Universal <span class="mathjax-formula">$L^s$</span>-rate-optimality of <span class="mathjax-formula">$L^r$</span>-optimal quantizers by dilatation and contraction

DOI:10.1051/ps:2008008 www.esaim-ps.org

UNIVERSAL L

-RATE-OPTIMALITY OF L

-OPTIMAL QUANTIZERS BY DILATATION AND CONTRACTION

Abass Sagna

Abstract. We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)_n≥1ofL^r-optimal quantizers of anR^d-valued random vectorX ∈L^r(P) defined in the probability space (Ω,A,P) with distributionPX =P. To be precise, we investigate theL^s-quantization rate of sequencesα^θ,μn =μ+θ(αn−μ) ={μ+θ(a−μ), a∈αn}whenθ∈R+, μ∈R, s∈(0, r) ors∈(r,+∞) and X ∈ L^s(P). We show that for a wide family of distributions, one may always find parameters (θ, μ) such that (α^θ,μn )_n≥1 is L^s-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ, μ) such that (α^θ,μ)_n≥1 also satisfies the so-called L^s-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptoticallyL^s-optimal. In both cases the sequence (α^θ,μ)_n≥1is incredibly close toL^s-optimality.

However we show (see Rem. 5.4) that this last sequence is not L^s-optimal (e.g. when s= 2, r= 1) for the exponential distribution.

Mathematics Subject Classification. 60G15, 60G35, 41A52.

Received July 12, 2007. Revised February 20, 2008.

1. Introduction

Let (Ω,A,P) be a probability space and let X : (Ω,A,P) −→ R^d be a random variable with distribution PX = P. The L^r(P)-optimal quantization problem of size n for P (or X) consists in the study of the best approximation of X by aσ(X)-measurable random vector taking at mostn values. ForX ∈L^r(P) this leads to the following optimization problem:

e_n,r(X) = inf{X−q(X)r, q:R^d→α, α⊂R^d, card(α)≤n}.

Letα⊂R^d be a subset (a codebook) of size n. A Borel partitionC_a(α)_a∈α ofR^d satisfying C_a(α)⊂ {x∈R^d :|x−a|= min

b∈αn

|x−b|},

Keywords and phrases. Rate-optimal quantizers, empirical measure theorem, dilatation, Lloyd algorithm.

1 Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UMR 7599, Universit´e Pierre et Marie Curie, Case 188, 4 place Jussieu, 75252 Cedex 05, Paris, France;[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2009

where| · | denotes a norm onR^d is called a Voronoi partition ofR^d (with respect toαand| · |).

The random variableX^αtaking values in the codebookαdefined by X^α=

a∈α

a1_{X∈Ca(α)}.

is called a Voronoi quantization of X. In other words, it is the nearest neighbour projection of X onto the codebook (also called grid)α.

For any Borel functionq:R^d→α,

|X−q(X)| ≥min

a∈αd(X, a) = d(X, α) =|X−X^α| Pa.s.

so that

e_n,r(X) = inf{X−X^αr, α⊂R^d, card(α)≤n}

= inf

α⊂R^d card(α)≤n

R^dd(x, α)^rdP(x) _1/r

. (1.1)

For all n≥ 1, the infimum in (1.1) is reached at one (at least) codebook α; α is then called aL^r-optimal n-quantizer. In addition, if card(supp(P))≥n then card(α) =n(see [5] or [8]). Moreover the quantization error,e_n,r(X), decreases to zero asn goes to infinity and the so-called Zador’s Theorem gives its convergence rate under a slightly stringent moment assumption onX.

Zador’s Theorem (see [5]): Suppose E|X|^r+η<+∞for some η >0 and letP=P_a+P_sbe the Lebesgue decomposition ofP with respect to the Lebesgue measureλ_d, whereP_a denotes the absolutely continuous part andP_sthe singular part ofP. Then

n→+∞lim n^r/d(e_n,r(P))^r=Q_r(P).

with

Q_r(P) =J_r,d

R^df^d+rd dλ_d ^d+r

d

=J_r,d f ^d

d+r ∈[0,+∞), J_r,d= inf

n≥1n^r/de^r_n,r(U([0,1]^d))∈(0,+∞), where U([0,1]^d) denotes the uniform distribution on the set [0,1]^d and f = ^dP_dλ^a

d. Furthermore, this theorem naturally suggests to set the following definitions.

