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Optimal Poisson quantisation
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 1123–1132 www.elsevier.com/locate/stapro
Optimal Poisson quantisation$ Ilya Molchanov, Nikolay Tontchev Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland Received 14 June 2006; received in revised form 3 January 2007; accepted 21 February 2007 Available online 12 March 2007
Abstract The quantisation problem for probability measures aims to represent a measure using a discrete measure supported by a finite set X. We consider a similar problem where X is a realisation of a finite Poisson point process, the objective function is given by the expected Lp -error, and the constraints are imposed on the total intensity of the process. r 2007 Elsevier B.V. All rights reserved. MSC: primary 60G55; secondary 60D05; 60E99; 94A20 Keywords: Intensity measure; Point process; Poisson process; Quantisation
1. Introduction The quantisation procedure aims to represent a probability measure P in Rd using a discrete measure supported by the points X ¼ fx1 ; . . . ; xn g Rd in order to minimise the Lr -quantisation error given by Z IðXÞ ¼ rðy; XÞr PðdyÞ. (1) Rd
Here r40 and rðy; XÞ is the distance from y 2 Rd to the nearest point of X with respect to any given norm k k on Rd . The Zador theorem proved formally by Graf and Luschgy (2000, Theorem 6.2) establishes that the minimum value of IðXÞ over all n-point sets X normalised by nr=d converges to the Ld=ðdþrÞ -norm of the density of the absolutely continuous part of P times a certain (generally unknown) constant. Further recent results can be found in Delattre et al. (2004) and Gruber (2004). A signal with distribution P can be transmitted by replacing the input signal with the point nearest to it from X, the later is sometimes called a codebook. It is possible that some codes from X may be wrongly transmitted or not transmitted at all. In this case X can be modelled as a point process, where possible transmission faults result in displacements or complete loss (thinning) of some points. It is natural to assume that X is a Poisson point process, because it appears as the limit in a number of thinning and displacement schemes for point processes, see Ellis (1986) and Daley and Vere-Jone (1998, Section 9.1). If X is a point process, then IðXÞ given $
Supported by Swiss National Science Foundation, Grant no. 200020-109217.
Corresponding author. Tel.: +41 31 631 8801; fax: +41 31 631 3870.
E-mail addresses: [email protected] (I. Molchanov), [email protected] (N. Tontchev). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.02.004
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by (1) is a random variable. Its expected value (if it exists) depends on the distribution of X. For instance, if X is a Poisson process, then this expected value is solely determined by the intensity measure of X. If X is a collection of n i.i.d. points or a Poisson process, then in many cases it is possible to determine the asymptotic behaviour of EIðXÞ if the number of points in X is growing or, alternatively, the intensity measure of X is multiplied by a factor that increases indefinitely. For the binomial point process Xn ¼ fX 1 ; . . . ; X n g with i.i.d. points drawn from a given absolutely continuous distribution n with density g, Cohort (2004) found the asymptotics of IðXn Þ as n ! 1. If P is absolutely continuous probability measure on Rd with density h and under certain integrability assumptions on h and a power of g, he showed that Z r hðuÞ r=d nr=d IðXn Þ ! kd G 1 þ du as n ! 1 (2) d Rd gðuÞr=d in L2 and almost surely, where kd is the volume of the unit ball in Rd and G is the Gamma-function. The righthand side of (2) is minimised for the probability density g given by gðxÞ ¼ gopt ðxÞ ¼ R
hd=ðdþrÞ ðxÞ hðyÞd=ðdþrÞ dy
,
so that the smallest value of the integral in (2) becomes khkd=ðdþrÞ , i.e. the Ld=ðdþrÞ -norm of h. However, a binomial point process Xn generated by the density gopt is not necessarily the optimal one among all n-point binomial processes that may be used to quantise P. Strictly speaking, one has to find the optimal density gn that minimises EIðXn Þ and then study the asymptotics for the corresponding minimal value of EIðXn Þ. In other words, the infimum of the limit of nr=d EIðXn Þ is not necessarily the limit of the infima of nr=d EIðXn Þ. Laws of large numbers for functionals of point processes have been considered in Penrose and Yukich (2003). These functionals F n ðf Þ depend on the total number of points and the distribution f of the typical point in the binomial process or the intensity nf in the Poisson case. However Penrose and Yukich (2003) does not contain results that would imply the convergence of minimisers or optimal values. By analysing the proofs from Penrose and Yukich (2003) (in particular Lemma 3.1) it is possible to justify the uniform convergence of the rescaled F n ðf n Þ for a sequence ff n ; nX1g if Z jf n ðyÞ f n ðxÞj dy ! 0 as n ! 1 (3) n kyxkpn1=d
for all x 2 Rd . If f n ðxÞ ! f ðxÞ as n ! 1 for all x, condition (3) would require that f n ðyÞ ! f ðxÞ as n ! 1 and y ! x. Incidentally, a similar double limit condition appears in Molchanov and Zuyev (2000), which deals with the minimisation problem for the expectation EF ðXÞ of a functional F calculated for a Poisson point process X. In this paper we consider the optimisation problem for the expected value of the functional (1) in case X is the Poisson point process with intensity measure m such that mðRd Þ ¼ n. Assuming that the density hðyÞ of P is Riemann integrable, we show that minimum expected error rescaled by nr=d converges to the Ld=ðdþrÞ -norm of h times a certain (unknown) constant I. Numerical computations suggest that I equals the constant r=d kd Gð1 þ r=dÞ from (2), but this still remains an open question. In Section 2 the main properties of the expected quantisation error are considered. The main difficulty here is the possibility that X may be empty, since the integral in (1) then would involve the distance from a point to the empty set. The control of this exceptional probability is an essential part of subsequent proofs. Section 3 establishes the asymptotic behaviour of the quantisation error for the uniform distribution P on the unit cube in Rd . The main idea here is the firewall construction from Delattre et al. (2004), and Graf and Luschgy (2000), which involves adding some points to X such that the total quantisation error becomes the sum of quantisation errors calculated for individual sets that partition the unit cube. The firewalls are formed by extra points located along the boundaries of the sets that form the partition. The main new feature of the proof is the fact that the random nature of the problem makes it impossible to construct firewalls almost surely. We use estimates for the coverage probabilities from Hall (1985) in order to adjust the number of extra points added such that they establish a firewall with a high probability. Section 4 extends the results for a
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general absolutely continuous measure P with Riemann integrable density and gives the weak limit for the sequence of normalised intensity measures. The proof follows the scheme of Molchanov and Tontchev (2007). Finally, some comments about the limiting constant are given. 2. Objective functional For any measurable set K Rd and a finite non-empty set X define Z rðy; XÞr dy, IðK; XÞ ¼
(4)
K
where r40 and rðy; XÞ is the distance from y 2 Rd to the nearest point from X. Define IðK; mÞ ¼ Em IðK; XÞ,
(5)
where Em is the expectation with respect to the distribution of the Poisson point process X with intensity m from the family M of all non-negative finite Borel measures on Rd . It is useful to write Em IðK; XÞ as EIðK; Pm Þ, where Pm denotes the Poisson process of intensity m. For nX1 define I n ðKÞ ¼
inf
m2M; mðRd Þ¼n
nr=d IðK; mÞ.
