Distortion mismatch in the quantization of probability measures

January 2008, Vol. 12, p. 127–153 www.esaim-ps.org

DOI: 10.1051/ps:2007044

DISTORTION MISMATCH IN THE QUANTIZATION OF PROBABILITY MEASURES

Siegfried Graf

, Harald Luschgy

and Gilles Pag` es

Abstract. We elucidate the asymptotics of the L^s-quantization error induced by a sequence ofL^r- optimaln-quantizers of a probability distributionP onR^d when s > r. In particular we show that under natural assumptions, the optimal rate is preserved as long ass < r+d(and for everys in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration onR^d and on the Wiener space.

Mathematics Subject Classification. 60G15, 60G35, 41A25.

Received June 2, 2006. Revised November 24, 2006.

1. Introduction

Optimal quantization is devoted to the best approximation in L^r_Rd(P) (r > 0) of a random vector X : (Ω,A,P) → R^d by random vectors taking finitely many values in R^d (endowed with a norm .). When X ∈ L^r(P), this leads for every n ≥ 1 to the following n-level L^r(P)-optimal quantization problem for the random vectorX defined by

en,r(X) := inf

X−q(X)_r, q:R^d→R^d,Borel function, card(q(R^d))≤n

. (1.1)

One shows that the above infimum can be taken over the Borel functions q:R^d→α:=q(R^d), α⊂R^d, cardα≤n which are someprojection following the nearest neighbour rule on their imagei.e.

q(x) =

a∈α

a1_Va(α)(x),

Keywords and phrases. Optimal quantization, Zador Theorem.

1 Universit¨at Passau, Fakult¨at f¨ur Informatik und Mathematik, 94030 Passau, Germany;[email protected]

2 Universit¨at Trier, FB IV-Mathematik, 54286 Trier, Germany;[email protected]

3Laboratoire de Probabilit´es et Mod`eles al´eatoires, UMR 7599, Universit´e Paris 6, case 188, 4, pl. Jussieu, 75252 Paris cedex 5, France;[email protected]

c EDP Sciences, SMAI 2008

Article published by EDP Sciences and available at http://www.esaim-ps.org or http://dx.doi.org/10.1051/ps:2007044

(Va(α))a∈α being a Borel partition ofR^dsatisfying Va(α)⊂

x∈R^d : x−a= min

b∈αx−b

. The setα=q(R^d) is (also) called a Voronoin-quantizer and one denotes

X^α:=q(X).

Then, ifd(x, α) := mina∈αx−adenotes the distance ofxto the setα, one has X−X^αr_r =Ed(X, α)^r=

R^dd(x, α)^rP_X(dx)

which shows thaten,r(X) actually only depends on the distributionP =P_X ofX so that

en,r(X) =en,r(P) = inf

card(α)≤n

d(x, α)^rdP(x)

1 r

. The first two basic results in optimal quantization theory are the following (see [6]):

– The above infimum is in fact a minimum: there exists for every n ≥ 1 (at least) one L^r(P)-optimal n-quantizerα^∗_n. If supp(P) is infinite, then card(α^∗_n) =n.

– Zador’s Theorem: IfX∈L^r+η(P)i.e.

R^dx^r+ηdP(x)<+∞for some η >0, then limn n^1den,r(P) = (Qr(P))^1r∈R+.

A more explicit expression is known for the real constantQr(P) (see (2.3) below). In particular,Qr(P)>0 if and only if P has an absolutely continuous part (with respect to the Lebesgue measure λd onR^d). When P has an absolutely continuous part, a sequence (αn)n≥1 ofn-quantizers isL^r-rate optimalforP if

lim sup

n

n^dr

R^dd(x, αn)^rP_X(dx)<+∞

and isasymptotically L^r-optimalif lim

n

R^dd(x, αn)^rPX(dx) en,r(P)^r = 1.

Our aim in this paper is to deeply investigate the (asymptotic)L^s-quantization error induced by a sequence (αn)n≥1ofL^r-optimaln-quantizers. It follows from the monotony ofs→ .sthat (αn)n≥1remains anL^s-rate optimal sequence as long ass≤r. As soon ass > rno such straightforward answer is available (except for the uniform distribution over the unit interval since the sequence ((^2k₂⁻¹_n )_1≤k≤n)n≥1isL^r-optimal for everyr >0).