A sequence ofn-quantizers (α_n)_n≥1 is

– L^r(P)-rate-optimal(orrate-optimalforX) if

lim sup

n→+∞n^1/d

R^dd(x, α_n)^rdP(x) _1/r

<+∞, – asymptotically L^r(P)-optimalif

n→+∞lim n^r/d

R^dd(x, α_n)^rdP(x) =Q_r(P),

– L^r(P)-optimalif for alln≥1,

e^r_n,r(P) =

R^dd(x, α_n)^rdP(x).

Optimal quantizers are used in many fields of application like Signal Processing (discretization of emitted signals) or more recently, Numerical Probability where they provide some simples quadrature formulae for the computation of expectations and conditional expectations. This approach has been extensively developed in Finance for the pricing of American options, swing options (commodities), portfolio management (stochastic control) or stochastic volatility estimation (non linear filtration); we refer for example to [9] for applications to stochastic control problems. The errors bounds in these quadrature formulae are always based on the mean quantization errorX−X^αswhere αis an optimalL^r-quantizer, usually withr≤s.

Motivated by this problem, the asymptotic behavior of the L^s-mean quantization error of a sequence of L^r-quantizers has been extensively investigated in [6]. A lower bound has been established which shows that if the distributionP is unbounded in the sense that the density functionf = _dλ^dP

d satisfiesλ_d(f >0) = +∞then for any sequence (α_n)_n≥1 of asymptoticallyL^r-optimal quantizers, lim inf

n n^1dX−X^αns= +∞, ∀s > r+d.

On the other hand, under natural assumptions in the tail of the distribution P, it is shown in [6] that for any sequence of L^r-optimal quantizers,∀s ∈(0, r+d), lim sup

n

n^1dX−X^αns <+∞ i.e.(α_n)_n≥1 remains L^s-rate-optimal as long ass < r+d.

The aim of this paper is to show that some simple transformation of the L^r-optimal quantizers, namely some dilatation-translation, makes possible to overcome the critical exponent r+d: we will establish that for a wide family of distributions, one can always find θ∈R₊ andμ∈R(depending onr, s anddbut not onn) such that (α^θ,μ_n )_n≥1 isL^s-rate-optimal. From a general upper bounds that we establish for such transformed sequences of quantizers we derive an heuristic to specify some explicit optimal (in a sense which will be elucidated later) scaling parameters (θ, μ) for several families of distributions (Gaussian Vector, exponential and gamma distributions). Some numerical computations carried with the Gaussian and the exponential distributions show that the resulting sequence of quantizers is very close toL^s-optimality.

So, one application could be to use these quantizers to initialize the procedures used for L^s-optimal (and local optimal) quantizers search whens= 2. Indeed, in the quadratic case,s= 2, several stochastic procedures like the Competitive Learning Vector Quantization algorithm or the randomized Lloyd’s I procedure have been designed. Both rely on the stationary property: X^α = E(X|X^α), satisfied by optimal (and locally optimal) quadratic quantizers. In one dimension, Newton’s method is used to compute the optimal quadratic quantizers. Thus a whole package of optimal n-quantizers of the N(0, I_d) distributions are available in the websitehttp://www.quantize.maths-fi.comford∈ {1, . . . ,10} andn∈ {1, . . . ,5000}.But, whens= 2, the natural extension of these procedures become more difficult to implement due to some loss of stability. When s >2 the procedures tend to explode more and more often while when 1≤s <2 the convergence phase becomes chaotic. In particular, the sensibility of the procedure to its initialization increases as s moves away from 2.

Thus, initializing theses procedures by the dilated-contractedL²-optimal (or locally optimal) quantizers would make them more stable and speed up the convergence. This is what we do to carry theL⁴-optimal quantizers of the one dimensional Gaussian distribution (used for numerical experiments in Sect. 5.1.2) by Newton’s method.

