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Approximation of distributions
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 298–303 www.elsevier.com/locate/stapro
Approximation of distributions Nacereddine Belilia,, Henri Heinichb a
LMI, place E. Blondel, 76131 Mont-Saint-Aignan Cedex, France INSA de Rouen, De´partement de Ge´nie Mathe´matique, place E. Blondel, 76131 Mont-Saint-Aignan Cedex, France
b
Received 4 April 2003; received in revised form 23 June 2005 Available online 7 September 2005
Abstract Let P be a Borel-probability on Rd and let C be a family of Borel-probabilities with finite second order moments. Under some conditions on C, we prove that there exists a best approximation to P in C for the Wasserstein distance. r 2005 Elsevier B.V. All rights reserved. Keywords: Best approximation; Wasserstein distance
1. Problem and notations Let ðM; dÞ be a metric space and C M: For xeC; the existence of a point y0 2 C such that dðx; y0 Þ ¼ dðx; CÞ9 inffdðx; yÞ; y 2 Cg is a classical problem of optimization. A sequence ðyn Þ C is an approximate sequence (for x and C) if dðx; yn Þ ! dðx; CÞ: Here we study this approximation problem for C a subset of the Borel-probabilities on Rd with the Wasserstein distance. We assume that all the random variables (r.v.) in this paper are defined on a rich enough probability space and, unless explicitly stated, all of them will be Rd valued, dX1. We denote by E½X and LðX Þ the expectation and probability-distribution of a r.v. X : For two Borel-probabilities P and Q on Rd ; ðPjQÞ is the set of the probabilities on Rd Rd whose marginal distributions are (respectively) P and Q: The square of Wasserstein’s distance is lðP; QÞ ¼ inffE½kX Y k2 ; LðX R; Y Þ 2 ðPjQÞg: This expression is interesting only if P and Q have finite second order moments i.e., s2 ðPÞ9 kxk2 dPðxÞ and s2 ðQÞo1. We say that ðX ; Y Þ is an optimal coupling for ðP; QÞ if lðP; QÞ ¼ E½kX Y k2 ¼ s2 ðPÞ þ s2 ðQÞ 2E½hX ; Y i; LðX ; Y Þ 2 ðPjQÞ: The existence of an optimal coupling is traditional, see for example Rachev and Ru¨schendorf (1998). For a class C of probabilities on Rd and a probability P on Rd , we consider the problem ðPÞ : the existence of Q0 2 C such that lðP; Q0 Þ ¼ lðP; CÞ9 infflðP; QÞ; Q 2 Cg. For example, when C ¼ Cu is the family of uniform probabilities defined on compact convex sets of Rd with a non-empty interior, Cuesta-Albertos et al. (2003) showed the existence of such a probability when P is Corresponding author.
E-mail addresses: [email protected] (N. Belili), [email protected] (H. Heinich). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.031
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absolutely continuous with respect to Lebesgue measure on Rd : The functional lðP; CÞ can be employed as an alternative parameter to measure the flatness (see Cuesta-Albertos et al., 2002). We are going to weaken the condition on P and to generalize the class Cu : Indeed, we only suppose that the dimension of the vector-space spanned by the support of P is d. For P we can take the empirical law of an nsample, if n is large enough, and we obtain a natural estimator by choosing Q 2 C such that lðP; QÞ ¼ lðP; CÞ. For this purpose, let us introduce some additional notations. L If a sequence of Borel-probabilities fQn g converges in distribution to Q; we write Qn ! Q: We denote by S P the support of P; a Borel-probability on Rd : If C is a subset of Borel-probabilities, C is its closure for the convergence in distribution. If f is a Borel-function on Rd ; f ðQÞ is the probability defined by f ðQÞðAÞ ¼ Qðff 2 AgÞ, particularly for a 2 R; aðQÞðAÞ ¼ Qðfx j ax 2 AgÞ. ¯ is the closure of A; A its interior, vsðAÞ the vector space generated by A and dimðvsðAÞÞ the For A Rd ; A dimension of this vector space. For a sequence fAn g Rd ; we write An ! A if 1An ðxÞ ! 1A ðxÞ; 8 x 2 Rd and An ! A, if there exists a subsequence such that Ani ! A. We denote by AðÞ the set fx j 9 y 2 A; kx ykpg. With abuse of notation, we will denote the Lebesgue measure on every Euclidean-space with the same symbol l. Finally a probability Q is symmetric (with respect to a base), if given X ¼ ðX i Þ such that LðX Þ ¼ Q; then, for all ¼ ði Þ 2 ðf1; þ1gÞd ; we have LðX 9 ði X i ÞÞ ¼ Q. 2. The main theorem Let P denote the set of orthogonal projections of Rd and for p 2 P; p? is the orthogonal of p. We say that a family C of Borel-probabilities on Rd satisfies the ðÞ condition if: (0) (1) (2) (3)
C is formed by probabilities having a finite second order moment. C is closed by translation: Q 2 C if and only if Qð aÞ 2 C; 8a 2 Rd : For each p 2 P and a 2 Rnf0g; ðap þ p? ÞðQÞ 2 C if Q 2 C. For each m 2 C; there is Q in C such that m ¼ p? ðQÞ and pðQÞ is symmetric, where p? is the orthogonal projection onto vsðSm Þ: In particular, if dimðvsðS m ÞÞ ¼ d; then m 2 C: (4) For all p 2 P and Q 2 C; dimðvsðSpðQÞ ÞÞ ¼ dim(pÞ 9 dimðpðRd ÞÞ. The above conditions allow to reduce the initial problem: ðPÞ is equivalent to the same problem by supposing the centered probabilities: R The problem R x dPðxÞ ¼ x dQðxÞ ¼ 0; 8Q 2 C. Next, we adopt this framework by still writing C for the centered family C. The main result is the following: Theorem. Let P be a probability on Rd such that dimðvsðS P ÞÞ ¼ d and C a class satisfying ðÞ condition. Then, there exists Q 2 C such that lðP; QÞ ¼ lðP; CÞ. Moreover, for all approximate sequence lðQ; Qn Þ ! 0. Proof. Obviously there exists an approximate sequence fQn g C; such that Qn converges in distribution to m and lðP; mÞplðP; CÞ. If dimðvsðS m ÞÞ ¼ d; property (3) implies that m 2 C and the theorem is proved. Let us suppose dimðvsðS m ÞÞod: Corollary 1 below ensures that dimðvsðSm ÞÞX1: Assertion (3) gives the existence of a probability Q 2 C such that m ¼ p? ðQÞ and pðQÞ is symmetric and different from d0 ; by assertion (4). Corollary 2 gives pðPÞ ¼ d0 and we obtain a contradiction with the assumption on P. The proof of the last statement is easy. & Let us note that the last assertion can be used for numerical simulations. Lemma 1. Let P and Q be two centered probabilities having finite second order moments such that Q is symmetric and lðP; QÞ ¼ s2 ðPÞ þ s2 ðQÞ. Then the supports of P and of Q are orthogonal.
