1/2≤𝐶|𝑠−𝑡|^1/2, (6.10) and therefore we can choose 𝑎=𝑏=1/2 to apply Theorem 5.1.

(iv) Fractional Ornstein-Uhlenbeck Process
The fractional Ornstein-Uhlenbeck process (𝑋_𝑡^(𝐻))_𝑡∈[0,1] for 𝐻∈(0,2) is a continuous stationary centered Gaussian process with the covariance function 𝔼𝑋_𝑠^(𝐻)𝑋_𝑡^(𝐻)=𝑒^{−𝛼|𝑡−𝑠|𝐻},𝛼>0. (6.11) In [22] the authors constructed an admissible sequence (𝑔_𝑗)^(𝐻)_𝑗∈ℕ for 𝐻∈(0,1] that satisfies the requirements of Proposition 4.1 with 𝜃=1/2+𝐻/2. Since the eigenvalues 𝜆_𝑗 of 𝐶_𝑋^(𝐻) in 𝐿₂([0,1]) satisfy 𝜆_𝑗≈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 (𝑋_𝑡)_𝑡∈[0,1] fulfilling the SDE 𝑋_𝑡=^𝑡₀𝑏𝑠,𝑋_𝑠𝑑𝑠+^𝑡₀𝜎𝑠,𝑋_𝑠𝑑𝑊_𝑠, (6.12) where the deterministic functions 𝑏,𝜎∶[0,1]×ℝ→ℝ satisfy the growth assumption |𝑏(𝑡,𝑥)|+|𝜎(𝑡,𝑥)|≤𝐶(1+|𝑥|). (6.13) Under some additional ellipticity assumption on 𝜎, the asymptotics of the quantization error in (𝐿_𝑝([0,1],‖⋅‖_𝑝) is then given by 𝑒_𝑛,𝑟𝑋,𝐿_𝑝([0,1])≈log(𝑛)^−1/2, (6.14) as 𝑛→∞ (see [10] and also [32]). Furthermore, one shows that for 0≤𝑠≤𝑡≤1𝔼𝑋_𝑡−𝑋_𝑠^𝑝1/𝑝≤𝐶(𝑡−𝑠)^1/2 (6.15) (see [17, Examples 3.1]) such that Theorem 5.1 can be applied with 𝑎=𝑏=1/2.