In fact, initializing this procedure to a n-tuple different from the dilated sequence usually makes the hessian matrix of the L⁴-quantization error singular (which makes the procedure very unstable), especially when the grid’s size becomes large (typically whenn≥400).

The paper is organized as follows. In Section 2 we establish a general lower bound for dilated-translated se- quences of quantizers. General upper bounds are also established in Section 3 for such sequences. In Section 4 we provide a necessary and sufficient condition ofL^s-rate-optimality for the dilated-translated sequences. Section 5 deals with some examples of distributions for which we give the set of parameters (θ, μ) such that the dilated- translated sequence isL^s-rate-optimal and try to find the couple (if any) which makes the resulted transformed sequence satisfy theL^s-empirical measure theorem. The last section is devoted to some applications.

Notations: •Letα_n be a set of npoints ofR^d . For everyμ∈R^d and everyθ >0 we denote α^θ,μ_n =μ+θ(α_n−μ) ={μ+θ(a−μ), a∈α_n}.

•Letf :R^d−→R^d be a Borel function and letμ∈R^d, θ >0. One notes byf_θ,μ(orf_θifμ= 0) the function defined byf_θ,μ(x) =f(μ+θ(x−μ)), x∈R^d.

• IfX ∼P,P_θ,μ will stand for the probability measure of the random variable ^X−μ_θ +μ, θ >0, μ∈R^d. In other words, it is the distribution image ofP byx−→ ^x−μ_θ +μ.Note that if P=f·λ_d then ^dP_dλ^θ,μ

d =θ^df_θ,μ.

• IfAis a matrixA stands for its transpose.

• Setx= (x₁, . . . , x_d); y= (y₁, . . . , y_d)∈R^d; we denote [x, y] = [x₁, y₁]×. . .×[x_d, y_d].

• Let| · |be a norm on R^d and letAbe a subset of R^d; we denote byB(x, r) the closed ball, centered tox with radiusr >0 and by d(x, A) the distance betweenxandA; both with respect to the norm| · |.

2. Lower estimate

Letr, s >0. Consider an asymptoticallyL^r(P)-optimal sequence of quantizers (α_n)_n≥1 . For everyμ∈R^d and any θ >0, we construct the sequence (α^θ,μ_n )_n≥1and try to lower bound asymptotically theL^s-quantization error induced by this sequence. This estimation provides a necessary condition of rate-optimality for the sequence (α^θ,μ_n )_n≥1. In the particular case whereθ= 1 andμ= 0 we get the same result as in [6].

Theorem 2.1. Let r, s ∈ (0,+∞), and let X be a random variable taking values in R^d with distribution P such that P_a = f.λ_d ≡ 0. Suppose that E|X|^r+η < ∞ for someη > 0. Let (α_n)_n≥1 be an asymptotically L^r(P)-optimal sequence of quantizers. Then, for every θ >0 and for every μ∈R^d,

lim inf

n→+∞n^s/d X−X^{αθ,μn s}_s≥Q^Inf_r,s(P, θ), (2.1) with

Q^Inf_r,s(P, θ) =θ^s+dJ_s,d

R^d

f^d+rd dλ_{d s/d}

{f >0}

f_θ,μf^−d+rs dλ_d.

Proof. Letm≥1 and

f_m^θ,μ=

m2^m−1 k,l=0

l 2^m1_E^m

k∩G^m_l ; with

E_k^m= k

2^m ≤f < k+ 1 2^m

∩B(0, m) andG^m_l = l

2^m ≤f_θ,μ< l+ 1 2^m

∩B(0, m).

The sequence (f_m^θ,μ)_m≥1 is non-decreasing and

m→+∞lim f_m^θ,μ=f_θ,μ λ_d-p.p.

Let

I_m={(k, l)∈ {0, . . . , m2^m−1}²:λ_d(E_k^m)>0;λ_d(G^m_l )>0}.