(6)
Note that a finite Poisson point process is empty with a positive probability. If X is empty, rðy; XÞ is not well defined and we set it to zero, i.e. define rðy; ;Þ ¼ 0 for every y 2 Rd . Alternatively, one can consider the conditional expectations given that Xa;. The both approaches yield the same asymptotics of I n as n ! 1. Lemma 1. For each convex K, I n ðKÞ equals the infimum of nr=d IðK; mÞ over all Borel measures m with total mass n which are supported by the closure of K. Proof. Consider any Borel measure m on Rd . Let m0 be the image of m under the metric projection on K, i.e. the map that maps every z 2 Rd to the point nearest to it from the closure of K. The convexity assumption implies that the distance between any y 2 K and z is not smaller than the distance between y and x if x is the metric projection of any z 2 Rd . If Pm0 is a Poisson process with intensity measure m0 , then it is possible to couple Pm0 and Pm using the metric projection map so that rðy; Pm0 Þprðy; Pm Þ a.s. for all y from the closure of K. Then IðK; m0 ÞpIðK; mÞ. & Lemma 2. If m and n are measures from M, then, for every Borel set K Rd , d
IðK; m þ nÞpIðK; mÞ þ ½En ðIðK; XÞ2 Þ1=2 eð1=2ÞmðR Þ .
(7)
Proof. Without loss of generality assume that K is a subset of the unit cube. Let Pmþn , Pm and Pn be independent Poisson processes with intensities m þ n, m and n, respectively. Since Pmþn coincides in distribution with Pm [ Pn and IðK; Pm [ Pn ÞpIðK; Pm Þ a.s. given that Pm a;, IðK; m þ nÞ ¼ Emþn ½IðK; XÞ; Xa; ¼ E½IðK; Pm [ Pn Þ; Pm [ Pn a; ¼ E½IðK; Pm [ Pn Þ; Pm a; þ E½IðK; Pm [ Pn Þ; Pm ¼ ;; Pn a; pEm IðK; XÞ þ ðE½IðK; Pn Þ; Pn a;2 Þ1=2 PðPm ¼ ;Þ1=2 d
¼ Em IðK; XÞ þ ðE½IðK; Pn Þ; Pn a;2 Þ1=2 eð1=2ÞmðR Þ , i.e. (7) holds.
&
Lemma 3. The functional IðK; mÞ is continuous in the weak topology for measures m supported by K.
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Proof. Let fnn ; nX1g be a sequence of measures supported by K and weakly converging to m 2 M. The functional IðK; nn Þ can be written as Z Z IðK; nn Þ ¼ Enn rðy; XÞr dy ¼ Enn rðy; XÞr dy Z Z
K
K
d r=2
expfnn ðBt1=r ðyÞÞgð1 expfnn ðBt1=r ðyÞc ÞgÞ dt dy,
¼ K
0
ð8Þ
where Bt1=r ðyÞ is the ball centred at y 2 Rd of radius t1=r and superscript c denotes the complement in Rd . Since nn ðBt1=r ðyÞÞ ! mðBt1=r ðyÞÞ for almost all t and the integrand in (8) is bounded by 1, the dominated convergence theorem yields that IðK; nn Þ ! IðK; mÞ. & 3. Quantisation of the uniform measure The following result concerns the asymptotic behaviour of the functional I n ðKÞ for K ¼ I ¼ ½0; 1d being the unit cube in Rd . Theorem 4. The limit of I n ðIÞ as n ! 1 exists and is a positive finite number. Proof. For a fixed mX1, define k ¼ ½ðn=mÞ1=d . Partition the unit cube I into the (possibly closed or semi-open) d-dimensional cubes F 1 ; F 2 ; . . . ; F kd of side length kd . Consider a measure m with total mass m, which is Pkd optimal for IðI; mÞ, i.e. mr=d IðI; m Þ ¼ I m ðIÞ. For i ¼ 1; . . . ; kd , let mi ¼ m z1 ¯ ¼ i¼1 mi , where zF i : F i and m I ! F i is an affine bijection. Then I n ðIÞpn
Z
r=d
r
Em¯ rðy; XÞ dy ¼ n I
i¼1
kd Z X i¼1
kd Z X
pnr=d
r=d
Fi
Em¯ rðy; XÞr dy
Fi
Emi rðy; XÞr dy þ R1 eð1=2Þm .
The last inequality follows from Lemma 2 with R1 ¼
kd Z X i¼1
Fi
½n2r=d Eni rðy; XÞ2r 1=2 dy,
(9)
where ni ¼ m¯ mi , i ¼ 1; . . . ; kd . Since frðy; XÞ4tg ¼ fX \ Bt ðyÞ ¼ ;g, the support of ni is contained in I and pffiffiffi rðy; XÞp d for all y 2 I, the integrand in (9) can be bounded as Eni rðy; XÞ2r ¼ Eni ðrðy; XÞ2r 1Xa; Þ Z 1 ¼ Pni frðy; XÞ4t1=2r ; Xa;g dt 0 Z pffiffi 2r ð dÞ
p
0
expfni ðBt1=2r ðyÞÞg dt.
ð10Þ
Since the support of m is contained in I, mj is supported by the closure of F j , for j ¼ 1; . . . ; kd . By changing the variable s ¼ t1=2r , the right-hand side of (10) turns into pffiffi Z pdffiffi k1 Z ðpþ1Þ d =k X 2r1 expfni ðBs ðyÞÞgs ds ¼ 2r expfni ðBs ðyÞÞgs2r1 ds. 2r pffiffi 0
p¼0
p d =k
pffiffiffi pffiffiffi The summand with p ¼ 0 in the sum can be bounded from above by ð d =kÞ2r . Since sXp d =k for pX1, at least p 1 of the cubes that partition the unit cube and F i itself are covered by Bs ðyÞ. Therefore,
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ni ðBs ðyÞÞ ¼ Z
P
jai mj ðBs ðyÞÞXðp
1Þm and
pffiffi ðpþ1Þ d =k 2r1
expfni ðBs ðyÞÞgs
pffiffi p d =k
1127
pffiffiffi!2r d dsp ðp þ 1Þ2r1 eðp1Þm . k
Hence "
k1 X 1þ ðp þ 1Þ2r1 eðp1Þm
n2r=d R1 p 2rd 2r k r
p¼1
"
r 1=2
pð2rd Þ
!#1=2
k1 X ðm þ 1Þ 1 þ ðp þ 1Þ2r1 eðp1Þm
#1=2
r
.
p¼1
It is easy to see from the ratio test that the sum in the upper bound for R1 converges as k ! 1. Thus, R1 pcðm þ 1Þr for some constant c, so that I n ðIÞpn
kd Z X
r=d
i¼1
Fi
Emi rðy; XÞr dy þ cðm þ 1Þr eð1=2Þm
kd
¼
X
nr=d kdr
Em rðy; XÞr dy þ cðm þ 1Þr eð1=2Þm I
i¼1 r=d r
¼n
Z
k IðI; m Þ þ cðm þ 1Þr eð1=2Þm .