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Proof. We use the following statement: Let m be a probability on M nþp whose marginals are m1 on M n and m2 on M p and let X a r.v. defined on ðO1 ; A1 ; P1 Þ with distribution m1 . There exists a probability space ðO ¼ O1 O2 ; A ¼ A1 A2 ; P ¼ P1 P2 Þ; where ðO2 ; A2 ; P2 Þ is another probability space, and a couple of r.v’s ðX ; Y Þ such that X ðo1 ; o2 Þ ¼ X ðo1 Þ; LðX ; Y Þ ¼ m. With the above notations, an immediate consequence is: For any r.v. X 0 ; such that LðX 0 Þ ¼ P; there exists Y 0 with distribution Q such that ðX 0 ; Y 0 Þ is an optimal coupling and therefore, E½hX 0 ; Y 0 i ¼ 0. For LðX ; Y Þ 2 ðPjQÞ we have E½hX ; Y ip0 and by changing Y into Y , we obtain E½hX ; Y i ¼ 0: Let X and X 0 be two independent r.v’s with the same distribution P, the probability P ¼ LðX X 0 Þ is symmetric. As before, for any r.v. X 0 with LðX 0 Þ ¼ P , there exists Y 0 , with LðY 0 Þ ¼ Q, such that lðP ; QÞ ¼ E½kX 0 Y 0 k2 . Putting X 0 ¼ X X 0 , one deduces that E½hX 0 ; Y i ¼ 0 8 Y with law Q, therefore lðP ; QÞ ¼ s2 ðP Þ þ s2 ðQÞ. Thus, we can suppose without loss of generality that P is symmetric. For a r.v. Z ¼ ðZ i Þ if signðZ i Þ ¼ 1fZi 40g 1fZi o0g , we put signðZÞ ¼ ðsignðZ i ÞÞ and jZj ¼ ðjZ i jÞ. Let X and Y be two independent r.v’s with laws P and Q respectively. Since LðsignðY ÞjX jÞ ¼ P and LðsignðY ÞjY jÞ ¼ Q; we have E½hjX j; jY ji ¼ 0, thus hX ; Y i ¼ 0 a.s. & Corollary 1. Let P be a probability on Rd such that the vector space generated by its support has dimension d. Then lðP; CÞos2 ðPÞ if class C verifies ðÞ condition. Proof. By negation, we suppose that lðP; CÞ ¼ s2 ðPÞ ¼ lðP; d0 Þ: For Q 2 C and a 2 R, aa0; we have aQ 2 C: It is clear that the minimum of the function a ! lðP; aQÞ is s2 ðPÞ ððE½hX 0 ; Y 0 iÞ2 Þ=s2 ðQÞ, where ðX 0 ; Y 0 Þ is optimal for ðP; QÞ: By taking Q symmetric, we deduce, as in Lemma 1, that hX ; Y i ¼ 0 a.s. for any couple ðX ; Y Þ such that LðX ; Y Þ 2 ðPjQÞ: Therefore, S P and S Q are orthogonal, which is a contradiction. & Corollary 2. Let P and Q be two probabilities on Rd and p 2 P with 1pdimðpÞod. Let us assume that for all real aa0, and u ¼ p? þ ap, we have lðP; p? ðQÞÞplðP; uðQÞÞ. Then, there exists a couple ðX ; Y Þ such that ðp? ðX Þ; p? ðY ÞÞ is an optimal coupling for ðp? ðPÞ; p? ðQÞÞ. Moreover, for this couple we have E½hpðX Þ; pðY Þi ¼ 0. Proof. The existence of a couple ðX ; Y Þ satisfying the conditions of this corollary results from the statement in Lemma 1. Noting Z1 ¼ pðZÞ; Z2 ¼ p? ðZÞ and LðY 2 Þ ¼ Q2 ; we have lðP; Q2 ÞplðP; uQÞps2 ðPÞ þ a2 s2 ðQ1 Þþ s2 ðQ2 Þ 2E½hX 2 ; Y 2 i 2aE½hX 1 ; Y 1 i. Taking the minimum over a, we obtain lðP; Q2 ÞplðP; Q2 Þ ðE½hX 1 ; Y 1 iÞ2 =s2 ðQ1 Þ and the proof is complete. & 3. Examples In this section we are going to show that the result can be applied to uniform probabilities on compact convex sets with non-empty interior and other examples. Notice that the only thing to be proved is that condition (3) holds, because the remaining ones are evident. 3.1. Uniform probabilities on compact convex sets with non-empty interior Let us recall that Cu denotes the set of uniform probabilities on compact convex sets with non-empty interior. We have seen that we can suppose that these probabilities are centered. It is easy to see that if S Q is the support of Q 2 Cu ; then 0 2 S Q : If fQn g 2 Cu converges in distribution to m; the Lemmas in the appendix ensure that the supports S n of Qn are contained in a compact K and S n ! S: L
Proposition 1. Let fQn g Cu be a sequence such that Qn ! m and S n ! S where Sn is the support of Qn . Then, if Sm is the support of m; we have S¯ ¼ S m . ¯ If B is a compact ball such that B \ S¯ ¼ ;, Lemma 3 ensures that Proof. Let us show that Sm is contained in S.
B \ Sn ¼ ; as soon as n is large enough. Consequently Qn ðBÞ ¼ 0 and thus mðBÞ ¼ 0. The complementary set c ¯ of S¯ is a countable union of open balls with compact closure, therefore mðS¯ Þ ¼ 0; and S m S.