Our main motivation for investigating this problem comes from the recent application of optimal quantization to numerical integration (see [11]) and to the computation of conditional expectation (e.g. for the pricing of American options, see [1]). Let us consider for the sake of simplicity the case of the error bound in the quantization based cubature formulae for numerical integration. Let F : R^d → R be a C¹ function with a Lipschitz continuous differentialDF. It follows from a simple Taylor expansion (see [11]) that for any random vectorX with distributionP =PX quantized byX^α(α⊂R^d)

E(F(X))−E(F(X^α))−E(DF(X^α).(X−X^α))≤[DF]_LipE|X−X^α|²

where [DF]_Lip denotes the Lipschitz coefficient ofDF. Ifαis anL²-optimal (orquadratic) quantizer then it is stationary (see [11] or [6]) so that

X^α=E(X|X^α) which makes the first order term vanish since

E(DF(X^α).(X−X^α)) =E(DF(X^α).E(X−X^α|X^α)) = 0.

Finally, if (αn)n≥1is a sequence ofquadraticoptimaln-quantizers

E(F(X))−E(F(X^αn))≤[DF]_Lip(en,2(P))²∼[DF]_LipQ₂(P)n^−2d. (1.2) Now, if the HessianD²F does exist, is ρ-H¨older (ρ∈(0,1]) and computable, the same approach yields

E(F(X))−E(F(X^αn))−E((X−X^αn)^∗D²F(X^αn)(X−X^αn))≤[D²F]ρX−X^αn2+₂₊^ρρ. (1.3) Consequently elucidating the asymptotic behaviour ofX−X^αn2+ρ= (

R^dd(x, αn)^2+ρdP(x))^2+ρ1 is necessary to evaluate to what extend the cubature formula in (1.3) does improve the former one (1.2). Similar problems occur when evaluating the error in the first order quantization based scheme designed for the pricing of multi- asset American options or for non-linear filtering (see [2, 13]). One also meets such mismatch problems in infinite dimensions when dealing with (product) functional quantization on the Wiener space in order to price path-dependent European options (see the example in Sect. 6 and [12]).

The paper is organized as follows: in Section 2, a lower bound for theL^s(P)-quantization rate of convergence of an asymptoticallyL^r-optimal sequence (αn)n≥1ofn-quantizers is established. In particular this result implies that for absolutely continuous distributionsPwith unbounded support, the quantization raten^−d1 inL^scannot be preserved as soon ass > r+d. Ifs≤r+d, then the lower bound can be finite. We conjecture that, when (αn)n≥1 is L^r-rate optimal the lower bound is in fact the sharp rate. In Section 3, several natural criteria on the distributionP are derived. They ensure that (αn)n≥1isL^s-rate optimal for a givens∈(r, r+d) or even for alls∈(r, r+d). Our criteria are applied to many parametrized families of distributions onR^d. We investigate by the same method in Section 4 the critical cases=r+dand the super-critical cases > r+d. In Section 5 we show that for compactly supported distributions on the real line the lower bound obtained in Section 2 does hold as a sharp rate. Finally, in Section 6 we apply our results to the evaluation of errors in numerical integration by quantization based cubature formulae in finite and infinite dimensions.

Notations : • · will denote a norm onR^d andB(x, r) will denote the closed ball centred atxwith radius r >0 (with respect to this norm),d(x, A) will denote the distance betweenx∈R^d and a subsetA⊂R^d.

• λd will denote the Lebesgue measure onR^d (equipped with its Borelσ-fieldB(R^d)).

•Let (an)n≥0and (bn)n≥0be two sequences of positive real numbers. The symbolanbn is foran=O(bn) andbn=O(an) whereas the symbol an ∼bn meansan=bn+o(bn) asn→ ∞.

• xis for the integral part of the real numberx.

• f ∝g means that the functionsf andg are proportional.

2. The lower estimate

In this section we derive an explicit lower bound in the (r, s)-problem for non-purely singular probability distributionsP and asymptoticallyL^r-optimal quantizers. This bound is expected to be best possible.