By choosing m sufficiently large we see that the upper limit of I n ðIÞ does not exceed I m ðIÞ þ e for any given e40. Thus, I n ðIÞ converges as n ! 1 to a non-negative real number denoted by I. To show that I40, notice that IðI; mÞ can be written in the form IðI; mÞ ¼ Em IðI; XÞXEm ½IðI; XÞ1jXjpn . Thus, I n ðIÞXnr=d inf IðI; XÞ Pfzn png, jXj¼n
where zn is Poisson distributed of mean n. Note that the infimum is taken over all deterministic X and so is the quantisation error of the uniform distribution. By Graf and Luschgy (2000, Theorem 6.2), the normalised quantisation error has a positive limit. It suffices to note that Pfzn png converges to a positive number. & 4. Quantisation of non-uniform measures Let h : Rd ! Rþ be a Riemann integrable function, so that h has a bounded support. Define Z hðyÞEm rðy; XÞr dy, Eðh; mÞ ¼ E n ðhÞ ¼ n
Rd r=d
inf
m2M;mðRd Þ¼n
Eðh; mÞ.
Theorem 5. The limit of E n ðhÞ exists and lim E n ðhÞ ¼ khkd=ðdþrÞ I.
n!1
(11) Pm
Proof. First show that (11) holds for the step-function hðyÞ ¼ i¼1 si 1F i ðyÞ, where mX1 is fixed and F 1 ; . . . ; F m partition the unit cube into sub-cubes of side lengths l 1 ; . . . ; l m . Define !1 m X d=ðdþrÞ d=ðdþrÞ ti ¼ si sj j¼1
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Pm and ni ¼ ½ti n for nX1 and i ¼ 1; . . . ; m. Note that i¼1 ni pn. For each i ¼ 1; . . . ; m, P choose 1mi to be a minimiser of IðI; mÞ over all measures with total mass ni , i.e. mi ðRd Þ ¼ ni . Define m¯ ¼ m i¼1 mi zF i and ni ¼ for all i ¼ 1; . . . ; m. Then m¯ mi z1 Fi Z m X r=d r=d E n ðhÞpn Eðh; mÞ si Em¯ rðy; XÞr dy ¯ ¼n i¼1
Fi
Z m X r=d pn si Em z1 rðy; XÞr dy þ R1 , i
Fi
i¼1
Fi
where R1 ¼ nr=d
m X
si eni =2
Z Fi
i¼1
½Eni rðy; XÞ2r 1=2 dy.
Since Eni rðy; XÞ2r is bounded for all i ¼ 1; . . . ; m and ni ¼ ½ti n, we have R1 ! 0 as n ! 1. The convergence holds even if ti ¼ 0 for some i, since then si ¼ 0 and the corresponding summand in the sum in R1 vanishes. Furthermore, Z m X si Em z1 rðy; XÞr dy þ R1 E n ðhÞpnr=d ¼ nr=d
i¼1 m X
Fi
si l rþd i
i
Z I
i¼1
pnr=d
m X
Fi
Emi rðy; XÞr dy þ R1
r=d
si l rþd ni i
I ni ðIÞ þ R1 .
i¼1
Therefore, lim E n ðhÞp
n!1
m X
si l rþd lim ðn=ni Þr=d lim I ni ðIÞ i n!1
i¼1
¼
m X
d=ðdþrÞ si
i¼1
¼
n!1
m X
!r=d d=ðdþrÞ sj
l rþd I i
j¼1
m X
!ðdþrÞ=d d=ðdþrÞ d si li
I ¼ khkd=ðdþrÞ I.
i¼1
To prove the reverse bound for the limit of E n ðhÞ as n ! 1, choose m to be a minimiser of Eðh; mÞ over all measures with the total mass n. Define mi to be the restriction of m onto F i and let ni ¼ mðF i Þ ¼ mi ðRd Þ. For 0obo1 and 0oaob=ðd 1Þ choose en ¼ na , nX1. Let F i;en and F ei n denote the inner and the outer cubes parallel to F i obtained by reducing (respectively enlarging) F i by en , so that their side lengths are l i en and n l i þ en , respectively. On each face of the cubes F i;2en , F i;en , F i , F ei n , F 2e choose independent homogeneous i b Poisson processes each having the intensity n with respect to the ðd 1Þ-dimensional Hausdorff measure of the faces. The superposition of these processes is the Poisson process Pgi with intensity measure gi . Then Z Z m m X X E n ðhÞXnr=d si Em rðy; XÞr dyXnr=d si Emþgi rðy; XÞr dy R2 , F i;2en
i¼1
i¼1
F i;2en
where R2 ¼ n
r=d
Z m X si i¼1
F i;2en
½Egi rðy; XÞ2r 1=2 emðR
d Þ=2
dypmsmax nr=d en=2
and smax ¼ maxðs1 ; . . . ; sm Þ. Note that R2 ! 0 as n ! 1.
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In the following we need a bound for the probability that the union of balls placed at the points of the point process Pgi covers the whole surfaces of the corresponding cubes. If P is the Poisson process on a ðd 1Þdimensional unit cube F 0 of intensity nb , then the probability that F 0 is not covered by the balls of radius en centred at every point of P satisfies b
PfF 0 is not coveredgpc minf1; ðnb þ ad2 nbðd1Þ Þen a g,
(12)
kd1 ed1 n
is the volume of the ðd 1Þ-dimensional ball of radius en , see Hall where c is a constant and a ¼ (1985, Theorem 3). n Denote by U i the event that all faces of F i;2en , F i;en , F i , F ei n or F 2e are covered by fBen ðxÞ; x 2 Pgi g. When i the event U i occurs, the set gi builds the firewall, i.e. the nearest point to y from Pmþgi necessarily belongs to the Poisson process Pmi þgi . Thus, Emþgi ðrðy; XÞr 1U i Þ ¼ Emi þgi ðrðy; XÞr 1U i Þ. Therefore, E n ðhÞXnr=d
Z m X si i¼1
F i;2en
i¼1
F i;2en
i¼1
F i;2en
Z m X ¼ nr=d si Z m X ¼ nr=d si
Emþgi ðrðy; XÞr 1U i Þ dy R2 Emi þgi ðrðy; XÞr 1U i Þ dy R2 Emi þgi ðrðy; XÞr Þ dy R2 R3 ,
where R3 ¼ n
r=d
Z m X si i¼1
F i;2en
Emi þgi ðrðy; XÞr 1U c Þ dypsmax mnr=d PðU ci Þ1=2 . i
By (12), the probability of the complement U ci is bounded by b
PðU ci Þp2dcðnb þ ad2 nbðd1Þ Þen a . Since ¼ kd1 nbaðd1Þ ! 1 nb a ¼ nb kd1 ed1 n we have n below by
q
PðU ci Þ
E n ðhÞXnr=d
! 0 as n ! 1 for every q40, whence R3 ! 0 as n ! 1. Therefore, E n ðhÞ is bounded from Z m X si i¼1
¼ nr=d
m X i¼1
Xnr=d
as n ! 1,
m X
F i;2en
Emi þgi ðrðy; XÞr Þ dy R2 R3
si ðl i 2en Þrþd
Z I
Eðmi þgi ÞzF i ðrðy; XÞr Þ dy R2 R3
si ðl i 2en Þrþd I ni þki ðIÞ R2 R3 ,
i¼1
where ki is the expected total number of points in gi , i.e. ki ¼ ½cnb . Since bo1, ki =n ! 0 as n ! 1. Therefore, r=d m X n rþd si l i lim I. lim E n ðhÞX n!1 n!1 ni þ k i i¼1 Choose a subsequence of n, such that ni =n ! vi 2 ½0; 1 as n ! 1 for all i ¼ 1; . . . ; m. Note that all vi ’s are positive; otherwise E n ðhÞ ! 1 as n ! 1, which contradicts the obtained upper bound on the limit of E n ðhÞ.