For every (k, l)∈I_m there exists compact setsK_k^m andL^m_l such that:

K_k^m⊂E^m_k ,L^m_l ⊂G^m_l ,λ_d(E_k^m\K_k^m)≤ 1

m⁴2^2m+1 andλ_d(G^m_l \L^m_l )≤ 1 m⁴2^2m+1· Then

(E_k^m∩G^m_l )\(K_k^m∩L^m_l ) = E_k^m∩G^m_l ∩((K_k^m)^c∪(L^m_l )^c)

⊂ (E_k^m\K_k^m)∪(G^m_l \L^m_l ).

Consequently,

λ_d(E_k^m∩G^m_l \K_k^m∩L^m_l ) ≤ λ_d(E_k^m\K_k^m) +λ_d(G^m_l \L^m_l )

≤ 1

m⁴2^2m+1 + 1 m⁴2^2m+1

= 1

m⁴2^2m· For everym≥1 and every (k, l)∈I_m, set

A^m_k,l:=K_k^m∩L^m_l ,

f˜_m^θ,μ:=

m2^m−1 k,l=0

l 2^m1_Am

k,l,

and

f˜_m:=

m2^m−1 k,l=0

k 2^m1_Am

k,l. We get

{f_m^θ,μ= ˜f_m^θ,μ} ⊂

k,l∈{0,...,m2^m−1}

(E_k^m∩G^m_l )\A^m_k,l .

Therefore, for everym≥1,

λ_d({f_m^θ,μ= ˜f_m^θ,μ})≤

m2^m−1 k,l=0

1

m⁴2^2m = 1 m² and finally

m≥1

1_{fθ,μ

m = ˜fm^θ,μ}<∞ λ_d-p.p.

As a consequentlyλ_d(dx)-p.p.,f_m^θ,μ(x) = ˜f_m^θ,μ(x) for large enoughm. Then ˜f_m^θ,μλd−→^p.p.f_θ,μ when m→+∞.Since in additionA^m_k,l⊂E_k^m∩G^m_l we obtain

f˜_m^θ,μ≤f_m^θ,μ≤f_θ,μ. (2.2)

Moreover, for everyn≥1,

n^s/dX−X^{αθ,μn s}_s = n^s/d

R^d

d(z, μ+θ(α_n−μ))^sf(z)λ_d(dz)

= n^s/d

R^d min

a∈αn|z−(μ+θ(a−μ))|^sf(z)λ_d(dz)

= θ^sn^s/d

R^d min

a∈αn|(z−μ)/θ+μ−a|^sf(z)λ_d(dz).

Making the change of variablex:= (z−μ)/θ+μyields:

n^s/d X−X^{αθ,μn s}_s = θ^s+dn^s/d

R^d

d(x, α_n)^sf_θ,μ(x)λ_d(dx)

≥ θ^s+dn^s/d

R^dd(x, α_n)^sf˜_m^θ,μλ_d(dx) (by (2.2))

= θ^s+dn^s/d

m2^m−1 k,l=0

l 2^m

A^m_k,l

d(x, α_n)^sλ_d(dx). (2.3)

Letm≥1 and (k, l)∈I_m. Define the closed sets ˜A^m_k,l by ˜A^m_k,l=∅ifλ_d( ˜A^m_k,l) = 0 and otherwise by A˜^m_k,l={x∈R^d: d(x, A^m_k,l)≤ε_m},

whereε_m∈(0,1] is chosen so that

A˜^m_k,l

f^d+rd dλ_d ≤

1 + 1/m

A^m_k,l

f^d+rd dλ_d.

Since ˜A^m_k,l is compactA˜^m_k,l⊂B(0, m+ 1)∀(k, l) , and

A^m_k,l⊂( ˜A^m_k,l)_εm/2:={x∈R^d : d(x, A^m_k,l)≤ε_m/2}={x∈R^d: d(x,( ˜A^m_k,l)^c)> ε_m/2}, there is (Ref. [1], Lem. 4.3) a finite “firewall” setβ_k,l^m such that ∀n≥1, ∀x∈( ˜A^m_k,l)_εm/2,

d(x, α_n∪β_k,l^m) = d(x,(α_n∪β_k,l^m)∩A˜^m_k,l).