Letr∈(0,∞). Let P be a probability measure on (R^d,B(R^d)) satisfying

R^dx^rdP(x)<+∞ (2.1)

and supp(P) is not finite. Thenen,r(P)∈(0,∞) for everynanden,r(P)→0 asn→ ∞. A sequence (αn)n≥1

of quantizers is called asymptoticallyL^r-optimal forP if cardαn≤nfor everynand

R^dd(x, αn)^rdP(x)∼en,r(P)^r as n→ ∞. (2.2) LetP^a =f.λd denote the absolutely continuous part ofP with respect toλd. Assume that

R^dx^r+ηdP(x)<

+∞for someη >0. Then by the Zador Theorem (see [6])

nlim→∞n^r/den,r(P)^r=Qr(P) (2.3)

where

Qr(P) :=Jr,d

R^df^d/(d+r)dλd

(d+r)/d

∈[0,∞) (2.4)

and

Jr,d:= inf

n≥1n^r/den,r(U([0,1]^d))^r∈(0,∞),

(U([0,1]^d) denotes the uniform distribution on the hyper-cube [0,1]^d). This theorem was first stated by Zador in [14, 15] and then generalized by Bucklew and Wise (see [3]), the first completely rigourous proof has been proposed by Graf & Luschgy in [6]. Note that the finiteness of

R^df^d/(d+r)dλd is a simple consequence of the H¨older Inequality and the moment assumption

R^dx^r+ηdP(x)<+∞: first note that

x≤1f^d+rd dλd <+∞

sinceλd(x ≤1)<+∞and _d+^d_r≤1. Then setting p= 1 +r/d,q= 1 +d/r,α= (r+η)d/(d+r)

B(0,1)^c

f^d+rd dλd =

x>1

x^−αx^αf^d+rd (x)dλd(x)

≤

x>1

x^−αqdλd(x)

₁/q

x^αpf(x)dλd(x)

1/p

=

x>1

x^−(1+η/r)ddλd(x)

₁/q

x^r+ηdP(x)

1/p

<+∞.

Furthermore, for probabilities P on R^d with P^a = 0, the empirical measures associated to an asymptotically L^r-optimal sequence (αn)n≥1ofn-quantizers satisfy (see [6] Th. 7.5 or [4] for this slight extension)

1 n

a∈α_n

δa

−→w Pr (2.5)

wherePr denotes theL^r-point density measure ofP defined by

Pr:=fr.λd with fr:= f^d/(d+r) f^d/(d+r)dλd

. (2.6)

Note that the limitQr(P) in the Zador Theorem reads Qr(P) =Jr,d

f_r^−r/ddP^a.

The quantity that naturally comes out in the (r, s)-problem,r, s∈(0,∞), is Qr,s(P) := Js,d

fr^−s/ddP^a (2.7)

= Js,d

R^df^d/(d+r)dλd s/d

{f >0}f^1−s/(d+r)dλd∈(0,+∞].

Theorem 1. Letr, s∈(0,∞). AssumeP^a = 0and

R^dx^r+ηdP(x)<+∞ for someη >0. Let(αn)n≥1 be an asymptotically L^r-optimal sequence of n-quantizers forP. Then

lim inf

n→∞ n^s/d

d(x, αn)^sdP(x)≥Qr,s(P). (2.8) Prior to the proof, let us provide a few comments on this lower bound.

Comments. • The main corollary that can be directly derived from Theorem 1 is that

{f >0}f^1−s/(d+r)dλd= +∞=⇒ lim

n→∞n^s/d

d(x, αn)^sdP(x) = +∞

since thenQr,s(P) = +∞.

By contraposition, a necessary condition for an asymptotically L^r-optimal sequence of quantizers (αn) to achieve the optimal raten^−s/d for the L^s-quantization error is thatQr,s(P)<+∞. But, under the moment assumption of Theorem 1 the following equivalence holds true

Qr,s(P)<+∞ ⇐⇒

f^−d+rs dP^a=

{f >0}f^1−d+rs dλd<+∞ (2.9) since

R^df^d/(d+r)dλd<+∞.

In turn, for probability measuresP satisfyingλd(f >0) = +∞a necessary condition for the right hand side of (2.9) to be satisfied is that

s < d+r. (2.10)

Indeed, if s≥d+r, the following chain of inequalities holds true

λd(f >0) =

1_{{f >0}}f⁻¹dP^a ≤ 1_{{f >0}}f⁻¹_d+r^s dP^a

d+r s

=

{f >0} f^1−s/(d+r)dλd

^d+r

s

where we used thatp→ .L^p(P^a)is non-decreasing sinceP^a(R^d)≤1.