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Conversely, let us suppose that dimðvsðSm ÞÞ ¼ pX1; and denote by p the orthogonal projection on this space. Let aob be two points of Rp ; Lemma 5 ensures that ðlðS n \ fapppbgÞÞ=lðS n Þ ¼ Qn ðfapppbgÞXk: Therefore, mðfapppbgÞXk i.e. S¯ Sm : & L
Let fQn g Cu ; such that Qn ! m; with 1p dimðvsðSÞÞod and S the support of m. The orthogonal projection on vsðSÞ is noted p? : We saw in Proposition 1, that for a subsequence (still noted S n Þ, the supports S n of Qn converge to the support S: Let us choose a point M n 2 S n such that an 9kpðM n Þk ¼ sup kpðS n Þk. The mapping un ¼ ð1=an Þp þ p? , transforms Qn into Qn 9un ðQn Þ 2 Cu . By extracting a subsequence if necessary, we can suppose that fun ðM n Þg converges. Let us show the convergence in distribution of the sequence fQn g. For each n; Sn is contained in the compact convex set K n 9fx j p? ðxÞ 2 p? ðSn Þ; kpðxÞkpan g: The mapping un transforms K n into K n , which is contained in fx j p? ðxÞ 2 p? ðSn Þ, kpðxÞkp1g. For nXNðÞ, we have K n fx j kpðxÞkp1; p? ðxÞ 2 p? ðSÞðÞg. Consequently, the sequence fQn g is relatively compact for the L
convergence in distribution. We can suppose that Qn ! Q; it is obvious that lðSQ Þ40 and, therefore, Q 2 Cu . L
Moreover, we have: p? ðQn Þ ¼ p? ðQn Þ ! p? ðQÞ ¼ m: Finally it is enough to symmetrize pðQÞ; which belongs to Cu ; to prove that Cu verifies assertion (3). The theorem is valid for the class Cu : Remark. A referee for this paper has given a shorter proof. He notices that m 2 Cu and fQn g Cu converges in distribution to m: It is obvious that the sequence of the supports of fQn g satisfies Lemma 4 and therefore m satisfies Theorem 3.1 in Cuesta et al. (2003). It is easy to show that condition (3) holds for Cu : Nevertheless, we have kept our proof because the method used is applicable to the following examples. 3.2. Other examples We will extend the property ensuring that a probability, m 2 Cu whose support generates a vector space with dimension pX1, is the projection of a probability include in Cu . If Qn converges to m; there exists a compact set K such that for all n; Qn ðKÞX1 : Let Sn be the support of ? Qn and S K n ¼ S n \ K: We can suppose that 1p dimðvsðSÞÞ; where S is the support of m and that S n ! S: Let p be the orthogonal projection on this space. The previous method ensures that fun ðQn Þ ¼ Qn g is relatively compact and the vector space generated by the support of the limit of any subsequence is Rd : In a similar way, it is enough to prove that condition (3) holds for the following examples:
The set consisting of the probabilities with uniform law on the boundary of a compact convex set with nonempty interior. The set consisting of the probabilities with uniform distribution on C 1 nC 2 ; where C i are compact convex sets with non-empty interior. The justification of the other conditions is easy. Now, let us consider a fourth example. Call Co the class of probabilities Q having finite second order moments such that QðyAþ ð1 yÞBÞX inf½QðAÞ; QðBÞ and whose supports generate Rd . If Q 2 Co ; then Q has a density g; verifying that fgXag is convex for all a 2 Rþ . As previously we can restrict ourselves to centered probabilities.
Proposition 2. Let fQn g Co ; be a sequence converging in law to mad0 . Let S m be the support of m and S an 9fgn Xag, where gn is the density of Qn and a a rational 40. Then, we have S an ! Sa and S m ¼ S, where S is the closure of [S a . Proof. The existence of the subsequence results from the preceding considerations. The proof of S m S is similar toR the one for the uniform laws. If K is a compact set contained in the complement of S; then Qn ðKÞ ¼ K gn ðxÞ dxpalðKÞ; 8 a40: Thus, for an open ball O contained in K, we have mðOÞ ¼ 0; and since the complement of S is a countable union of such balls, mðSÞ ¼ 1. This shows our inclusion.
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Conversely, let us suppose mad0 : Let p be the orthogonal projection on the vector space generated by the
support of m: IfR K ¼ K p K p? is the product of two compact sets and is contained in S a , by Lemma 4, we have, Qn ðKÞ ¼ K gn ðxÞ dxXalðKÞXkðaÞlðK p Þ where kðaÞ is a constant 40: We deduce that mðK p Þ ¼ mðKÞ40:
Moreover, by Lemma 3, if a is small enough and K a compact include in S ; we have K S a : Thus for x 2 S ;
there exists a basis of compact neighborhoods having the previous form and contained in Sa , therefore x 2 Sm : So, we have S S m ; which completes the proof. & The method used for Cu can be adapted to Co and we easily show that Co verifies all the conditions of the class C: The previous approximation theorem is valid for Co : 4. Appendix. Some properties of convex sets In this part we study the properties and the convergence of convex sets. Other properties can be found in Dharmadhikari and Joag-Dev (1989), in particular for the class C0 : Lemma 2. Let fC n g Rd be a sequence of convex sets containing the point 0. There exists a convex set C such that C n ! C: Moreover, if C n ! C; then, for any a40, the following assertions hold (i) CðaÞ lim C n ðaÞ CðaÞ ¼ CðaÞ, (ii) C ¼ fx j 9 xn 2 C pn ; xn ! xg. Proof. Let D be a countable dense set of lines, i.e. {[ D; D 2 D} is dense in Rd . The intervals I D n 9C n \ D converge, for a subsequence to I D in R: By the diagonal process, there exists a subsequence fni g such that I D ni converges in R: It is easy to see that if D is a line passing through the origin, then I Dni ! I D : Therefore C ¼ [I D
is convex and C ni ! C. Now, let us prove assertion (i) for a sequence fC n g such that C n ! C: There exists a subsequence fni g such that the set C a 9 lim C ni ðaÞ exists, from the first part of the Lemma. TThus, there is a subsequence fnij g such that f1C ni ðaÞ g and f1C ni g converge for all rational a. j
j
