QUANTIZATION BALLS AND ASYMPTOTICS OF QUANTIZATION RADII FOR PROBABILITY DISTRIBUTIONS WITH RADIAL EXPO-NENTIAL TAILS

In this paper, we provide the sharp asymptotics for the quantization radius (maximal radius) for a sequence of optimal quantizers for random variables X in ( R d , (cid:107)·(cid:107) ) with radial exponential tails. This result sharpens and generalizes the results developed for the quantization radius in [ 4 ] for d ≥ 2, where the weak asymptotics is established for similar distributions in the Euclidean case. Furthermore, we introduce quantization balls , which provide a more general way to describe the asymptotic geometric structure of optimal codebooks, and extend the terminology of the quantization radius.


Introduction and results
We consider (R d , • ) equipped with an arbitrary norm and a random variable X in R d such that for some r > 0 the r-th moment E X r is finite.For some natural number n ∈ N, the quantization problem consists in finding a set α = {a 1 , . . ., a n } that minimizes is called r-optimal n-quantizer.In the present setting, the existence of optimal quantizers is guaranteed (see [1,Lemma 4.10]).For a given n-quantizer α, the nearest neighbor projection is given 284 Electronic Communications in Probability by where the Voronoi partition {C a (α), a ∈ α} is defined as a Borel partition of R d satisfying The random variable π α (X ) is called the α-quantization of X .One easily verifies that π α (X ) is the best quantization of X in α ⊂ R d , that is for every random variable Y with values in α we have One typically is interested in the behavior of e n,r (X , R d ) when n tends to infinity.By finiteness of E X r , we deduce that e n,r (X , R d ) is finite as well.Considering the dense and countable subset {q 1 , q 2 , . . .
and derive that e n,r (X , R d ) tends to zero as n tends to infinity.Naturally, one may now ask for a more precise description of this convergence.To answer this question, it will be convenient to write a n b n for sequences (a n ) n∈N and Let E X r+δ < ∞ for some δ > 0 and h be the non-vanishing Lebesgue-continuous part of the density of the distribution P X .Then, the sharp asymptotics of the quantization error is given by The numbers Q r (d) ∈ (0, ∞), depending on r, d and • , are usually unknown, unless in some special cases, for example This result is known as the Zador Theorem.Its final proof was completed by Graf and Luschgy and can be found in [1,Chapter 6].
As good as the asymptotics of the quantization error can be estimated, as difficult it is to describe the geometric structure of optimal codebooks, or to give at least some asymptotic results on this.For a random variable X with an unbounded support, one easily sees that any sequence α n of n-codebooks satisfying Again, one may ask to describe this behavior more precisely, e.g. for sequences (α n ) n∈N of roptimal n-quantizers for X .Pagès and Sagna [4]  In this paper, we proof this conjecture and extend the results for distributions having radial exponential tails to distributions having the density Furthermore, we provide the sharp asymptotics of the quantization radius (which will be our terminology for the maximal radius) of a sequence of r-optimal n-quantizers in (R d , • ), for arbitrary norms • , • 0 and arbitrary positive r.This result extends the formulation of the conjecture to arbitrary norms as well as cases in which the quantizing norm • does not coincide with the norm • 0 in equation (1.3).The latter extension seems to be more important since it includes, for example, normal distributions having a regular non-unit covariance matrix.Furthermore, we introduce quantization balls which provide a more general way to describe the asymptotic geometric structure of optimal codebooks and extend the terminology of the quantization radius.The paper is organized as follows: In section 2 we introduce the basic notations and provide some technical support for the following section.In section 3 we formulate the main theorem and give the proof for the lower an upper bound (section 3.1 and 3.2).We also transfer the result to the important example of a general normal distribution in (R d , • 2 ).In section 4 we provide some numerical illustrations of the results.

Notations:
• Throughout this paper, we consider a probability space (Ω, , P), (R d , • ) for some d ∈ N equipped with arbitrary norm • and a Borel random variable X in (R d , • ) with finite r-th moment E X r < ∞ for some r > 0.
• For an arbitrary norm • 0 and any s > 0 we set • For an arbitrary norm • 0 we will denote by dist
2. Let (α n ) n∈N be a sequence of r-optimal n-quantizers for the random variable where conv denotes the convex hull and If is independent of (α n ) n∈N we call the quantization ball.
3. The survival function for the random variable X is given by

4.
For s ≥ 0 we define the generalized survival function by Since we will focus on distributions with radial exponential tails in the following, we provide relevant results for the survival functions and the (r, r + ν)-distribution properties for random variables having this density type.
Lemma 2.2.Let P X = f λ d with f having the shape for constants θ , k > 0, c > −d and the norming constant Then, for every s ≥ 0 )), we have )).We obtain

Quantization Balls and Asymptotics of Quantization Radii for Probability Distributions with Radial Exponential Tails 287
Integration by parts yields and thus the assertion with C θ ,c,k, • := The proof of the following result is based on the proof in [4] of a similar result in the Euclidean case.