On the other hand, still whens < d+r, the following criterion holds for the finiteness ofQr,s(P):

∃ϑ >0,

R^dx^{ds/(d+r−s)+ϑ}dP(x)<+∞ =⇒Qr,s(P)<+∞. (2.11)

Set ρ= 1− _d+^s_r∈(0,1) and u= _d+^ds_r−s+ϑ. Then (2.11) follows from the regular H¨older inequality applied with ˜p=_ρ¹ =_d+^d+_r−^r_s and ˜q=₁₋¹_ρ =_d+^s_r−s,

B(0,1)^c

f^ρdλd ≤

B(0,1)^c

(f(x)^ρx^uρ)^p˜dλd(x)

₁/p˜

B(0,1)^c

x^−uρq˜dλd(x) ₁/q˜

=

B(0,1)^c

f(x)x^udλd(x) ρ

B(0,1)^c

x^{−uρ/(1−ρ)}dλd(x) ₁₋ρ

<+∞

using the moment assumption in (2.11) anduρ/(1−ρ) =d+ϑ₁₋^ρ_ρ > d.

• It is generally not true in the general setting of Theorem 1 that lim

n→∞n^s/d

d(x, αn)^sdP(x) =Qr,s(P) (see Counter-Example 2 in Sect. 3). However, one may reasonably conjecture that this limiting result holds true for sequences (αn) of exactly L^r-optimal n-quantizers. Our result in one dimension for compactly supported distributions (see Sect. 5) supports this conjecture.

• In any case, note that (2.8) improves the obvious lower bound lim inf

n→∞ n^s/d

d(x, αn)^sdP(x)≥lim inf

n→∞ n^s/den,s(P)^s≥Qs(P).

(The right inequality needs no moment assumption onPas can be checked from the proof of the Zador Theorem, see [6].) In fact, one even has that, for everyr, s∈(0,+∞),

Qr,s(P)≥Qs(P).

Furthermore, this inequality is strict whenr=s(except if f isλd-a.e.constant on{f >0}). Let us provide a short proof of this fact. Setp= (d+s)/s >1,q= (d+s)/d >anda=ds/(d+r)(d+s),b= (d+r−s)d/(d+ r)(d+s). Then the H¨older inequality yields

(Qs(P))^d+sd =

f^d/(d+s)dλd=

{f >0}f^af^bdλd

≤

f^apdλd

1/p

{f >0}f^bqdλd

₁/q

(“<” iff^apandf^bq are not proportional)

=

f^d/(d+r)dλd

s/(d+s)

{f >0}f^1−d+rs dλd

d/(d+s)

= (Qr,s(P))^d+sd .

Proof of Theorem 1. First keep in mind that, the r+η-moment assumption on P implies the finiteness of f^d+rd dλd. The existence of at least one asymptoticallyL^r-optimal sequence (αn)n≥1follows from the existence of anr-moment forP. For every integerm≥1, set

fm:=

m2^m−1 k=0

k 2^m1_Em

k with E^m_k =

k

2^m ≤f < k+ 1 2^m

∩B(0, m).

The sequence (fm)m≥1 is non-decreasing and converges tof1_{{0≤f <+∞}}=f λd-a.e.

LetIm:={k∈ {0, . . . , m2^m−1} : λd(E^m_k )>0}.

For everyk∈Im, there exists a closed set A^m_k ⊂E_k^m satisfying λd(E_k^m\A^m_k )≤ 1

m³2^m·

Let εm∈(0,1] be a positive real number such that the closed setsA^m_k :={x∈R^d : d(x, A^m_k )≤εm}, k∈Im,

satisfy

A^m

k

f^d+rd dλ_d≤(1 + 1/m)

A^m

k

f^d+rd dλd<+∞.

Set

fm:=

m2^m−1 k=0

k 2^m1

Amk. It is clear that

{fm=fm} ⊂

0≤k≤m2^m−1

(E^mk \A^mk ).