o simplify the writing and without loss of generality, we can suppose that for any rational a; the sets fC n ðaÞg and fC n g respectively converge to C a and C. Let x be a point of CðaÞ; there exists y 2 C such that ky xkpa; and for n large enough, y 2 C n : We obtain x 2 C n ðaÞ and this implies x 2 C a ; i.e. CðaÞ C a . Second inclusion. Let x be a point of C a ; we have x 2 C n ðaÞ; for all n large enough, and there exists yn 2 C n with kyn xkpa: The sequence fyn g is bounded and, for a subsequence, yni ! y. Then for any rational Z; kyni ynj kpZ if i and j are large enough. Consequently y 2 C nj ðZÞ; therefore y 2 lim C nj ðZÞ ¼ C Z , we deduce that for Z small enough, x 2 C Z ðaÞ: Now, when Z ! 0; it is easy to see that fC Z g decreases to C and x 2 CðaÞ ¼ CðaÞ: This shows part (i). To establish part (ii), we notice that for x 2 C; there exists fxn g C; xn ! x: Since xn 2 C p for pXPðxn Þ ¼ pn ; the direct implication is established. Conversely, if xn 2 C pn and xn ! x; then for all rational Z40 and n large enough, we have x 2 C pn ðZÞ: Thus, x 2 lim C pn ðZÞ ¼ C Z and we obtain x 2 C: & Lemma 3. Let fC n g Rd be a sequence of convex sets containing the point 0 such that C n ! C: Then for all
compact K such that K \ C ¼ ; (respectively, K C the interior of C in vsðCÞÞ, there exists N such that, if
nXN; then K \ C n ¼ ; (resp. K C n Þ: Proof. By negation, we suppose that there exists a subsequence fni g and fxi g such that xi 2 K \ C ni and xi ! x. Assertion (ii) of Lemma 2 implies x 2 C \ K; which is contradictory. Let us establish the last statement. Let x be a point of K C ; we want to show that there exists nx 2 N such that x 2 C n ; if nXnx : For that, let us note p ¼ dimðvsðCÞÞ; Dx the line joining the origin to x; x1 the extremity of the interval C \ Dx such that x1 ¼ ð1 þ 2aÞx; a40. The cases a ¼ 0 and a ¼ 1 offer no difficulty. There exists Z40 and p linearly independent points y ¼ ðyi Þ; i ¼ 1 p; such that, yi 2 qðCÞ; where qðCÞ is the boundary of C, kyi x1 k ¼ Z and dðx; HðyÞÞ9 inf½kx uk; u 2 HðyÞ4a; where HðyÞ is the hyperplane passing by y: We also introduce the points yn ¼ ðyni Þ; where yni is the extremity of the interval C n \ Dyi nearest
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to yi . As yni ! yi when n ! þ1, we obtain sup½ku vk; u 2 HðyÞ \ C; v 2 Hðyn Þ \ C n ! 0,and consen quently dðx; Hðy ÞÞ4a=2 for n large enough: nXnx : Hence, there is a ball Bx centered in x; such that Bx C n ; 8nXnx . As K is compact there exists a finite sequence of balls Bi centered in xi 2 K such that K [Bi and Bi C n ; 8nXnxi : It suffices to take n0 ¼ supi ½ni to obtain K C n for nXn0 : & The two following lemmas constitute the junction with the properties of class C. Lemma 4. Let fC n g Rd be a sequence of convex sets and K be a convex compact set. If there is a constant a40 such that lðK \ C n ÞXalðC n Þ40, then there is a compact set K such that, for all n; C n K . Proof. We may suppose that the C n ’s are closed and we proceed by negation. Suppose that there is a sequence fxn g, xn 2 C n , such that kxn k ! 1. If Sn is the cone generated by K \ C n with summit xn ; then K \ C n K \ S n : Let yn be an interior point of C n \ K: Let Dn be the line passing by yn and xn ; we denote by zn the point of Dn \ C n nearest to xn : There exists a linear form u such that sup uðK \ C n Þ ¼ uðzn Þ; let F n denote the set S n \ fuXuðzn Þg: It is obvious that F n C n and lðF n ÞXbn lðK \ C n Þ; where bn ! 1: This contradicts the initial assumption. & For the next lemma, we introduce the following notations: Let x ¼ ðxi Þ and y ¼ ðyi Þ be two points of Rd , we write xpy if xi pyi ; 8i; and xoy if xi oyi ; 8i: In the same way, we note for xpy; ½x; y9 fz j xi pzi pyi ; 8ig. Lemma 5. Let K be a convex compact of Rd ; p be an orthogonal projection of Rd and a; b two points in pðRd Þ with
aob: Then, there exists a constant k40 such that, for any convex compact C; C K; and ½a; b pðC Þ, we have lðC \ fp 2 ½a; bgÞXklðCÞ: Proof. Let us suppose dim(pÞ ¼ 1. Introduce the following notations: a a ¼ inf pðCÞ ¼ pðMÞ; b þ b ¼ sup pðCÞ ¼ pðNÞ; M and N 2 C; thus infða; bÞ40: We denote DM -respectively DN —the cone with summit M; base C a 9C \ fp ¼ ag and height b þ b a -resp. the cone with summit N; base C b 9C \ fp ¼ bg and height a a þ b-. As C DM [ DN ; we have lðCÞpk1 ðlðC a Þ þ lðC b ÞÞ, where k1 is a constant which depends only on a; b; a and b: The cylinder with base C a and C b and height b a is contained in C \ fapppbg; thus lðC \ fapppbgÞXk2 ðlðC a Þ þ lðC b ÞÞ: Finally, we obtain lðCÞpk:lðC \ fapppbgÞ; this proves the lemma for dim(pÞ ¼ 1 and for oneC K: We can suppose that p ¼ ðpi Þ; i ¼ 1; . . . ; p, where p ¼ dimðpÞ and pi 2 P; pi pj ¼ 0 if iaj: Let C be a convex compact in Rp ; we denote by a ¼ ðai 9 inf pi ðCÞÞ; b ¼ ðbi 9 sup pi ðCÞÞ; then aoaobob: We repeat the preceding procedure for each pi ; ai et bi ; and the result follows for one C K: To prove the Lemma, we proceed by negation. We suppose that there exists a sequence fC n g K such that ðlðC n \ fapppbgÞÞ=lðC n Þ ! 0: Taking a subsequence if necessary, by Lemma 2, we have C n ! C: By
Lemma 3, the relations ½a; b pðC n Þ give ½a; b pðC Þ and lðC n \ fapppbgÞ ! lðC \ fapppbgÞ: Thus, C verifies the relation lðC \ fp 2 ½a; bgÞXk:lðCÞ; for a suitable k40; which is a contradiction. & Acknowledgements We are grateful to the referee who read this paper and made a very interesting report. References Cuesta-Albertos, J.A., Matra`n, C., Rodrı´ guez-Rodrı´ guez, J., 2002. Shape of a distribution through the Wasserstein distance. In: Distributions with Given Marginals and Statistical Modelling. Kluwer Academic Publishers, Dordrecht, pp. 51–62. Cuesta-Albertos, J.A., Matra`n, C., Rodrı´ guez-Rodrı´ guez, J., 2003. Approximation to probabilities through uniform laws on convex sets. J. Theoret. Probab. 16, 363–376. Dharmadhikari, S., Joag-Dev, K., 1989. Unimodality, Convexity, and Applications. Academic-Press, New York. Rachev, S.T., Ru¨schendorf, L., 1998. Mass Transportation Problems. Springer, New York.