This last equality holds in particular for everyx∈A^m_k,l sinceA^m_k,l⊂( ˜A^m_k,l)_εm/2. Now setβ^m=

k,lβ_k,l^m and n^m_k,l= card((α_n∪β^m)∩A˜^m_k,l).The empirical measure theorem (see (5.3)) yields lim sup

n

card(α_n∩A˜^m_k,l)

n =

αn∩A˜^m_k,lf^d+rd dλ_d f^d+rd dλ_d ≤

A˜^m_k,lf^d+rd dλ_d f^d+rd dλ_d · Moreover

n^m_k,l

n ∼card(α_n∩A˜^m_k,l)

n when n→+∞

then

lim inf

n→+∞

n n^m_k,l ≥

f^d+rd dλ_d

A˜^m_k,lf^d+rd dλ_d ≥ m m+ 1

f^d+rd dλ_d

A^m_k,lf^d+rd dλ_d· (2.4) On the other hand,

A^m_k,l

d(x, α_n)^sλ_d(dx) ≥

A^m_k,l

d(x,(α_n∪β_k,l^m)∩A˜^m_k,l)^sλ_d(dx)

= λ_d(A^m_k,l)

d(x,(α_n∪β^m_k,l)∩A˜^m_k,l)^s1_Am

k,l(x) λ_d(dx) λ_d(A^m_k,l)

≥ λ_d(A^m_k,l)e^s_nm

k,l,s(U(A^m_k,l)),

where U(A) = 1_A/λ_d(A) denotes the uniform distribution in the Borel setA when λ_d(A)= 0. Then we can write for every (k, l)∈I_m,

lim inf

n→+∞n^s/d

A^m_k,l

d(x, α_n)^sλ_d(dx)≥λ_d(A^m_k,l) lim inf

n

n n^m_k,l

_s/d lim inf

n n^s/de^s_n,s(U(A^m_k,l)), since

lim inf

n n^s/de^s_n,s(U(A^m_k,l))≥J_s,d·λ_d(A^m_k,l)^s/d. Owing to equation (2.4), one has

lim inf

n→+∞n^s/d

A^m_k,l

d(x, α_n)^sλ_d(dx)≥λ_d(A^m_k,l)

⎛

⎝ m m+ 1

f^d+rd dλ_d

A^m_k,lf^d+rd dλ_d

⎞

⎠

s/d

J_s,d·λ_d(A^m_k,l)^s/d.

However, on the setsA^m_k,l, the statement ¹_f ≥_k+1

2^m

₋₁

holds sincef < ^k+1_2m onE_k^m. Hence

lim inf

n→+∞n^s/d

A^m_k,l

d(x, α_n)^sλ_d(dx)≥J_s,d

m+ 1 m

f^d+rd λ_d(dx) _s/d

k+ 1 2^m

_−d+r^d _·^s_d

λ_d(A^m_k,l).

It follows from equation (2.3) and the super-additivity of the liminf that for everym≥1,

lim inf

n n^s/dX−X^{αθ,μn s}_s ≥ θ^s+dJ_s,d

m+ 1 m

f^d+rd λ_d(dx)

s/d m2^m−1 k,l=0

l 2^m

k+ 1 2^m

_−d+r^s

λ_d(A^m_k,l)

≥ θ^s+dJ_s,d

m+ 1 m

f^d+rd λ_d(dx) _s/d

{f >0}

f˜_m^θ,μ( ˜f_m+ 2^−m)^−d+rs dλ_d.

Finally, applying Fatou’s Lemma yields lim inf

n→+∞n^s/d X−X^{αθ,μn s}_s≥θ^s+dJ_s,d

R^df^d+rd dλ_{d s/d}

{f >0}f_θ,μf^−d+rs dλ_d.

3. Upper estimate

Letr, s >0. Let (α_n)_n≥1 be an (asymptotically)L^r(P)-optimal sequence of quantizers. In this section we will provide some sufficient conditions ofL^s(P)-rate-optimality for the sequence (α^θ,μ_n )_n≥1.