Hence

λd({fm=fm})≤

m2^m−1 k=0

1

m³2^m = 1 m²

so that

m≥1

1_{f

m=f_m}<+∞ λd-a.e.

i.e., for λd-a.e. x, fm(x) = fm(x) for large enoughm so that fm converges to f λd-a.e.. Finally, as a result fm≤fm≤f andfmconverges tof λd-a.e. Then, for everyn≥1,

n^sd

R^d(d(x, αn))^sdP(x) ≥ n^sd

R^d(d(x, αn))^sfm(x)dλd(x)

= n^sd

m2^m−1 k=0

k 2^m

A^m

k

(d(x, αn))^sdλd(x). (2.12)

Since all the sets A^m_k, k = 0, . . . , m2^m−1 are bounded (as subsets of B(0, m+ 1)), there exists for every m ≥1 and every k∈ {0, . . . , m2^m−1} a finite “firewall”β_k^m ⊂R^d (see [6] or Lem. 4.3 in [4] and note that A^m_k ⊂(A^m_k)ε_m/2:={x∈R^d : d(x,(A^m_k)^c)> εm/2}) such that

∀n≥1, ∀x∈A^mk, d(x, αn∪βk^m) =d(x,(αn∪βk^m)∩A^mk ).

Setβ^m=∪0≤k≤m2^m−1β^m_k . Then, for everyk∈ {0, . . . , m2^m−1}, for everyx∈A^m_k ,

d(x, αn)≥d(x, αn∪β_k^m) =d(x,(αn∪β_k^m)∩A^m_k )≥d(x,(αn∪β^m)∩A^m_k ).

Set temporarilyn^m_k := card((αn∪β^m)∩A^m_k). First note that it is clear that n^m_k

n ∼card(αn∩A^m_k)

n asn→ ∞.

It follows from the asymptoticL^r-optimality of the sequence (αn) and the empirical measure theorem (see (2.5)) that

lim sup

n

card(αn∩A^m_k)

n ≤

A^m_k f^d+rd dλd

f^d+rd dλd

(2.13)

so that

lim inf

n

n n^m_k ≥

f^d+rd dλd

A^m_k f^d+rd dλd

≥ m m+ 1

f^d+rd dλd

A^m

k f^d+rd dλd

. On the other hand, for everyk∈Im,

A^m

k

(d(x, αn))^sdλd(x)≥

A^m

k

d(x,(αn∪β^m)∩A^m_k)^sdλd(x)≥λd(A^m_k )e^s_nm

k,s(U(A^m_k ))

whereU(A^m_k ) denotes the uniform distribution overA^m_k (note that the inequality is trivial whenλd(A^m_k ) = 0).

Then one may apply Zador’s theorem which yields, combined with (2.13), lim inf

n n^ds

A^m_k

(d(x, αn))^sdλd(x) ≥ λd(A^m_k )×lim inf

n

n n^m_k

s d×lim

n^m

k

((n^m_k )^sde_nm

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

≥ λd(A^m_k )×

⎛

⎝ m m+ 1

f^d+rd dλd

A^m_k f^d+rd dλd

⎞

⎠

s d

Js,d×(λd(A^m_k))^ds

≥ Js,d

m m+ 1

sd

f^d+rd dλd

sd

⎛

⎝ λd(A^m_k )

A^m

k f^d+rd dλd

⎞

⎠

ds

λd(A^mk )

≥ Js,d

m m+ 1

s

d

f^d+rd dλd

s d

k+ 1 2^m

−_d+r^s

λd(A^m_k ) with the convention ⁰₀ = 0.

Consequently, using (2.12) and the super-additivity of lim inf yield that, for everym≥1,

lim inf

n n^sd

R^d(d(x, αn))^sdP(x) ≥ Js,d

m m+ 1

s

d

R^df^d+rd dλd

s dm2^m−1

k=0

k 2^m

k+ 1 2^m

−_d+r^s

λd(A^m_k)

= Js,d

m m+ 1

s

d

R^df^d+rd dλd

s d

{f >0}

fm(fm+ 2^−m)^−d+rs dλd.

Now, by Fatou’s Lemma, one concludes by lettingmgo to infinity that lim inf

n n^sd

R^d(d(x, αn))^sdP(x)≥Js,d

R^df^d+rd dλd

sd

{f >0}f^1−d+rs dλd.

3. The upper estimate

Let r, s ∈ (0,∞). In this section we investigate whether the upper bound

d(x, αn)^sdP(x) = O(n^−s/d) for L^r-optimal n-quantizersαn holds true. (This is of course less precise than the lower bound given in the previous section.) The reason for the restriction to (exactly) optimaln-quantizers (whens > r) instead of only asymptotically optimaln-quantizers will become clear soon. Seee.g.the subsequent Example 2. First note that the L^p(P)-norms being non-decreasing as a function ofp, the above upper bound trivially holds for s∈(0, r]

since

n^s/d

d(x, αn)^sdP(x)≤

n^r/d

d(x, αn)^rdP(x)

sr

.