The main theorem
For this section, let X be a random variable in (R d , • ) with P X = f λ d and f having the shape for constants θ , k > 0, c > −d, an arbitrary norm • 0 and a norming constant K. Note, that , which holds as a maximum if c ≥ 0, g is increasing on [0, y * ] and decreasing on [ y * , ∞).Furthermore, we set for n ∈ N For any sequence of r-optimal n-quantizers x , n → ∞, and the quantization ball is given by Remark 3.2.We see that the asymptotics of the quantization radius and the quantization ball are independent of the sequence of quantizers (α n ) n∈N .Furthermore, the quantization ball is independent of the choice of r and, except for a scaling factor, of the underlying norm • .
. Let P X be a centered d-dimensional normal distribution with regular covariance matrix Σ and corresponding non increasing ordered eigenvalues Its density is given by ), where • 2 denotes the Euclidean norm in R d .Thus, it has the form (3.1) The operator norm of the natural embedding j : is given as the root of the biggest eigenvalue λ 1 of the covariance matrix.Using Theorem 3.1 we obtain the asymptotics of the quantization radius for any sequence of r-optimal n-quantizers and the quantization ball is given as the normalized unit ball in (R d , • 0 ) The proof of Theorem 3.1 consists of two parts, one for the lower and one for the upper bound.
For convenience we will denote by Quantization Balls and Asymptotics of Quantization Radii for Probability Distributions with Radial Exponential Tails 289

Lower bound
Proposition 3.4.Let (α n ) n∈N be a sequence of r-optimal n-quantizers for the random variable X in (R d , • ).Then, for every δ ∈ (0, 1) and every sequence Proof.Assume that there is a δ ∈ (0, 1) and a sequence does not hold.Then, there exists an ε > 0 and a strictly increasing subsequence

.4)
Case 1: There is a subsequence of (n j ) j∈N , for convenience also denoted (n j ) j∈N , such that b n j φ(n j ) 0 − ε 2C ≥ y * for all j ∈ N.Then, by monotonicity of g and (3.4) Taking the (r +ν)-th root and the negative logarithm, we obtain by using the (r, r +ν)-distribution property which yields a contradiction by letting ν go to d.

Case 2:
There is an for all j ≥ N , which is a contradiction since inf x∈A f (x) = min{g( ε 4C ), g( y * + ε C )} > 0 and the left hand side converges to zero as j tends to infinity.
The following corollary provides the lower bound for the main theorem.

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Corollary 3.5.We use the notations from Proposition 3.4.For every δ ∈ (0, 1) there exists an n δ ∈ N such that for all n ≥ n δ .As an immediate consequence

.6)
The lower bound for the quantization radius is given by Proof.Assume that (3.5) does not hold.Then, there exists a strictly increasing sequence We show that there are By using a separation theorem, we find for every k ∈ N a continuous linear f n k , such that Since B • 0 (0, 1) is compact and f n k continuous, there is f n k ( y).
Quantization Balls and Asymptotics of Quantization Radii for Probability Distributions with Radial Exponential Tails 291

Upper bound
Proposition 3.6.Let (α n ) n∈N be a sequence of r-optimal n-quantizers for X in (R d , • ).Then, for every δ > 0 there exists an n δ ∈ N such that for all n ≥ n δ .
Proof.Assume that there is a strictly increasing subsequence (3.9) Step 1: We show that for some ε > 0, k 0 ∈ N and all k ≥ k 0 .With (3.9), we obtain dist which implies in view of equivalence of the norms dist On the other hand, it holds with ε := δ for all sequences Combining (3.10) and (3.11) we obtain Electronic Communications in Probability for all k ≥ k 0 .
Step 2: By choosing in Proposition 3.4 b n = 0 for n ∈ N, we find a sequence c n k ∈ α n k with ε k := c n k → 0. By using (3.12) and Proposition 2.4, we get for some constant K > 0 as k → ∞.Applying Lemma 2.2 with s = 0 and s = r leads to as k → ∞.Finally, taking the negative logarithm we get This is equivalent to which yields a contradiction.
The following corollary provides the upper bound for the main theorem.It follows from Proposition 3.6, the definition of lim sup and a similar argumentation as for (3.6) and (3.7).
Corollary 3.7.We use the notations from Proposition 3.6.For every δ > 0 exists an n for all n ≥ n δ .This immediately leads to The upper bound for the quantization radius is given by x , n → ∞.
The definition of lim inf and lim sup,

Numerical illustration
Finally, we want to illustrate some of our results.For the computation of the optimal codebooks presented below, we used the CLVQ-Algorithm, see [3].We consider the Euclidean R 2 , r = 2 and X d = N (0, Σ) with eigenvalues λ 1 = 1 and λ 2 = 1 4 .The figures show the 2-optimal n-quantizers for n = 50, 250, 1000.The two ellipses in the figures are the scaled quantization balls ρ(α n ) and φ(n) with φ(n) as in section 3.As already mentioned in [4] for the unit-covariance case, we see that in this case the quantization radius ρ(α n ) seems for finite n to be smaller than its asymptotic equivalent φ(n) as well.Furthermore, we observe that for small n the convex hull of α n does not completely fill the ellipse ρ(α n ), whereas for growing n almost the whole ellipse seems to be filled by conv(α n ).