Examples 6.2 (Lévy processes). Let (𝑋_𝑡)_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 exp(_ℝ𝑖𝑢𝑥)ℙ^𝑋1(𝑑𝑥)=exp−𝜓(𝑢),𝑢∈ℝ, (6.16) is characterized by the Lévy-Khintchine formula𝜓(𝑢)1=𝑖𝑎𝑢+2𝜎²𝑢²+_ℝ1−𝑒^𝑖𝑢𝑥+𝑖𝑢𝑥𝜒_(|𝑥|<1)Π(𝑑𝑥), (6.17) 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 𝔼𝑋_𝑡−𝑋_𝑠^𝑝=𝔼𝑋_𝑡−𝑠^𝑝, (6.18) and it is further known that the latter moment is finite if and only if _(|𝑥|≥1)|𝑥|^𝑝Π(𝑑𝑥)<∞. (6.19) Furthermore, by the Lévy-Ito decomposition, 𝑋 can be written as the sum of independent Lévy processes 𝑋=𝑋⁽¹⁾+𝑋⁽²⁾+𝑋⁽³⁾, (6.20) 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 of Lévy 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 extended Millar'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]𝔼𝑋_𝑡^𝑝≤𝐶𝑡=𝐶𝑡^1/𝑝𝑝. (6.21) Combining (6.18) and (6.21), we can choose 𝜌 in (5.2) as 𝜌_1,𝑝(𝑥)=𝑥^1/𝑝. For 𝑝∈[1,2) we have by using (6.21) with 𝑝=2𝔼𝑋_𝑡^𝑝≤𝔼𝑋_𝑡^2𝑝/2≤(𝐶𝑡)^𝑝/2=𝐶^1/2𝑡^1/2𝑝, (6.22) and thus we can choose 𝜌_1,𝑝(𝑥)=𝐶𝑥^1/2. Combining these facts, we get 𝜌_1,𝑝(𝑥)=𝐶𝑥^1/2∨𝑝 for 𝑝≥1. (ii)We consider the Compound Poisson process 𝑋_𝑡=^𝐾𝑡_𝑘=1𝑍_𝑘, (6.23) where 𝐾 denotes a standard Poisson process with intensity 𝜆=1 and (𝑈_𝑘)_𝑘∈ℕ is an i.i.d sequence of random variables with ‖𝑍₁‖_{𝐿𝑝(ℙ)}<∞. Then, one shows that 𝔼^𝐾𝑡_𝑘=1𝑍_𝑘^𝑝≤𝑡𝑍₁^𝑝_{𝐿𝑝(ℙ)}exp(−𝑡)^∞_𝑘=1𝑡^𝑘−1𝑘^𝑝𝑡𝑘!≤𝐶^1/𝑝𝑝, (6.24) so that (5.2) is satisfied with 𝜙_2,𝑝(𝑥)=𝑥^1/𝑝. (iii) We consider a Brownian motion with drift. Using Examples 6.1 (i) and Lemma 2.1 we can choose 𝜌 in (5.2) as 𝜌_3,𝑝(𝑥)=𝜌₃(𝑥)=𝑥^1/2 for all 𝑝≥1. (2)Lévy Processes with Nonvanishing Brownian Component Let 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 log(𝑛)^−1/2≈𝐶𝑒_𝑛,𝑟𝑊,𝐿_𝑝≲𝑒_𝑛,𝑟𝑋,𝐿_𝑝,𝑛⟶∞ (6.25) 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 