Definition 3.1. Letθ >0, μ∈R^d and let P be a probability distribution such thatP =f·λ_d. The couple (θ, μ) is saidP-admissibleif

{f >0} ⊂μ(1−θ) +θ{f >0} λ_d-p.p. (3.1) One remarks that when supp(P) =R^d then every couple (θ, μ) isP-admissible. Indeed, everyx∈R^d can be written x=μ(1−θ) +θz withz=^{x−μ(1−θ)}_θ andf(z)>0.

Theorem 3.1. Letr, s∈(0,+∞), s < r and letX be a random variable taking values inR^dwith distributionP such thatP =f·λd. Suppose that(θ, μ)isP-admissible for someθ >0;μ∈R, andE|X|^r+η <∞, for someη >

0. Let(α_n)_n≥1 be an asymptoticallyL^r-optimal sequence of n-quantizers. If

{f >0}f

r−sr

θ,μ f^−r−ss dλ_d<+∞ (3.2)

then, (α^θ,μ_n )_n≥1 is L^s(P)-rate-optimal and lim sup

n→+∞n^s/d X−X^{αθ,μn s}_s≤θ^s+d(Q_r(P))^s/r

{f >0}f

r−sr

θ,μ f^−r−ss dλ_{d 1−r}^s

. (3.3)

Remark 3.1. Note that ifθ= 1 andμ= 0 then

{f >0}f

r−sr

θ,μ f^−r−ss dλ_d =

{f >0}f^r−sr f^−r−ss dλ_d=

{f >0}fdλ_d= 1.

Which gives the expected result since X−X^αns≤ X−X^αnr.

Proof. LetP^θdenote the distribution of the random variableθX. P^θ is absolutely continuous with respect to λ_d, withp.d.f g_θ(x) =θ^−df(^x_θ).

For everyn≥1,

n^s/dX−X^{αθ,μn s}_s = n^s/d

R^dd(x, α^θ,μ_n )^sdP(x)

= n^s/d

{f >0}min

a∈αn|x−μ(1−θ)−θa|^sf(x)dλ_d(x).

Making the change of variablez:=x−μ(1−θ) gives n^s/dX−X^{αθ,μn s}_s=n^s/d

{f >0}−μ(1−θ)d(z, θα_n)^sf(z+μ(1−θ))dλ_d(z)

≤n^s/d

θ{f >0}d(z, θα_n)^sf(z+μ(1−θ))g_θ⁻¹(z)dP^θ(z) (3.4)

≤n^s/d

R^dd(z, θα_n)^rdP^θ(z)

_s/r

θ{f >0}

f(z+μ(1−θ))g_θ⁻¹(z)_r−s^r dP^θ(z)

^r−s

r

≤

n^r/dθX−θX^θαnr_r

_s/r

θ{f >0}

f(z+μ(1−θ))^r−sr g_θ^−r−ss (z)dλ_d(z) ^r−s

r

where we used the P-admissibility of (θ, μ) in the first inequality. The second inequality derives from H¨older inequality applied withp=r/s >1 andq= 1−s/r.

Moreover

θX−θX^θαnr_r=E

a∈αminn|θX−θa|^r

=θ^rX−X^αnr_r. (3.5) Then

n^s/d X−X^{αθ,μn s}_s≤θ^s

n^r/dX−X^αnr_rs/r

θ{f >0}f(z+μ(1−θ))^r−sr g⁻

r−ss

θ (z)dλ_d(z) ^r−s

r

.

Owing to the asymptoticallyL^r(P)-optimality of (α_n) and making again the change of variablex:=z/θyields

lim sup

n→+∞n^s/d X−X^{αθ,μn s}_s≤θ^s(Q_r(P))^s/r

θ^r−sds

θ{f >0}f(z+μ(1−θ))^r−sr f(z/θ)^−r−ss dλ_d(z) ^r−s_r

=θ^s(Q_r(P))^s/r

θ^r−srd

{f >0}f_θ,μ(x)^r−sr f(x)^−r−ss dλ_d(x) ^r−s

r

=θ^s+d(Q_r(P))^s/r

{f >0}

f_θ,μ(x)^r−sr f(x)^−r−ss dλ_d(x) ^r−s

r

.