Statistics & Probability Letters 76 (2006) 298–303 www.elsevier.com/locate/stapro
Approximation of distributions Nacereddine Belilia,, Henri Heinichb a
LMI, place E. Blondel, 76131 Mont-Saint-Aignan Cedex, France INSA de Rouen, De´partement de Ge´nie Mathe´matique, place E. Blondel, 76131 Mont-Saint-Aignan Cedex, France
b
Received 4 April 2003; received in revised form 23 June 2005 Available online 7 September 2005
Abstract Let P be a Borel-probability on Rd and let C be a family of Borel-probabilities with finite second order moments. Under some conditions on C, we prove that there exists a best approximation to P in C for the Wasserstein distance. r 2005 Elsevier B.V. All rights reserved. Keywords: Best approximation; Wasserstein distance
1. Problem and notations Let ðM; dÞ be a metric space and C M: For xeC; the existence of a point y0 2 C such that dðx; y0 Þ ¼ dðx; CÞ9 inffdðx; yÞ; y 2 Cg is a classical problem of optimization. A sequence ðyn Þ C is an approximate sequence (for x and C) if dðx; yn Þ ! dðx; CÞ: Here we study this approximation problem for C a subset of the Borel-probabilities on Rd with the Wasserstein distance. We assume that all the random variables (r.v.) in this paper are defined on a rich enough probability space and, unless explicitly stated, all of them will be Rd valued, dX1. We denote by E½X and LðX Þ the expectation and probability-distribution of a r.v. X : For two Borel-probabilities P and Q on Rd ; ðPjQÞ is the set of the probabilities on Rd Rd whose marginal distributions are (respectively) P and Q: The square of Wasserstein’s distance is lðP; QÞ ¼ inffE½kX Y k2 ; LðX R; Y Þ 2 ðPjQÞg: This expression is interesting only if P and Q have finite second order moments i.e., s2 ðPÞ9 kxk2 dPðxÞ and s2 ðQÞo1. We say that ðX ; Y Þ is an optimal coupling for ðP; QÞ if lðP; QÞ ¼ E½kX Y k2 ¼ s2 ðPÞ þ s2 ðQÞ 2E½hX ; Y i; LðX ; Y Þ 2 ðPjQÞ: The existence of an optimal coupling is traditional, see for example Rachev and Ru¨schendorf (1998). For a class C of probabilities on Rd and a probability P on Rd , we consider the problem ðPÞ : the existence of Q0 2 C such that lðP; Q0 Þ ¼ lðP; CÞ9 infflðP; QÞ; Q 2 Cg. For example, when C ¼ Cu is the family of uniform probabilities defined on compact convex sets of Rd with a non-empty interior, Cuesta-Albertos et al. (2003) showed the existence of such a probability when P is Corresponding author.
E-mail addresses: [email protected] (N. Belili), [email protected] (H. Heinich). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.031
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absolutely continuous with respect to Lebesgue measure on Rd : The functional lðP; CÞ can be employed as an alternative parameter to measure the flatness (see Cuesta-Albertos et al., 2002). We are going to weaken the condition on P and to generalize the class Cu : Indeed, we only suppose that the dimension of the vector-space spanned by the support of P is d. For P we can take the empirical law of an nsample, if n is large enough, and we obtain a natural estimator by choosing Q 2 C such that lðP; QÞ ¼ lðP; CÞ. For this purpose, let us introduce some additional notations. L If a sequence of Borel-probabilities fQn g converges in distribution to Q; we write Qn ! Q: We denote by S P the support of P; a Borel-probability on Rd : If C is a subset of Borel-probabilities, C is its closure for the convergence in distribution. If f is a Borel-function on Rd ; f ðQÞ is the probability defined by f ðQÞðAÞ ¼ Qðff 2 AgÞ, particularly for a 2 R; aðQÞðAÞ ¼ Qðfx j ax 2 AgÞ. ¯ is the closure of A; A its interior, vsðAÞ the vector space generated by A and dimðvsðAÞÞ the For A Rd ; A dimension of this vector space. For a sequence fAn g Rd ; we write An ! A if 1An ðxÞ ! 1A ðxÞ; 8 x 2 Rd and An ! A, if there exists a subsequence such that Ani ! A. We denote by AðÞ the set fx j 9 y 2 A; kx ykpg. With abuse of notation, we will denote the Lebesgue measure on every Euclidean-space with the same symbol l. Finally a probability Q is symmetric (with respect to a base), if given X ¼ ðX i Þ such that LðX Þ ¼ Q; then, for all ¼ ði Þ 2 ðf1; þ1gÞd ; we have LðX 9 ði X i ÞÞ ¼ Q. 2. The main theorem Let P denote the set of orthogonal projections of Rd and for p 2 P; p? is the orthogonal of p. We say that a family C of Borel-probabilities on Rd satisfies the ðÞ condition if: (0) (1) (2) (3)
C is formed by probabilities having a finite second order moment. C is closed by translation: Q 2 C if and only if Qð aÞ 2 C; 8a 2 Rd : For each p 2 P and a 2 Rnf0g; ðap þ p? ÞðQÞ 2 C if Q 2 C. For each m 2 C; there is Q in C such that m ¼ p? ðQÞ and pðQÞ is symmetric, where p? is the orthogonal projection onto vsðSm Þ: In particular, if dimðvsðS m ÞÞ ¼ d; then m 2 C: (4) For all p 2 P and Q 2 C; dimðvsðSpðQÞ ÞÞ ¼ dim(pÞ 9 dimðpðRd ÞÞ. The above conditions allow to reduce the initial problem: ðPÞ is equivalent to the same problem by supposing the centered probabilities: R The problem R x dPðxÞ ¼ x dQðxÞ ¼ 0; 8Q 2 C. Next, we adopt this framework by still writing C for the centered family C. The main result is the following: Theorem. Let P be a probability on Rd such that dimðvsðS P ÞÞ ¼ d and C a class satisfying ðÞ condition. Then, there exists Q 2 C such that lðP; QÞ ¼ lðP; CÞ. Moreover, for all approximate sequence lðQ; Qn Þ ! 0. Proof. Obviously there exists an approximate sequence fQn g C; such that Qn converges in distribution to m and lðP; mÞplðP; CÞ. If dimðvsðS m ÞÞ ¼ d; property (3) implies that m 2 C and the theorem is proved. Let us suppose dimðvsðS m ÞÞod: Corollary 1 below ensures that dimðvsðSm ÞÞX1: Assertion (3) gives the existence of a probability Q 2 C such that m ¼ p? ðQÞ and pðQÞ is symmetric and different from d0 ; by assertion (4). Corollary 2 gives pðPÞ ¼ d0 and we obtain a contradiction with the assumption on P. The proof of the last statement is easy. & Let us note that the last assertion can be used for numerical simulations. Lemma 1. Let P and Q be two centered probabilities having finite second order moments such that Q is symmetric and lðP; QÞ ¼ s2 ðPÞ þ s2 ðQÞ. Then the supports of P and of Q are orthogonal.