𝑋_𝑡⁽²⁾=∑^𝐾𝑡_𝑘=1𝑍_𝑘it holds that‖𝑍₁‖_{𝐿𝑝∨𝑟(ℙ)}<∞. Therefore, we receive the mean pathwise regularity for 𝑋, all 𝑝,𝑟≥1, and some constant 𝐶<∞𝜌_𝑝(𝑥)∶=𝐶𝑥^{1/2∨𝑟∨𝑝}. (6.26) Thus, we can apply Theorem 5.1 with 𝑎=1/2∨𝑝∨𝑟 and 𝑏=1/2. (3)Compound Poisson Processes For a Compound Poisson process 𝑋 we know that the rate for the asymptotic quantization error under suitable assumptions is given by 𝑒_𝑛,𝑟𝑋,𝐿_𝑝≈exp−𝜅log(𝑛)loglog(𝑛),𝑛⟶∞; (6.27) 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 𝑋−𝑉_𝑚(𝑋)_{𝐿𝑝([0,1])𝐿𝑟(ℙ)}=𝑜exp−𝜅log(𝑛)loglog(𝑛), (6.28) 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 with 𝛼∈(0,2)These are Lévy processes satisfying the self-similarity property 𝑋=_𝑡^𝑑𝑡^1/𝛼𝑋₁, (6.29) and furthermore 𝔼𝑋₁^𝛼=∞,sup_𝑟𝔼𝑋₁^𝑟<∞=𝛼. (6.30) Thus, we can choose 𝜌(𝑥)=𝐶𝑥^1/𝛼 for any 𝑝≥1 and constants 𝐶_𝑝<∞. The asymptotics of the quantization error for 𝑋 is given by 𝑒_𝑛,𝑟𝑋,𝐿_𝑝≈log(𝑛)^−1/𝛼,𝑛⟶∞ (6.31) 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 an 𝑛-quantizer 𝛼_𝑛 as an element of 𝐸^𝑛 (again denoted by 𝛼_𝑛) instead of being a subset of 𝐸. Then, 𝑟-optimality of an 𝑛-quantizer 𝛼_𝑛 for the random variable 𝑋 in the separable Banach space 𝐸 reads 𝛼_𝑛=argmin_{𝑎𝛼=1,…,𝑎𝑛∈𝐸𝑛}𝐷^𝑋_𝑛,𝑟(𝛼),𝐷^𝑋_𝑛,𝑟(𝛼)∶=𝔼min_{1≤𝑖≤𝑛}𝑋−𝑎_𝑖^𝑟, (7.1) with 𝐷^𝑋_𝑛,𝑟(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 ℝ^𝑑 is smooth. Let 𝑟>1, and assume that any Voronoi diagram {𝑉_𝑎𝑖(𝛼),𝑎_𝑖∈𝛼,1≤𝑖≤𝑛} with 𝑉_𝑎𝑖(𝛼)∶={𝑥∈ℝ^𝑑∶‖𝑥−𝑎_𝑖‖=min_𝑎∈𝛼‖𝑥−𝑎‖} satisfies ℙ^𝑋(𝑉_𝑎𝑖(𝛼)∩𝑉_𝑎𝑗(𝛼))=0 for 𝑖≠𝑗. Then, the distortion function is differentiable at every admissible 𝑛-tuple 𝛼=(𝑎₁,…,𝑎_𝑛) (i.e., 𝑎_𝑖≠𝑎_𝑗 for 𝑖≠𝑗) with ∇𝐷^𝑋_𝑛,𝑟(𝛼)=𝑟𝔼𝜒_{𝐶𝑎𝑖(𝛼)⧵{𝑎𝑖}}(𝑋)𝑋−𝑎_𝑖^𝑟−1∇‖⋅‖𝑋−𝑎_𝑖∈ℝ^𝑑𝑛, (7.2) where {𝐶_𝑎𝑖(𝛼)∶1≤𝑖≤𝑛} denotes any Voronoi partition induced by 𝛼={𝑎₁,…,𝑎_𝑛}.