The next theorem provides a less accurate asymptotic upper bound than the previous one since, beyond the restriction on the distribution of X, we need now the sequence (α_n) to be (exactly) L^r(P)-optimal. Before giving the theorem, recall first the following result established in [6] and related to the maximal function ψ_b:R^d−→R+∪ {+∞}defined by

ψ_b(x) = sup

n≥1

λ_d(B(x, bd(x, α_n)))

P(B(x, bd(x, α_n)))· (3.6)

Proposition 3.1. Let b ∈ 0,1/2

, X ∼P, withP_a = 0, such that E|X|^r+η <∞, for someη >0.Let (α_n) be an L^r(P)-optimal sequence of quantizers. Then ∀x∈R^d,∀n≥1,

n^1/dd(x, α_n)≤C(b)ψ_b(x)^1/(d+r) (3.7)

whereC(b) denotes a real constant not depending onn.

Theorem 3.2. Letr, s∈(0,+∞)and letX be a random variable taking values inR^d with distributionP such that P =f ·λ_d. Suppose that E|X|^r+η<∞for someη >0 and P_θ,μP

i.e. P_θ,μ is absolutely continuous with respect toP

for someθ >0, μ∈R^d. Let(α_n)_n≥1be anL^r(P)-optimal sequence of quantizers and suppose that the maximal function defined previously satisfies

ψ^s/(d+r)_b ∈ L¹(P_θ,μ) for some b∈(0,1/2). (3.8) Then,

lim sup

n

n^s/d X−X^{αθ,μn s}_s≤C(b)θ^s+d

{f >0}f_θ,μf^−d+rs dλ_d<+∞ (3.9) whereC(b) is a positive real constant not depending onθ, μandn.

Notice that this theorem does not require (θ, μ) to be P-admissible.

Proof. It follows from the definition ofψ_b that (becausef is a limit which is less than the sup) f^−d+rs ≤ ψ

d+rs

b P_θ,μ-a.s.

Then, under Assumption (3.8),

f^−d+rs dP_θ,μ=

{f >0}f_θ,μf^−d+rs dλ_d<+∞.

For alln≥1,

n^s/dX−X^{αθ,μn s}_s = n^s/d

R^dd(z, α^θ,μ_n )^sf(z)dλ_d(z)

= n^s/dθ^s

R^d min

a∈αn|(z−μ)/θ+μ−a|^sf(z)dλ_d(z) We make the change of variablex:= (z−μ)/θ+μ. Then

n^s/d X−X^{αθ,μn s}_s = n^s/dθ^s+d

R^dd(x, α_n)^sf(μ+θ(x−μ))dλ_d(x)

= n^s/dθ^s

R^dd(x, α_n)^sdP_θ,μ(x).

Moreover, it is established in [6] that

lim sup

n

n^s/dd(·, αn)^s≤C(b)f^−d+rs .

Hence, from inequality (3.7) and under Assumption (3.8), we can apply the Lebesgue dominated convergence theorem to the above inequalities to get

lim sup

n

n^s/d

d(x, α_n)^sdP_θ,μ(x)≤

lim sup

n

n^s/dd(x, α_n)^sdP_θ,μ(x)

≤C(b)

f^−d+rs dP_θ,μ(x).

=θ^dC(b)

{f >0}f_θ,μ(x)f^−d+rs (x)dλ_d(x).

For a given distribution, Assumption (3.8) is not easy to verify. But whens=r+d, the lemma and criterions below provide a sufficient condition so that Assumption (3.8) is satisfied. In the rest of this section we extend some of the results obtained in [6].

Let P =f ·λ_d be an absolutely continuous distribution. Let r, s ∈ (0,+∞) and (θ, μ) be a P-admissible couple of parameters. We will need the following hypotheses:

(H1) for allM >0,

sup

z∈B(0,M)

f(μ+θ(z−μ))

f(z) 1{f(z)>0}<+∞. (3.10) (H2) There existsb∈(0,1/2), M∈(0,+∞) such that

B(0,M)^c

sup

t≤2b|x|

λ_d(B(x, t)) P(B(x, t))

_s/(d+r)

dP_θ,μ<+∞. (3.11)

(H3) λ_d(· ∩supp(P))P and supp(P) is a finite union of closed convex sets.