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Therefore, m X
lim E n ðhÞX
n!1
i¼1
si l rþd lim i n!1
n ni þ k i
r=d I¼
m X
r=d
si l rþd vi i
I.
i¼1
By (Graf and Luschgy (2000, Lemma 6.8)), the right-hand side is minimised for vi ¼ ti , i ¼ 1; . . . ; m. Thus,
lim E n ðhÞX
n!1
m X
si
m P j¼1
r=ðdþrÞ si
i¼1
¼
m X
!r=d d=ðdþrÞ sj
l rþd I i
!ðrþdÞ=d
d=ðdþrÞ d si li
I ¼ khkd=ðdþrÞ I.
i¼1
A general function h from Theorem 5 can be approximated from above and below by two step-functions h phph . Since Eðh ; XÞpEðh; XÞpEðh ; XÞ, the obtained result for step-functions yields that the limit of E n ðhÞ lies between kh kd=ðdþrÞ and kh kd=ðdþrÞ . Because h is Riemann integrable, the both bounds can be made arbitrarily close to each other. & The following theorem describes the asymptotic behaviour of the normalised measures that minimise the objective functional Eðh; mÞ. Similarly to Lemma 3, it is easy to see that Eðh; mÞ is weak continuous, so that its infimum over all measures with a given total mass is achieved. An analogue of Lemma 1 shows that it is possible to assume that the minimising measure is supported by the closure of K. Theorem 6. Assume that h : Rd ! Rþ is Riemann integrable supported by K. If mn 2 M minimises Eðh; mÞ over all measures with total mass n and mn is supported by the closure of K, then n1 mn converges weakly as n ! 1 to the probability measure L with density proportional to hðyÞd=ðdþrÞ . Proof. Without loss of generality assume that K I. Assume that n1 mn converges weakly to some measure n and let n1 mn ðF 0 Þ ! nðF 0 Þ ¼ n1 for a cube F 0 , such that nðqF 0 Þ ¼ 0. Hence nðF c0 Þ ! 1 nðF 0 Þ ¼ n2 , where F c0 is the complement of F 0 in Rd . The idea of the proof is to show that L and n coincide on F 0 . Denote lðF Þ ¼ kh1F kd=ðdþrÞ . It is straightforward to check that lðF Þ ¼ LðF ÞðdþrÞ=d lðIÞ. Choose a Poisson process Pg with intensity proportional to nb as in the proof of Theorem 5 for the cubes a n F 0;2en , F 0;en , F 0 , F e0n , F 2e 0 , where en ¼ n , nX1. Then Z Z hðyÞEmn rðy; XÞr dy þ hðyÞEmn rðy; XÞr dy Eðh; mn Þ ¼ Fc
F0
Z X
F0
hðyÞEmn þg rðy; XÞr dy þ
Z0 Fc
hðyÞEmn þg rðy; XÞr dy R1 ,
0
where "Z R1 ¼ F0
hðyÞðEg rðy; XÞ2r Þ1=2 dy þ
#
Z Fc
hðyÞðEg rðy; XÞ2r Þ1=2 dy eð1=2Þn .
0
Note that the both integrals are finite and nr=d R1 ! 0 as n ! 1.
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n Let U 0 denote the event that all faces of F 0;2en , F 0;en , F 0 , F e0n , F 2e 0 are covered by fBen ðxÞ : x 2 Pg g. Since 2en c c F 0;2en F 0 and ðF 0 Þ F 0 , Z Z r Eðh; mn ÞX hðyÞEmn þg rðy; XÞ dy þ hðyÞEmn þg rðy; XÞr dy R1
ðF 2en Þc
F 0;2en
0
Z
r
X F 0;2en
If
m0n
and
m00n
Z
hðyÞEmn þg ðrðy; XÞ 1U 0 Þ dy þ
ðF 2en Þc 0
hðyÞEmn þg ðrðy; XÞr 1U 0 Þ dy R1 .
are the restrictions of mn onto F 0 and F c0 , respectively, then
Emn þg ðrðy; XÞr 1U 0 Þ ¼ Em0n þg ðrðy; XÞr 1U 0 Þ;
y 2 F 0;2en ,
Emn þg ðrðy; XÞr 1U 0 Þ ¼ Em00n þg ðrðy; XÞr 1U 0 Þ;
n c y 2 ðF 2e 0 Þ .
Therefore,
Z
Eðh; mn ÞX F 0;2en
where
Z R2 ¼ p
Z
F 0;2en
F 0;2en
Z ðF 2en Þc
hðyÞEm00n þg rðy; XÞr dy R1 R2 ,
0
hðyÞEm0n þg ðrðy; XÞr 1U c Þ dy þ 0
Z ðF 2en Þc 0
hðyÞEm00n þg ðrðy; XÞr 1U c Þ dy 0
hðyÞðEm0n þg ðrðy; XÞ2r Þ1=2 dy Pm0n þg ðU c0 Þ1=2
Z
þ
hðyÞEm0n þg rðy; XÞr dy þ
ðF 2en Þc 0
hðyÞðEm00n þg ðrðy; XÞ2r Þ1=2 dy Pm00n þg ðU c0 Þ1=2
pc1 ðnb þ nbðd2Þ naðd1Þ Þen
baðd1Þ
for a certain constant c1 . Thus, nr=d R2 ! 0 as n ! 1. Let n1 ¼ m0n ðRd Þ and n2 ¼ m00n ðRd Þ and k be the total intensity of Pg . Since k=n ! 0 as n ! 1, r=d Z n ðn1 þ kÞr=d hðyÞEm0n þg rðy; XÞr dy lim nr=d Eðh; mn ÞX lim n!1 n!1 n1 þ k F0 r=d Z n ðn2 þ kÞr=d hðyÞEm0 0n þg rðy; XÞr dy. þ lim n!1 n2 þ k Fc 0
Since ðni þ kÞ=n ! ni 2 ð0; 1 for i ¼ 1; 2, lim E n ðhÞX lim nr=d Eðh; mn Þ
n!1
n!1 r=d r=d Xn1 lðF 0 ÞI þ n2 lðF c0 ÞI r=d r=d ¼ ðn1 LðF 0 ÞðdþrÞ=d þ n2 LðF c0 ÞðdþrÞ=d ÞlðIÞI XðLðF 0 Þ þ LðF c0 ÞÞlðIÞI ¼ lim E n ðhÞ. n!1
Therefore, n1 ¼ LðF 0 Þ and n2 ¼ LðF c0 Þ.