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Proof. We use the following statement: Let m be a probability on M nþp whose marginals are m1 on M n and m2 on M p and let X a r.v. defined on ðO1 ; A1 ; P1 Þ with distribution m1 . There exists a probability space ðO ¼ O1 O2 ; A ¼ A1 A2 ; P ¼ P1 P2 Þ; where ðO2 ; A2 ; P2 Þ is another probability space, and a couple of r.v’s ðX ; Y Þ such that X ðo1 ; o2 Þ ¼ X ðo1 Þ; LðX ; Y Þ ¼ m. With the above notations, an immediate consequence is: For any r.v. X 0 ; such that LðX 0 Þ ¼ P; there exists Y 0 with distribution Q such that ðX 0 ; Y 0 Þ is an optimal coupling and therefore, E½hX 0 ; Y 0 i ¼ 0. For LðX ; Y Þ 2 ðPjQÞ we have E½hX ; Y ip0 and by changing Y into Y , we obtain E½hX ; Y i ¼ 0: Let X and X 0 be two independent r.v’s with the same distribution P, the probability P ¼ LðX X 0 Þ is symmetric. As before, for any r.v. X 0 with LðX 0 Þ ¼ P , there exists Y 0 , with LðY 0 Þ ¼ Q, such that lðP ; QÞ ¼ E½kX 0 Y 0 k2 . Putting X 0 ¼ X X 0 , one deduces that E½hX 0 ; Y i ¼ 0 8 Y with law Q, therefore lðP ; QÞ ¼ s2 ðP Þ þ s2 ðQÞ. Thus, we can suppose without loss of generality that P is symmetric. For a r.v. Z ¼ ðZ i Þ if signðZ i Þ ¼ 1fZi 40g 1fZi o0g , we put signðZÞ ¼ ðsignðZ i ÞÞ and jZj ¼ ðjZ i jÞ. Let X and Y be two independent r.v’s with laws P and Q respectively. Since LðsignðY ÞjX jÞ ¼ P and LðsignðY ÞjY jÞ ¼ Q; we have E½hjX j; jY ji ¼ 0, thus hX ; Y i ¼ 0 a.s. & Corollary 1. Let P be a probability on Rd such that the vector space generated by its support has dimension d. Then lðP; CÞos2 ðPÞ if class C verifies ðÞ condition. Proof. By negation, we suppose that lðP; CÞ ¼ s2 ðPÞ ¼ lðP; d0 Þ: For Q 2 C and a 2 R, aa0; we have aQ 2 C: It is clear that the minimum of the function a ! lðP; aQÞ is s2 ðPÞ ððE½hX 0 ; Y 0 iÞ2 Þ=s2 ðQÞ, where ðX 0 ; Y 0 Þ is optimal for ðP; QÞ: By taking Q symmetric, we deduce, as in Lemma 1, that hX ; Y i ¼ 0 a.s. for any couple ðX ; Y Þ such that LðX ; Y Þ 2 ðPjQÞ: Therefore, S P and S Q are orthogonal, which is a contradiction. & Corollary 2. Let P and Q be two probabilities on Rd and p 2 P with 1pdimðpÞod. Let us assume that for all real aa0, and u ¼ p? þ ap, we have lðP; p? ðQÞÞplðP; uðQÞÞ. Then, there exists a couple ðX ; Y Þ such that ðp? ðX Þ; p? ðY ÞÞ is an optimal coupling for ðp? ðPÞ; p? ðQÞÞ. Moreover, for this couple we have E½hpðX Þ; pðY Þi ¼ 0. Proof. The existence of a couple ðX ; Y Þ satisfying the conditions of this corollary results from the statement in Lemma 1. Noting Z1 ¼ pðZÞ; Z2 ¼ p? ðZÞ and LðY 2 Þ ¼ Q2 ; we have lðP; Q2 ÞplðP; uQÞps2 ðPÞ þ a2 s2 ðQ1 Þþ s2 ðQ2 Þ 2E½hX 2 ; Y 2 i 2aE½hX 1 ; Y 1 i. Taking the minimum over a, we obtain lðP; Q2 ÞplðP; Q2 Þ ðE½hX 1 ; Y 1 iÞ2 =s2 ðQ1 Þ and the proof is complete. & 3. Examples In this section we are going to show that the result can be applied to uniform probabilities on compact convex sets with non-empty interior and other examples. Notice that the only thing to be proved is that condition (3) holds, because the remaining ones are evident. 3.1. Uniform probabilities on compact convex sets with non-empty interior Let us recall that Cu denotes the set of uniform probabilities on compact convex sets with non-empty interior. We have seen that we can suppose that these probabilities are centered. It is easy to see that if S Q is the support of Q 2 Cu ; then 0 2 S Q : If fQn g 2 Cu converges in distribution to m; the Lemmas in the appendix ensure that the supports S n of Qn are contained in a compact K and S n ! S: L
Proposition 1. Let fQn g Cu be a sequence such that Qn ! m and S n ! S where Sn is the support of Qn . Then, if Sm is the support of m; we have S¯ ¼ S m . ¯ If B is a compact ball such that B \ S¯ ¼ ;, Lemma 3 ensures that Proof. Let us show that Sm is contained in S.
B \ Sn ¼ ; as soon as n is large enough. Consequently Qn ðBÞ ¼ 0 and thus mðBÞ ¼ 0. The complementary set c ¯ of S¯ is a countable union of open balls with compact closure, therefore mðS¯ Þ ¼ 0; and S m S.