The same argument shows that when this rate holds for somes >0, then it holds for everys ∈(0, s].

For a sequence (αn)n≥1 of finite codebooks in R^d and b ∈ (0,∞) we introduce the maximal function ψ_b:R^d →R+∪ {∞}by

ψ_b(x) := sup

n≥1

λd(B(x, bd(x, αn)))

P(B(x, bd(x, αn))) (3.1)

(with the interpretation ⁰₀ := 0). Note thatψ_b is Borel-measurable and depends on the underlying norm onR^d. The theorem below provides a criterion based on these maximal functions that ensures theL^s-rate optimality of L^r-optimal n-quantizers. In Corollaries 1, 3, 4 we derive more applicable criteria which only involve the distributionP.

Theorem 2. Let r, s∈(0,∞). Assume P^a = 0and

x^r+ηdP(x)<+∞for some η >0. For every n≥1, letαn be anL^r-optimaln-quantizer forP. Assume that the maximal function associated with the sequence(αn) satisfies

ψ_b^s/(d+r)∈L¹(P) (3.2)

for someb∈(0,1/2). Then

sup

n

n^s/d

d(x, αn)^sdP(x)<+∞. (3.3)

Proof. Let y ∈ R^d and set δ = δn = d(y, αn). For every x ∈ B(y, δ/2) and a∈ αn, we have x−a ≥ y−a − x−y ≥δ/2 and hence

d(x, αn)≥δ/2≥ x−y, x∈B(y, δ/2).

Letβ=βn=αn∪ {y}. Then

d(x, β) =x−y, x∈B(y, δ/2).

Consequently, for everyb∈(0,1/2),

en,r(P)^r−en+1,r(P)^r ≥

d(x, αn)^rdP(x)−

d(x, β)^rdP(x)

≥

B(y,δb)

(d(x, αn)^r−d(x, β)^r)dP(x)

=

B(y,δb)

(d(x, αn)^r− x−y^r)dP(x)

≥

B(y,δb)

((δ/2)^r−(bδ)^r)dP(x)

= ((1/2)^r−b^r)δ^rP(B(y, bδ)).

One derives that

d(y, αn)^r≤ C(b)

P(B(y, bd(y, αn))(en,r(P)^r−en+1,r(P)^r) (3.4) for everyy∈R^d, b∈(0,1/2), n≥1, where C(r, b) = ((1/2)^r−b^r)⁻¹. Note that en,r(P)^r−en+1,r(P)^r>0 for everyn∈N(see [6]).

Now we estimate the increments en,r(P)^r−en+1,r(P)^r. (This extends a corresponding estimate in [7] to distributions with possibly unbounded support.)

Seten,r =en,r(P) for convenience. Let{Va :a∈αn+1} with Va =Va(αn+1) be a Voronoi partition ofR^d with respect to αn+1. ThenP(Va)>0 for alla∈αn+1 and cardαn+1=n+ 1 (see [6]),

card

a∈αn+1:

V_a

x−a^rdP(x)> 4e^rn+1,r

n+ 1

≤ n+ 1 4

and

card

a∈αn+1 :P(Va)> 4 n+ 1

≤ n+ 1 4 · This implies that

βn+1:=

a∈αn+1:

V_a

x−a^rdP(x)≤4e^r_n+1,r

n+ 1 , P(Va)≤ 4 n+ 1

satisfies cardβn+1 ≥ (n+ 1)/2. Choose a closed hyper-cube K = [−m, m]^d such that Pr(K) > 3/4. The empirical measure theorem (see (2.5) above or [4, 6] for details) implies

klim→∞

card(αk∩K)

k =Pr(K)

sincePr(∂K) =λd(∂K) = 0. We deduce card(αn+1∩K)≥3(n+ 1)/4 and hence card(βn+1∩K)≥(n+ 1)/4 for large enoughn. Since one can find a tessellation ofKinto [(n+ 1)/8]∨1 cubes of diameter less thanC₁n^−1/d, there exista₁, a₂∈βn+1,a₁=a₂ such that

a₁−a₂ ≤C₁(r)n^−1/d for everyn≥3. Letγ=αn+1\ {a1}. Using

d(x, γ)≤ x−a₂ ≤ x−a₁+a₁−a₂, one obtains

en,r(P)^r−en+1,r(P)^r ≤

d(x, γ)^rdP(x)−

d(x, αn+1)^rdP(x)