&
5. Conclusions An interesting open problem is to find explicitly the constant I from Theorem 4, i.e. the limit of I n ðIÞ as n ! 1. Note that I depends on r, the power of rðy; XÞ in (4). We conjecture that I ¼ lim nr=d IðI; nlÞ, n!1
(13)
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where l is the Lebesgue measure on I. Note that munif ¼ nl does not necessarily minimise nr=d IðI; mÞ among all n r=d measures of total mass n and therefore Iplimn!1 n IðI; nlÞ. Theorem 6 provides an argument in favour of (13), since the intensity mn minimising nr=d IðI; mÞ weakly converges to the uniform distribution. Numerical computations of the optimal Poisson measures using the steepest descent algorithm described in Molchanov and Zuyev (2002) (on the unit interval with r ¼ 2) have shown that the difference between n2 Ið½0; 1; munif n Þ and I n ð½0; 1Þ decreases to zero as n ! 1. It is also interesting to generalise (5) by replacing the distance function rðy; XÞ with a more general nonnegative function Zðy; XÞ, as it has been done in Molchanov and Tontchev (2007) for the case of deterministic X. For instance, Zðy; XÞ may be defined to be the volume of the Voronoi cell generated by X and containing y, or it may represent the error of approximation of a convex surface by its convex downward triangulation. The main difficulty here is caused by possible bad (or exceptional) configurations X which result in the infinite values for Zðy; XÞ and the resulting expectation IðK; mÞ. For instance if Zðy; XÞ is the volume of the Voronoi cell and K ¼ I, then with a positive probability there appear configurations of points with at least one unbounded Voronoi cell. Recall that in case of the quantisation error given by (4) these exceptional configurations are only those which result from the empty X, and it is much easier to control the probability of their appearance as it has been done in Lemma 2. References Cohort, P., 2004. Limit theorems for random normalized distortion. Ann. Appl. Probab. 14, 118–143. Daley, D.J., Vere-Jones, D., 1988. An Introduction to the Theory of Point Processes. Springer, New York. Delattre, S., Graf, S., Luschgy, H., Page`s, G., 2004. Quantization of probability distributions under norm-based distortion measures. Statist. Decision 22, 261–282. Ellis, S.P., 1986. A limit theorem for spatial point processes. Adv. Appl. Probab. 18, 646–659. Graf, S., Luschgy, H., 2000. Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730, Springer, Berlin. Gruber, P.M., 2004. Optimum quantization and its applications. Adv. Math. 186, 456–497. Hall, P., 1985. On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13, 991–1002. Molchanov, I., Tontchev, N., 2007. Optimal approximation and quantisation. J. Math. Anal. Appl. 325, 1410–1429. Molchanov, I.S., Zuyev, S.A., 2000. Variational analysis of functionals of a Poisson process. Math. Oper. Res. 25, 485–508. Molchanov, I.S., Zuyev, S.A., 2002. Steepest descent algorithms in space of measures. Statist. Comput. 12, 115–123. Penrose, M.D., Yukich, J.E., 2003. Weak laws in geometric probability. Ann. Appl. Probab. 13, 277–303.
Statistics & Probability Letters 77 (2007) 1123–1132 www.elsevier.com/locate/stapro
Optimal Poisson quantisation$ Ilya Molchanov, Nikolay Tontchev Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland Received 14 June 2006; received in revised form 3 January 2007; accepted 21 February 2007 Available online 12 March 2007
Abstract The quantisation problem for probability measures aims to represent a measure using a discrete measure supported by a finite set X. We consider a similar problem where X is a realisation of a finite Poisson point process, the objective function is given by the expected Lp -error, and the constraints are imposed on the total intensity of the process. r 2007 Elsevier B.V. All rights reserved. MSC: primary 60G55; secondary 60D05; 60E99; 94A20 Keywords: Intensity measure; Point process; Poisson process; Quantisation
1. Introduction The quantisation procedure aims to represent a probability measure P in Rd using a discrete measure supported by the points X ¼ fx1 ; . . . ; xn g Rd in order to minimise the Lr -quantisation error given by Z IðXÞ ¼ rðy; XÞr PðdyÞ. (1) Rd
Here r40 and rðy; XÞ is the distance from y 2 Rd to the nearest point of X with respect to any given norm k k on Rd . The Zador theorem proved formally by Graf and Luschgy (2000, Theorem 6.2) establishes that the minimum value of IðXÞ over all n-point sets X normalised by nr=d converges to the Ld=ðdþrÞ -norm of the density of the absolutely continuous part of P times a certain (generally unknown) constant. Further recent results can be found in Delattre et al. (2004) and Gruber (2004). A signal with distribution P can be transmitted by replacing the input signal with the point nearest to it from X, the later is sometimes called a codebook. It is possible that some codes from X may be wrongly transmitted or not transmitted at all. In this case X can be modelled as a point process, where possible transmission faults result in displacements or complete loss (thinning) of some points. It is natural to assume that X is a Poisson point process, because it appears as the limit in a number of thinning and displacement schemes for point processes, see Ellis (1986) and Daley and Vere-Jone (1998, Section 9.1). If X is a point process, then IðXÞ given $
Supported by Swiss National Science Foundation, Grant no. 200020-109217.
Corresponding author. Tel.: +41 31 631 8801; fax: +41 31 631 3870.
E-mail addresses: [email protected] (I. Molchanov), [email protected] (N. Tontchev). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.02.004
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by (1) is a random variable. Its expected value (if it exists) depends on the distribution of X. For instance, if X is a Poisson process, then this expected value is solely determined by the intensity measure of X. If X is a collection of n i.i.d. points or a Poisson process, then in many cases it is possible to determine the asymptotic behaviour of EIðXÞ if the number of points in X is growing or, alternatively, the intensity measure of X is multiplied by a factor that increases indefinitely. For the binomial point process Xn ¼ fX 1 ; . . . ; X n g with i.i.d. points drawn from a given absolutely continuous distribution n with density g, Cohort (2004) found the asymptotics of IðXn Þ as n ! 1. If P is absolutely continuous probability measure on Rd with density h and under certain integrability assumptions on h and a power of g, he showed that Z r hðuÞ r=d nr=d IðXn Þ ! kd G 1 þ du as n ! 1 (2) d Rd gðuÞr=d in L2 and almost surely, where kd is the volume of the unit ball in Rd and G is the Gamma-function. The righthand side of (2) is minimised for the probability density g given by gðxÞ ¼ gopt ðxÞ ¼ R
hd=ðdþrÞ ðxÞ hðyÞd=ðdþrÞ dy
,