Lemma 3.1. Let P = f ·λ_d and r > 0 such that

|x|^rP(dx) < +∞. Assume (α_n)_n≥1 is a sequence of quantizers such that

d(x, α_n)^rdP → 0. Let (θ, μ) be a P-admissible couple of parameters for which (H1) holds.

(a) If p∈(0,1)then for every b >0, ψ_b^p∈L¹_loc(P_θ,μ).

(b) If p∈(1,+∞] and if furthermore(H3) holds then for every b >0, f^−p∈L¹_loc(P) =⇒ψ^p_b ∈L¹_loc(P_θ,μ).

Proof. It follows from theP-admissibility of (θ, μ) that

P_θ,μ(dz) =θ^df(μ+θ(z−μ))λ_d(dz) =g_θ(z)P(dz), whereg_θ(z) =θd f(μ+θ(z−μ))

f(z) 1{f(z)>0}. Theng_θis locally bounded by (H1).

(a) Ifp∈(0,1), it follows from Lemma 1 in [6] thatψ_b^p∈L¹_loc(P). Hence ψ^p_b ∈L¹_loc(P_θ,μ) sinceg_θis locally bounded.

(b) If p∈ (1,+∞) it follows from Lemma 2 in [6] that if f^−p ∈ L¹_loc(P) then ψ^p_b ∈ L¹_loc(P_θ,μ) since g_θ is

locally bounded.

Corollary 3.1. (Distributions with unbounded support) Let r >0, s ∈ (0,+∞), s =r+d and let X be a random variable with probability measure P =f ·λ_d such that E|X|^r+η <+∞ for some η >0. Let(θ, μ) be P-admissible and suppose that(H1), (H2) hold.

(a) If s∈(0, r+d)then Assumption (3.8) of Theorem 3.2holds true.

(b) If s ∈ (r+d,+∞), and if furthermore, (H3) holds and f^−r+ds ∈ L¹_loc(P) then Assumption (3.8) of Theorem3.2holds true.

Proof. Let x₀ ∈ supp(P). We know from [1] that d(x₀, α_n) → 0. Then following the lines of the proof of Corollary 2 in [6] one has for|x| > N =|x₀|+ sup_n≥1d(x₀, α_n), d(x, α_n)≤2|x| for every n ≥1. Thus for everyb >0, x∈B(0, N)^c,

ψ_b(x)≤ sup

t≤2b|x|

λ_d(B(x, t)) P(B(x, t))·

Now, coming back to the core of our proof, it follows from(H2) that (forbcoming from(H2)),

B(0,M∨N)^c

ψ^s/(d+r)_b dP_θ,μ<+∞.

Since

ψ^s/(d+r)_b dP_θ,μ=

B(0,M∨N)

ψ^s/(d+r)_b dP_θ,μ+

B(0,M∨N)^cψ_b^s/(d+r)dP_θ,μ, it remains to show that the first term in the right hand side of this last equality is finite.

(a) If s ∈(0, r+d) it follows from Lemma 3.1, (a) that the first term in the right hand side of the above equality is finite. As a consequence,ψ^r+ds ∈L¹(P_θ,μ).

(b) Ifs > r+d, the first term in the right hand side of the above equality still finite owing to Lemma3.1, (b).

Consequently, Assumption (3.8) of Theorem3.2holds true provided(H3)holds andf^−r+ds ∈L¹_loc(P).

We next give two useful criterions ensuring that Hypothesis(H2)holds. The first one is useful for distribu- tions with radial tails and the second one for distributions which does not satisfy this last assumption.

Criterion 3.1. Let X ∼P. Suppose thatP =f·λ_d and E|X|^r+η<+∞for someη >0.

(a) Let r, s >0 andf =h(| · |)onB_|·|(0, N)^c with h: (R,+∞)→R+, R∈R+, a decreasing function and

| · | any norm onR^d. Suppose that(θ, μ)is a couple ofP-admissible parameters such that

f(cx)^−d+rs dP_θ,μ(x)<+∞ (3.12)