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Conversely, let us suppose that dimðvsðSm ÞÞ ¼ pX1; and denote by p the orthogonal projection on this space. Let aob be two points of Rp ; Lemma 5 ensures that ðlðS n \ fapppbgÞÞ=lðS n Þ ¼ Qn ðfapppbgÞXk: Therefore, mðfapppbgÞXk i.e. S¯ Sm : & L
Let fQn g Cu ; such that Qn ! m; with 1p dimðvsðSÞÞod and S the support of m. The orthogonal projection on vsðSÞ is noted p? : We saw in Proposition 1, that for a subsequence (still noted S n Þ, the supports S n of Qn converge to the support S: Let us choose a point M n 2 S n such that an 9kpðM n Þk ¼ sup kpðS n Þk. The mapping un ¼ ð1=an Þp þ p? , transforms Qn into Qn 9un ðQn Þ 2 Cu . By extracting a subsequence if necessary, we can suppose that fun ðM n Þg converges. Let us show the convergence in distribution of the sequence fQn g. For each n; Sn is contained in the compact convex set K n 9fx j p? ðxÞ 2 p? ðSn Þ; kpðxÞkpan g: The mapping un transforms K n into K n , which is contained in fx j p? ðxÞ 2 p? ðSn Þ, kpðxÞkp1g. For nXNðÞ, we have K n fx j kpðxÞkp1; p? ðxÞ 2 p? ðSÞðÞg. Consequently, the sequence fQn g is relatively compact for the L
convergence in distribution. We can suppose that Qn ! Q; it is obvious that lðSQ Þ40 and, therefore, Q 2 Cu . L
Moreover, we have: p? ðQn Þ ¼ p? ðQn Þ ! p? ðQÞ ¼ m: Finally it is enough to symmetrize pðQÞ; which belongs to Cu ; to prove that Cu verifies assertion (3). The theorem is valid for the class Cu : Remark. A referee for this paper has given a shorter proof. He notices that m 2 Cu and fQn g Cu converges in distribution to m: It is obvious that the sequence of the supports of fQn g satisfies Lemma 4 and therefore m satisfies Theorem 3.1 in Cuesta et al. (2003). It is easy to show that condition (3) holds for Cu : Nevertheless, we have kept our proof because the method used is applicable to the following examples. 3.2. Other examples We will extend the property ensuring that a probability, m 2 Cu whose support generates a vector space with dimension pX1, is the projection of a probability include in Cu . If Qn converges to m; there exists a compact set K such that for all n; Qn ðKÞX1 : Let Sn be the support of ? Qn and S K n ¼ S n \ K: We can suppose that 1p dimðvsðSÞÞ; where S is the support of m and that S n ! S: Let p be the orthogonal projection on this space. The previous method ensures that fun ðQn Þ ¼ Qn g is relatively compact and the vector space generated by the support of the limit of any subsequence is Rd : In a similar way, it is enough to prove that condition (3) holds for the following examples:
The set consisting of the probabilities with uniform law on the boundary of a compact convex set with nonempty interior. The set consisting of the probabilities with uniform distribution on C 1 nC 2 ; where C i are compact convex sets with non-empty interior. The justification of the other conditions is easy. Now, let us consider a fourth example. Call Co the class of probabilities Q having finite second order moments such that QðyAþ ð1 yÞBÞX inf½QðAÞ; QðBÞ and whose supports generate Rd . If Q 2 Co ; then Q has a density g; verifying that fgXag is convex for all a 2 Rþ . As previously we can restrict ourselves to centered probabilities.
Proposition 2. Let fQn g Co ; be a sequence converging in law to mad0 . Let S m be the support of m and S an 9fgn Xag, where gn is the density of Qn and a a rational 40. Then, we have S an ! Sa and S m ¼ S, where S is the closure of [S a . Proof. The existence of the subsequence results from the preceding considerations. The proof of S m S is similar toR the one for the uniform laws. If K is a compact set contained in the complement of S; then Qn ðKÞ ¼ K gn ðxÞ dxpalðKÞ; 8 a40: Thus, for an open ball O contained in K, we have mðOÞ ¼ 0; and since the complement of S is a countable union of such balls, mðSÞ ¼ 1. This shows our inclusion.
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Conversely, let us suppose mad0 : Let p be the orthogonal projection on the vector space generated by the
support of m: IfR K ¼ K p K p? is the product of two compact sets and is contained in S a , by Lemma 4, we have, Qn ðKÞ ¼ K gn ðxÞ dxXalðKÞXkðaÞlðK p Þ where kðaÞ is a constant 40: We deduce that mðK p Þ ¼ mðKÞ40:
Moreover, by Lemma 3, if a is small enough and K a compact include in S ; we have K S a : Thus for x 2 S ;
there exists a basis of compact neighborhoods having the previous form and contained in Sa , therefore x 2 Sm : So, we have S S m ; which completes the proof. & The method used for Cu can be adapted to Co and we easily show that Co verifies all the conditions of the class C: The previous approximation theorem is valid for Co : 4. Appendix. Some properties of convex sets In this part we study the properties and the convergence of convex sets. Other properties can be found in Dharmadhikari and Joag-Dev (1989), in particular for the class C0 : Lemma 2. Let fC n g Rd be a sequence of convex sets containing the point 0. There exists a convex set C such that C n ! C: Moreover, if C n ! C; then, for any a40, the following assertions hold (i) CðaÞ lim C n ðaÞ CðaÞ ¼ CðaÞ, (ii) C ¼ fx j 9 xn 2 C pn ; xn ! xg. Proof. Let D be a countable dense set of lines, i.e. {[ D; D 2 D} is dense in Rd . The intervals I D n 9C n \ D converge, for a subsequence to I D in R: By the diagonal process, there exists a subsequence fni g such that I D ni converges in R: It is easy to see that if D is a line passing through the origin, then I Dni ! I D : Therefore C ¼ [I D
is convex and C ni ! C. Now, let us prove assertion (i) for a sequence fC n g such that C n ! C: There exists a subsequence fni g such that the set C a 9 lim C ni ðaÞ exists, from the first part of the Lemma. TThus, there is a subsequence fnij g such that f1C ni ðaÞ g and f1C ni g converge for all rational a. j
j