so that the smallest value of the integral in (2) becomes khkd=ðdþrÞ , i.e. the Ld=ðdþrÞ -norm of h. However, a binomial point process Xn generated by the density gopt is not necessarily the optimal one among all n-point binomial processes that may be used to quantise P. Strictly speaking, one has to find the optimal density gn that minimises EIðXn Þ and then study the asymptotics for the corresponding minimal value of EIðXn Þ. In other words, the infimum of the limit of nr=d EIðXn Þ is not necessarily the limit of the infima of nr=d EIðXn Þ. Laws of large numbers for functionals of point processes have been considered in Penrose and Yukich (2003). These functionals F n ðf Þ depend on the total number of points and the distribution f of the typical point in the binomial process or the intensity nf in the Poisson case. However Penrose and Yukich (2003) does not contain results that would imply the convergence of minimisers or optimal values. By analysing the proofs from Penrose and Yukich (2003) (in particular Lemma 3.1) it is possible to justify the uniform convergence of the rescaled F n ðf n Þ for a sequence ff n ; nX1g if Z jf n ðyÞ f n ðxÞj dy ! 0 as n ! 1 (3) n kyxkpn1=d
for all x 2 Rd . If f n ðxÞ ! f ðxÞ as n ! 1 for all x, condition (3) would require that f n ðyÞ ! f ðxÞ as n ! 1 and y ! x. Incidentally, a similar double limit condition appears in Molchanov and Zuyev (2000), which deals with the minimisation problem for the expectation EF ðXÞ of a functional F calculated for a Poisson point process X. In this paper we consider the optimisation problem for the expected value of the functional (1) in case X is the Poisson point process with intensity measure m such that mðRd Þ ¼ n. Assuming that the density hðyÞ of P is Riemann integrable, we show that minimum expected error rescaled by nr=d converges to the Ld=ðdþrÞ -norm of h times a certain (unknown) constant I. Numerical computations suggest that I equals the constant r=d kd Gð1 þ r=dÞ from (2), but this still remains an open question. In Section 2 the main properties of the expected quantisation error are considered. The main difficulty here is the possibility that X may be empty, since the integral in (1) then would involve the distance from a point to the empty set. The control of this exceptional probability is an essential part of subsequent proofs. Section 3 establishes the asymptotic behaviour of the quantisation error for the uniform distribution P on the unit cube in Rd . The main idea here is the firewall construction from Delattre et al. (2004), and Graf and Luschgy (2000), which involves adding some points to X such that the total quantisation error becomes the sum of quantisation errors calculated for individual sets that partition the unit cube. The firewalls are formed by extra points located along the boundaries of the sets that form the partition. The main new feature of the proof is the fact that the random nature of the problem makes it impossible to construct firewalls almost surely. We use estimates for the coverage probabilities from Hall (1985) in order to adjust the number of extra points added such that they establish a firewall with a high probability. Section 4 extends the results for a
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general absolutely continuous measure P with Riemann integrable density and gives the weak limit for the sequence of normalised intensity measures. The proof follows the scheme of Molchanov and Tontchev (2007). Finally, some comments about the limiting constant are given. 2. Objective functional For any measurable set K Rd and a finite non-empty set X define Z rðy; XÞr dy, IðK; XÞ ¼
(4)
K
where r40 and rðy; XÞ is the distance from y 2 Rd to the nearest point from X. Define IðK; mÞ ¼ Em IðK; XÞ,
(5)
where Em is the expectation with respect to the distribution of the Poisson point process X with intensity m from the family M of all non-negative finite Borel measures on Rd . It is useful to write Em IðK; XÞ as EIðK; Pm Þ, where Pm denotes the Poisson process of intensity m. For nX1 define I n ðKÞ ¼
inf
m2M; mðRd Þ¼n
nr=d IðK; mÞ.
(6)
Note that a finite Poisson point process is empty with a positive probability. If X is empty, rðy; XÞ is not well defined and we set it to zero, i.e. define rðy; ;Þ ¼ 0 for every y 2 Rd . Alternatively, one can consider the conditional expectations given that Xa;. The both approaches yield the same asymptotics of I n as n ! 1. Lemma 1. For each convex K, I n ðKÞ equals the infimum of nr=d IðK; mÞ over all Borel measures m with total mass n which are supported by the closure of K. Proof. Consider any Borel measure m on Rd . Let m0 be the image of m under the metric projection on K, i.e. the map that maps every z 2 Rd to the point nearest to it from the closure of K. The convexity assumption implies that the distance between any y 2 K and z is not smaller than the distance between y and x if x is the metric projection of any z 2 Rd . If Pm0 is a Poisson process with intensity measure m0 , then it is possible to couple Pm0 and Pm using the metric projection map so that rðy; Pm0 Þprðy; Pm Þ a.s. for all y from the closure of K. Then IðK; m0 ÞpIðK; mÞ. & Lemma 2. If m and n are measures from M, then, for every Borel set K Rd , d
IðK; m þ nÞpIðK; mÞ þ ½En ðIðK; XÞ2 Þ1=2 eð1=2ÞmðR Þ .
(7)
Proof. Without loss of generality assume that K is a subset of the unit cube. Let Pmþn , Pm and Pn be independent Poisson processes with intensities m þ n, m and n, respectively. Since Pmþn coincides in distribution with Pm [ Pn and IðK; Pm [ Pn ÞpIðK; Pm Þ a.s. given that Pm a;, IðK; m þ nÞ ¼ Emþn ½IðK; XÞ; Xa; ¼ E½IðK; Pm [ Pn Þ; Pm [ Pn a; ¼ E½IðK; Pm [ Pn Þ; Pm a; þ E½IðK; Pm [ Pn Þ; Pm ¼ ;; Pn a; pEm IðK; XÞ þ ðE½IðK; Pn Þ; Pn a;2 Þ1=2 PðPm ¼ ;Þ1=2 d
¼ Em IðK; XÞ þ ðE½IðK; Pn Þ; Pn a;2 Þ1=2 eð1=2ÞmðR Þ , i.e. (7) holds.
&
Lemma 3. The functional IðK; mÞ is continuous in the weak topology for measures m supported by K.
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Proof. Let fnn ; nX1g be a sequence of measures supported by K and weakly converging to m 2 M. The functional IðK; nn Þ can be written as Z Z IðK; nn Þ ¼ Enn rðy; XÞr dy ¼ Enn rðy; XÞr dy Z Z
K
K
d r=2
expfnn ðBt1=r ðyÞÞgð1 expfnn ðBt1=r ðyÞc ÞgÞ dt dy,
¼ K
0
ð8Þ
where Bt1=r ðyÞ is the ball centred at y 2 Rd of radius t1=r and superscript c denotes the complement in Rd . Since nn ðBt1=r ðyÞÞ ! mðBt1=r ðyÞÞ for almost all t and the integrand in (8) is bounded by 1, the dominated convergence theorem yields that IðK; nn Þ ! IðK; mÞ. & 3. Quantisation of the uniform measure The following result concerns the asymptotic behaviour of the functional I n ðKÞ for K ¼ I ¼ ½0; 1d being the unit cube in Rd . Theorem 4. The limit of I n ðIÞ as n ! 1 exists and is a positive finite number. Proof. For a fixed mX1, define k ¼ ½ðn=mÞ1=d . Partition the unit cube I into the (possibly closed or semi-open) d-dimensional cubes F 1 ; F 2 ; . . . ; F kd of side length kd . Consider a measure m with total mass m, which is Pkd optimal for IðI; mÞ, i.e. mr=d IðI; m Þ ¼ I m ðIÞ. For i ¼ 1; . . . ; kd , let mi ¼ m z1 ¯ ¼ i¼1 mi , where zF i : F i and m I ! F i is an affine bijection. Then I n ðIÞpn
Z
r=d
r
Em¯ rðy; XÞ dy ¼ n I
i¼1
kd Z X i¼1
kd Z X
pnr=d
r=d
Fi
Em¯ rðy; XÞr dy
Fi
Emi rðy; XÞr dy þ R1 eð1=2Þm .