Remark 7.2. When 𝑟=1, the above result extends to admissible 𝑛-tuples with ℙ^𝑋({𝑎₁,…,𝑎_𝑛})=0. Furthermore, if the norm is just smooth on a set 𝐴∈ℬ(𝐸) with ℙ^𝑋(𝐴)=1, then the result still holds true. This is, for example, the case for (𝐸,‖⋅‖)=(ℝ^𝑑,‖⋅‖_∞) and random variables 𝑋 with ℙ^𝑋(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 let 𝑛∈ℕ,𝑚=𝑚(𝑛)∈ℕ,𝑟≥1,𝑋,𝑉_𝑚, and 𝜙_𝑚 be given. The procedure looks as follows.

Step 1. Calculation of the Distribution of the ℝ^𝑚-Valued Random Variable 𝜁∶=𝜙_𝑚(𝑉_𝑚(𝑋)). This step strongly depends on the shape of the random variable 𝑋 and the operators 𝑉_𝑚.
Being in the setting of Section 3.1 one starts with an orthonormal system (ℎ_𝑖)_𝑖∈ℕ in 𝐻 providing 𝜙_𝑚𝑉_𝑚(𝑋)=𝜙_𝑚𝑉_𝑚^∞_𝑖=1ℎ_𝑖ℎ_𝑖,𝑋=𝜙_𝑚^𝑚_𝑖=1ℎ_𝑖ℎ_𝑖,𝑋=^𝑚_𝑖=1𝑒_𝑖ℎ_𝑖,𝑋, (7.3) where (𝑒_𝑖)_{1≤𝑖≤𝑚} denote the unit vectors in ℝ^𝑚. Thus, the covariance matrix of the random variable 𝜁 admits the representation 𝔼𝜁𝜁^⟂=𝔼ℎ_𝑖,𝑋ℎ_𝑗,𝑋_{1≤𝑖,𝑗,≤𝑚}=𝐶_𝑋(ℎ_𝑖),ℎ_{𝑗1≤𝑖,𝑗,≤𝑚}, (7.4) with 𝐶_𝑋 being the covariance operator of 𝑋.
Similarly, we get for Gaussian random variables in the framework of Section 3.2𝔼𝜁𝜁^⟂=𝜆_𝑖^(𝑚)𝜆_𝑗^(𝑚)_{1≤𝑖,𝑗,≤𝑚}, (7.5) in the setting of Section 4𝔼𝜁𝜁^⟂=𝔼𝑋_𝑡𝑖^(𝑚)𝑋_𝑡𝑗^(𝑚)_{1≤𝑖,𝑗,≤𝑚}=𝛿_𝑡𝑖𝐶_𝑋𝛿_{𝑡𝑗1≤𝑖,𝑗,≤𝑚}, (7.6) and in the setting of Section 5𝔼𝜁𝜁^⟂=𝑚^1−2/𝑝𝑓_𝑗^(𝑚)𝐶_𝑋𝑓_𝑖^(𝑚)_{1≤𝑖,𝑗,≤𝑚}, (7.7) with 𝑓_𝑗^(𝑚) associated with ∫_[0,1]⋅𝑓_𝑗^(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 ^𝑡𝑖−1_𝑡𝑖^𝐾𝑡_𝑗=1𝑍_𝑘𝑑𝑡, (7.8) which is still possible.

Step 2. Use a (stochastic) optimization algorithm to solve the stationarity equation ∇𝐷^𝜁_𝑛,𝑟(𝛼)=0∈ℝ^𝑚, (7.9) for 𝜁=𝜙_m(𝑉_𝑚(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 ℙ^𝜁 needs to be simulated, which is the case for the above described examples.

Step 3. Reconstruct the quantizer𝛽=(𝑏₁,…,𝑏_𝑛) for the random variable 𝑋 by setting 𝑏_𝑖∶=𝜙_𝑚⁻¹𝑎_𝑖∈𝐹_𝑚⊂𝐸, (7.10) for 1≤𝑖≤𝑛 with 𝛼=(𝑎₁,…,𝑎_𝑛) being some solution of the stationarity (7.9).

Illustration
For illustration purposes, we will concentrate on the case described in Section 3.1 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]: 𝑋_𝑡^𝑑(𝐻)=^∞_𝑛=1√2𝑐_𝐻𝐽_1−𝐻𝑥_𝑛sin𝑥_𝑛𝑡𝑥_𝑛^1+𝐻𝜁¹_𝑛+^∞_𝑛=1√2𝑐_𝐻𝐽_−𝐻𝑦_𝑛1−cos𝑦_𝑛𝑡𝑦_𝑛^1+𝐻𝜁²_𝑛, (7.11) where 𝑐_𝐻 is given as 𝑐²_𝐻()Γ()∶=sin𝜋𝐻1+2𝐻𝜋, (7.12)𝐽_1−𝐻 and 𝐽_−𝐻 are Bessel functions with corresponding parameters, and 𝑥_𝑛 and 𝑦_𝑛 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 parameter 𝐻=0.3 in a 4-dimensional subspace.

Figure 2 

The 10-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.5 in a 4-dimensional subspace.

Figure 3 

The 10-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.7 in a 4-dimensional subspace.

Figure 4 

The 10-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.3 in an 8-dimensional subspace.

Figure 5 

The 10-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.5 in an 8-dimensional subspace.

Figure 6 

The 10-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.7 in an 8-dimensional subspace.

Figure 7 

The 30-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.3 in an 8-dimensional subspace.

Figure 8 

The 30-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.5 in an 8-dimensional subspace.

Figure 9 

The 30-quantizer for the fractional Brownian motion with Hurst parameter 𝐻=0.7 in an 8-dimensional subspace.