o simplify the writing and without loss of generality, we can suppose that for any rational a; the sets fC n ðaÞg and fC n g respectively converge to C a and C. Let x be a point of CðaÞ; there exists y 2 C such that ky xkpa; and for n large enough, y 2 C n : We obtain x 2 C n ðaÞ and this implies x 2 C a ; i.e. CðaÞ C a . Second inclusion. Let x be a point of C a ; we have x 2 C n ðaÞ; for all n large enough, and there exists yn 2 C n with kyn xkpa: The sequence fyn g is bounded and, for a subsequence, yni ! y. Then for any rational Z; kyni ynj kpZ if i and j are large enough. Consequently y 2 C nj ðZÞ; therefore y 2 lim C nj ðZÞ ¼ C Z , we deduce that for Z small enough, x 2 C Z ðaÞ: Now, when Z ! 0; it is easy to see that fC Z g decreases to C and x 2 CðaÞ ¼ CðaÞ: This shows part (i). To establish part (ii), we notice that for x 2 C; there exists fxn g C; xn ! x: Since xn 2 C p for pXPðxn Þ ¼ pn ; the direct implication is established. Conversely, if xn 2 C pn and xn ! x; then for all rational Z40 and n large enough, we have x 2 C pn ðZÞ: Thus, x 2 lim C pn ðZÞ ¼ C Z and we obtain x 2 C: & Lemma 3. Let fC n g Rd be a sequence of convex sets containing the point 0 such that C n ! C: Then for all
compact K such that K \ C ¼ ; (respectively, K C the interior of C in vsðCÞÞ, there exists N such that, if
nXN; then K \ C n ¼ ; (resp. K C n Þ: Proof. By negation, we suppose that there exists a subsequence fni g and fxi g such that xi 2 K \ C ni and xi ! x. Assertion (ii) of Lemma 2 implies x 2 C \ K; which is contradictory. Let us establish the last statement. Let x be a point of K C ; we want to show that there exists nx 2 N such that x 2 C n ; if nXnx : For that, let us note p ¼ dimðvsðCÞÞ; Dx the line joining the origin to x; x1 the extremity of the interval C \ Dx such that x1 ¼ ð1 þ 2aÞx; a40. The cases a ¼ 0 and a ¼ 1 offer no difficulty. There exists Z40 and p linearly independent points y ¼ ðyi Þ; i ¼ 1 p; such that, yi 2 qðCÞ; where qðCÞ is the boundary of C, kyi x1 k ¼ Z and dðx; HðyÞÞ9 inf½kx uk; u 2 HðyÞ4a; where HðyÞ is the hyperplane passing by y: We also introduce the points yn ¼ ðyni Þ; where yni is the extremity of the interval C n \ Dyi nearest
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to yi . As yni ! yi when n ! þ1, we obtain sup½ku vk; u 2 HðyÞ \ C; v 2 Hðyn Þ \ C n ! 0,and consen quently dðx; Hðy ÞÞ4a=2 for n large enough: nXnx : Hence, there is a ball Bx centered in x; such that Bx C n ; 8nXnx . As K is compact there exists a finite sequence of balls Bi centered in xi 2 K such that K [Bi and Bi C n ; 8nXnxi : It suffices to take n0 ¼ supi ½ni to obtain K C n for nXn0 : & The two following lemmas constitute the junction with the properties of class C. Lemma 4. Let fC n g Rd be a sequence of convex sets and K be a convex compact set. If there is a constant a40 such that lðK \ C n ÞXalðC n Þ40, then there is a compact set K such that, for all n; C n K . Proof. We may suppose that the C n ’s are closed and we proceed by negation. Suppose that there is a sequence fxn g, xn 2 C n , such that kxn k ! 1. If Sn is the cone generated by K \ C n with summit xn ; then K \ C n K \ S n : Let yn be an interior point of C n \ K: Let Dn be the line passing by yn and xn ; we denote by zn the point of Dn \ C n nearest to xn : There exists a linear form u such that sup uðK \ C n Þ ¼ uðzn Þ; let F n denote the set S n \ fuXuðzn Þg: It is obvious that F n C n and lðF n ÞXbn lðK \ C n Þ; where bn ! 1: This contradicts the initial assumption. & For the next lemma, we introduce the following notations: Let x ¼ ðxi Þ and y ¼ ðyi Þ be two points of Rd , we write xpy if xi pyi ; 8i; and xoy if xi oyi ; 8i: In the same way, we note for xpy; ½x; y9 fz j xi pzi pyi ; 8ig. Lemma 5. Let K be a convex compact of Rd ; p be an orthogonal projection of Rd and a; b two points in pðRd Þ with
aob: Then, there exists a constant k40 such that, for any convex compact C; C K; and ½a; b pðC Þ, we have lðC \ fp 2 ½a; bgÞXklðCÞ: Proof. Let us suppose dim(pÞ ¼ 1. Introduce the following notations: a a ¼ inf pðCÞ ¼ pðMÞ; b þ b ¼ sup pðCÞ ¼ pðNÞ; M and N 2 C; thus infða; bÞ40: We denote DM -respectively DN —the cone with summit M; base C a 9C \ fp ¼ ag and height b þ b a -resp. the cone with summit N; base C b 9C \ fp ¼ bg and height a a þ b-. As C DM [ DN ; we have lðCÞpk1 ðlðC a Þ þ lðC b ÞÞ, where k1 is a constant which depends only on a; b; a and b: The cylinder with base C a and C b and height b a is contained in C \ fapppbg; thus lðC \ fapppbgÞXk2 ðlðC a Þ þ lðC b ÞÞ: Finally, we obtain lðCÞpk:lðC \ fapppbgÞ; this proves the lemma for dim(pÞ ¼ 1 and for oneC K: We can suppose that p ¼ ðpi Þ; i ¼ 1; . . . ; p, where p ¼ dimðpÞ and pi 2 P; pi pj ¼ 0 if iaj: Let C be a convex compact in Rp ; we denote by a ¼ ðai 9 inf pi ðCÞÞ; b ¼ ðbi 9 sup pi ðCÞÞ; then aoaobob: We repeat the preceding procedure for each pi ; ai et bi ; and the result follows for one C K: To prove the Lemma, we proceed by negation. We suppose that there exists a sequence fC n g K such that ðlðC n \ fapppbgÞÞ=lðC n Þ ! 0: Taking a subsequence if necessary, by Lemma 2, we have C n ! C: By
Lemma 3, the relations ½a; b pðC n Þ give ½a; b pðC Þ and lðC n \ fapppbgÞ ! lðC \ fapppbgÞ: Thus, C verifies the relation lðC \ fp 2 ½a; bgÞXk:lðCÞ; for a suitable k40; which is a contradiction. & Acknowledgements We are grateful to the referee who read this paper and made a very interesting report. References Cuesta-Albertos, J.A., Matra`n, C., Rodrı´ guez-Rodrı´ guez, J., 2002. Shape of a distribution through the Wasserstein distance. In: Distributions with Given Marginals and Statistical Modelling. Kluwer Academic Publishers, Dordrecht, pp. 51–62. Cuesta-Albertos, J.A., Matra`n, C., Rodrı´ guez-Rodrı´ guez, J., 2003. Approximation to probabilities through uniform laws on convex sets. J. Theoret. Probab. 16, 363–376. Dharmadhikari, S., Joag-Dev, K., 1989. Unimodality, Convexity, and Applications. Academic-Press, New York. Rachev, S.T., Ru¨schendorf, L., 1998. Mass Transportation Problems. Springer, New York.