The last inequality follows from Lemma 2 with R1 ¼
kd Z X i¼1
Fi
½n2r=d Eni rðy; XÞ2r 1=2 dy,
(9)
where ni ¼ m¯ mi , i ¼ 1; . . . ; kd . Since frðy; XÞ4tg ¼ fX \ Bt ðyÞ ¼ ;g, the support of ni is contained in I and pffiffiffi rðy; XÞp d for all y 2 I, the integrand in (9) can be bounded as Eni rðy; XÞ2r ¼ Eni ðrðy; XÞ2r 1Xa; Þ Z 1 ¼ Pni frðy; XÞ4t1=2r ; Xa;g dt 0 Z pffiffi 2r ð dÞ
p
0
expfni ðBt1=2r ðyÞÞg dt.
ð10Þ
Since the support of m is contained in I, mj is supported by the closure of F j , for j ¼ 1; . . . ; kd . By changing the variable s ¼ t1=2r , the right-hand side of (10) turns into pffiffi Z pdffiffi k1 Z ðpþ1Þ d =k X 2r1 expfni ðBs ðyÞÞgs ds ¼ 2r expfni ðBs ðyÞÞgs2r1 ds. 2r pffiffi 0
p¼0
p d =k
pffiffiffi pffiffiffi The summand with p ¼ 0 in the sum can be bounded from above by ð d =kÞ2r . Since sXp d =k for pX1, at least p 1 of the cubes that partition the unit cube and F i itself are covered by Bs ðyÞ. Therefore,
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ni ðBs ðyÞÞ ¼ Z
P
jai mj ðBs ðyÞÞXðp
1Þm and
pffiffi ðpþ1Þ d =k 2r1
expfni ðBs ðyÞÞgs
pffiffi p d =k
1127
pffiffiffi!2r d dsp ðp þ 1Þ2r1 eðp1Þm . k
Hence "
k1 X 1þ ðp þ 1Þ2r1 eðp1Þm
n2r=d R1 p 2rd 2r k r
p¼1
"
r 1=2
pð2rd Þ
!#1=2
k1 X ðm þ 1Þ 1 þ ðp þ 1Þ2r1 eðp1Þm
#1=2
r
.
p¼1
It is easy to see from the ratio test that the sum in the upper bound for R1 converges as k ! 1. Thus, R1 pcðm þ 1Þr for some constant c, so that I n ðIÞpn
kd Z X
r=d
i¼1
Fi
Emi rðy; XÞr dy þ cðm þ 1Þr eð1=2Þm
kd
¼
X
nr=d kdr
Em rðy; XÞr dy þ cðm þ 1Þr eð1=2Þm I
i¼1 r=d r
¼n
Z
k IðI; m Þ þ cðm þ 1Þr eð1=2Þm .
By choosing m sufficiently large we see that the upper limit of I n ðIÞ does not exceed I m ðIÞ þ e for any given e40. Thus, I n ðIÞ converges as n ! 1 to a non-negative real number denoted by I. To show that I40, notice that IðI; mÞ can be written in the form IðI; mÞ ¼ Em IðI; XÞXEm ½IðI; XÞ1jXjpn . Thus, I n ðIÞXnr=d inf IðI; XÞ Pfzn png, jXj¼n
where zn is Poisson distributed of mean n. Note that the infimum is taken over all deterministic X and so is the quantisation error of the uniform distribution. By Graf and Luschgy (2000, Theorem 6.2), the normalised quantisation error has a positive limit. It suffices to note that Pfzn png converges to a positive number. & 4. Quantisation of non-uniform measures Let h : Rd ! Rþ be a Riemann integrable function, so that h has a bounded support. Define Z hðyÞEm rðy; XÞr dy, Eðh; mÞ ¼ E n ðhÞ ¼ n
Rd r=d
inf
m2M;mðRd Þ¼n
Eðh; mÞ.
Theorem 5. The limit of E n ðhÞ exists and lim E n ðhÞ ¼ khkd=ðdþrÞ I.
n!1
(11) Pm
Proof. First show that (11) holds for the step-function hðyÞ ¼ i¼1 si 1F i ðyÞ, where mX1 is fixed and F 1 ; . . . ; F m partition the unit cube into sub-cubes of side lengths l 1 ; . . . ; l m . Define !1 m X d=ðdþrÞ d=ðdþrÞ ti ¼ si sj j¼1
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Pm and ni ¼ ½ti n for nX1 and i ¼ 1; . . . ; m. Note that i¼1 ni pn. For each i ¼ 1; . . . ; m, P choose 1mi to be a minimiser of IðI; mÞ over all measures with total mass ni , i.e. mi ðRd Þ ¼ ni . Define m¯ ¼ m i¼1 mi zF i and ni ¼ for all i ¼ 1; . . . ; m. Then m¯ mi z1 Fi Z m X r=d r=d E n ðhÞpn Eðh; mÞ si Em¯ rðy; XÞr dy ¯ ¼n i¼1
Fi
Z m X r=d pn si Em z1 rðy; XÞr dy þ R1 , i
Fi
i¼1
Fi
where R1 ¼ nr=d
m X
si eni =2
Z Fi
i¼1
½Eni rðy; XÞ2r 1=2 dy.
Since Eni rðy; XÞ2r is bounded for all i ¼ 1; . . . ; m and ni ¼ ½ti n, we have R1 ! 0 as n ! 1. The convergence holds even if ti ¼ 0 for some i, since then si ¼ 0 and the corresponding summand in the sum in R1 vanishes. Furthermore, Z m X si Em z1 rðy; XÞr dy þ R1 E n ðhÞpnr=d ¼ nr=d
i¼1 m X
Fi
si l rþd i
i
Z I
i¼1
pnr=d
m X
Fi
Emi rðy; XÞr dy þ R1
r=d
si l rþd ni i
I ni ðIÞ þ R1 .
i¼1
Therefore, lim E n ðhÞp
n!1
m X
si l rþd lim ðn=ni Þr=d lim I ni ðIÞ i n!1
i¼1
¼
m X
d=ðdþrÞ si
i¼1
¼
n!1
m X
!r=d d=ðdþrÞ sj
l rþd I i
j¼1
m X
!ðdþrÞ=d d=ðdþrÞ d si li
I ¼ khkd=ðdþrÞ I.
i¼1
To prove the reverse bound for the limit of E n ðhÞ as n ! 1, choose m to be a minimiser of Eðh; mÞ over all measures with the total mass n. Define mi to be the restriction of m onto F i and let ni ¼ mðF i Þ ¼ mi ðRd Þ. For 0obo1 and 0oaob=ðd 1Þ choose en ¼ na , nX1. Let F i;en and F ei n denote the inner and the outer cubes parallel to F i obtained by reducing (respectively enlarging) F i by en , so that their side lengths are l i en and n l i þ en , respectively. On each face of the cubes F i;2en , F i;en , F i , F ei n , F 2e choose independent homogeneous i b Poisson processes each having the intensity n with respect to the ðd 1Þ-dimensional Hausdorff measure of the faces. The superposition of these processes is the Poisson process Pgi with intensity measure gi . Then Z Z m m X X E n ðhÞXnr=d si Em rðy; XÞr dyXnr=d si Emþgi rðy; XÞr dy R2 , F i;2en
i¼1
i¼1
F i;2en
where R2 ¼ n
r=d
Z m X si i¼1
F i;2en
½Egi rðy; XÞ2r 1=2 emðR
d Þ=2
dypmsmax nr=d en=2
and smax ¼ maxðs1 ; . . . ; sm Þ. Note that R2 ! 0 as n ! 1.