=

a∈γ

V_a

x−a^rdP(x) +

V_a1

d(x, γ)^rdP(x)−

a∈α_n+1

V_a

x−a^rdP(x)

=

V_a1

(d(x, γ)^r− x−a₁^r)dP(x)

≤ (2^r−1)

V_a1

x−a₁^rdP(x) + 2^ra1−a₂^rP(Va₁)

≤ 4(2^r−1)e^r_n+1,r

n+ 1 +4·2^rC₁(r)^rn^−r/d

n+ 1 ·

Consequently, using (2.3), for everyn∈N,

en,r(P)^r−en+1,r(P)^r≤C₂(r)n^−(d+r)/d (3.5) whereC₂(r) denotes a finite constant independent ofn. Combining (3.4) and (3.5), we get

n^s/dd(x, αn)^s ≤ C₃(r, b)^s

λd(B(x, bd(x, αn)) P(B(x, bd(x, αn))

s/(d+r)

(3.6)

≤ C₃(r, b)^sψ_b(x)^s/(d+r) (3.7)

for every x∈ R^d, n∈N, b∈(0,1/2) and some finite constantC₃(r, b). The proof is completed by integrating

both sides with respect toP.

Application to pointwise convergence rate. In the situation of Theorem 2, assumingP =P^a, but without assuming (3.2) (so thatsis not involved in that statement), one can deduce from (3.6) that

lim sup

n→∞ n^1/dd(x, αn)≤C₃(r, b)f^−1/(d+r)<+∞ P(dx)-a.s. (3.8) sinced(x, αn)→0 P(dx)-a.s.(see [4]) implies in turn

P(B(x, bd(x, αn))

λd(B(x, bd(x, αn)) →f(x) P(dx)-a.s. as n→ ∞

by the differentiation of measures. This improves considerably for absolutely continuous distributions and (exactly)L^r-optimal quantizers an a.s. result in [4].

Next we observe that in cases∈(0, d+r) a local version of condition (3.2) is always satisfied.

Lemma 1. Assume

x^rdP(x) < +∞ for some r ∈ (0,∞). Let (αn) be a sequence of finite codebooks in R^d satisfying

d(x, αn)^rdP(x)→0. Then the associated maximal functions ψ_b are locally in L^p(P) for every p∈(0,1)i.e.

∀M, b∈(0,∞),

B(0,M)

ψ^p

bdP <+∞.

Proof. LetM, b∈(0,∞) and set A= supp(P). Then maxx∈B(0,M)∩Ad(x, αn)→0 (see [4]) and hence C(M) := sup

n≥1 max

x∈B(0,M)∩Ad(x, αn)<+∞.

One derives that

B(x, bd(x, αn))⊂B(0, bC(M) +M)

for every x∈B(0, M)∩A, n∈N. Define the Hardy-Littlewood maximal functionϕ: R^d →R+∪ {∞}with respect to the finite measuresλd(· ∩B(0, bC(M) +M)) andP by

ϕ(x) =ϕ_b,M(x) := sup

ρ>0

λd(B(x, ρ)∩B(0, bC(M) +M))

P(B(x, ρ)) ·

Then

ψ_b(x)≤ϕ(x)

for everyx∈B(0, M)∩A. From the Besicovitch covering theorem follows the maximal inequality P(ϕ > ρ)≤ C₁λd(B(0, bC(M) +M))

ρ

for everyρ >0 where the finite constant C₁ only depends on dand the underlying norm. (See [10], Th. 2.19.

The result in [10] is stated for Euclidean norms but it obviously extends to arbitrary norms since any two norm onR^d are equivalent.) Consequently,

B(0,M)∩A

ψ^p_bdP ≤

ϕ^pdP = _∞

0

P(ϕ^p> t)dt≤1 + _∞

1

P(ϕ^p> t)dt

≤ 1 +C_{2 ∞}

1

t^−1/pdt <+∞

whereC₂=C₁λd(B(0, bC(M) +M).