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In the following we need a bound for the probability that the union of balls placed at the points of the point process Pgi covers the whole surfaces of the corresponding cubes. If P is the Poisson process on a ðd 1Þdimensional unit cube F 0 of intensity nb , then the probability that F 0 is not covered by the balls of radius en centred at every point of P satisfies b
PfF 0 is not coveredgpc minf1; ðnb þ ad2 nbðd1Þ Þen a g,
(12)
kd1 ed1 n
is the volume of the ðd 1Þ-dimensional ball of radius en , see Hall where c is a constant and a ¼ (1985, Theorem 3). n Denote by U i the event that all faces of F i;2en , F i;en , F i , F ei n or F 2e are covered by fBen ðxÞ; x 2 Pgi g. When i the event U i occurs, the set gi builds the firewall, i.e. the nearest point to y from Pmþgi necessarily belongs to the Poisson process Pmi þgi . Thus, Emþgi ðrðy; XÞr 1U i Þ ¼ Emi þgi ðrðy; XÞr 1U i Þ. Therefore, E n ðhÞXnr=d
Z m X si i¼1
F i;2en
i¼1
F i;2en
i¼1
F i;2en
Z m X ¼ nr=d si Z m X ¼ nr=d si
Emþgi ðrðy; XÞr 1U i Þ dy R2 Emi þgi ðrðy; XÞr 1U i Þ dy R2 Emi þgi ðrðy; XÞr Þ dy R2 R3 ,
where R3 ¼ n
r=d
Z m X si i¼1
F i;2en
Emi þgi ðrðy; XÞr 1U c Þ dypsmax mnr=d PðU ci Þ1=2 . i
By (12), the probability of the complement U ci is bounded by b
PðU ci Þp2dcðnb þ ad2 nbðd1Þ Þen a . Since ¼ kd1 nbaðd1Þ ! 1 nb a ¼ nb kd1 ed1 n we have n below by
q
PðU ci Þ
E n ðhÞXnr=d
! 0 as n ! 1 for every q40, whence R3 ! 0 as n ! 1. Therefore, E n ðhÞ is bounded from Z m X si i¼1
¼ nr=d
m X i¼1
Xnr=d
as n ! 1,
m X
F i;2en
Emi þgi ðrðy; XÞr Þ dy R2 R3
si ðl i 2en Þrþd
Z I
Eðmi þgi ÞzF i ðrðy; XÞr Þ dy R2 R3
si ðl i 2en Þrþd I ni þki ðIÞ R2 R3 ,
i¼1
where ki is the expected total number of points in gi , i.e. ki ¼ ½cnb . Since bo1, ki =n ! 0 as n ! 1. Therefore, r=d m X n rþd si l i lim I. lim E n ðhÞX n!1 n!1 ni þ k i i¼1 Choose a subsequence of n, such that ni =n ! vi 2 ½0; 1 as n ! 1 for all i ¼ 1; . . . ; m. Note that all vi ’s are positive; otherwise E n ðhÞ ! 1 as n ! 1, which contradicts the obtained upper bound on the limit of E n ðhÞ.
ARTICLE IN PRESS I. Molchanov, N. Tontchev / Statistics & Probability Letters 77 (2007) 1123–1132
1130
Therefore, m X
lim E n ðhÞX
n!1
i¼1
si l rþd lim i n!1
n ni þ k i
r=d I¼
m X
r=d
si l rþd vi i
I.
i¼1
By (Graf and Luschgy (2000, Lemma 6.8)), the right-hand side is minimised for vi ¼ ti , i ¼ 1; . . . ; m. Thus,
lim E n ðhÞX
n!1
m X
si
m P j¼1
r=ðdþrÞ si
i¼1
¼
m X
!r=d d=ðdþrÞ sj
l rþd I i
!ðrþdÞ=d
d=ðdþrÞ d si li
I ¼ khkd=ðdþrÞ I.
i¼1
A general function h from Theorem 5 can be approximated from above and below by two step-functions h phph . Since Eðh ; XÞpEðh; XÞpEðh ; XÞ, the obtained result for step-functions yields that the limit of E n ðhÞ lies between kh kd=ðdþrÞ and kh kd=ðdþrÞ . Because h is Riemann integrable, the both bounds can be made arbitrarily close to each other. & The following theorem describes the asymptotic behaviour of the normalised measures that minimise the objective functional Eðh; mÞ. Similarly to Lemma 3, it is easy to see that Eðh; mÞ is weak continuous, so that its infimum over all measures with a given total mass is achieved. An analogue of Lemma 1 shows that it is possible to assume that the minimising measure is supported by the closure of K. Theorem 6. Assume that h : Rd ! Rþ is Riemann integrable supported by K. If mn 2 M minimises Eðh; mÞ over all measures with total mass n and mn is supported by the closure of K, then n1 mn converges weakly as n ! 1 to the probability measure L with density proportional to hðyÞd=ðdþrÞ . Proof. Without loss of generality assume that K I. Assume that n1 mn converges weakly to some measure n and let n1 mn ðF 0 Þ ! nðF 0 Þ ¼ n1 for a cube F 0 , such that nðqF 0 Þ ¼ 0. Hence nðF c0 Þ ! 1 nðF 0 Þ ¼ n2 , where F c0 is the complement of F 0 in Rd . The idea of the proof is to show that L and n coincide on F 0 . Denote lðF Þ ¼ kh1F kd=ðdþrÞ . It is straightforward to check that lðF Þ ¼ LðF ÞðdþrÞ=d lðIÞ. Choose a Poisson process Pg with intensity proportional to nb as in the proof of Theorem 5 for the cubes a n F 0;2en , F 0;en , F 0 , F e0n , F 2e 0 , where en ¼ n , nX1. Then Z Z hðyÞEmn rðy; XÞr dy þ hðyÞEmn rðy; XÞr dy Eðh; mn Þ ¼ Fc
F0
Z X
F0
hðyÞEmn þg rðy; XÞr dy þ
Z0 Fc
hðyÞEmn þg rðy; XÞr dy R1 ,
0
where "Z R1 ¼ F0
hðyÞðEg rðy; XÞ2r Þ1=2 dy þ
#
Z Fc
hðyÞðEg rðy; XÞ2r Þ1=2 dy eð1=2Þn .
0
Note that the both integrals are finite and nr=d R1 ! 0 as n ! 1.
ARTICLE IN PRESS I. Molchanov, N. Tontchev / Statistics & Probability Letters 77 (2007) 1123–1132
1131
n Let U 0 denote the event that all faces of F 0;2en , F 0;en , F 0 , F e0n , F 2e 0 are covered by fBen ðxÞ : x 2 Pg g. Since 2en c c F 0;2en F 0 and ðF 0 Þ F 0 , Z Z r Eðh; mn ÞX hðyÞEmn þg rðy; XÞ dy þ hðyÞEmn þg rðy; XÞr dy R1
ðF 2en Þc
F 0;2en
0
Z
r
X F 0;2en
If
m0n
and
m00n
Z
hðyÞEmn þg ðrðy; XÞ 1U 0 Þ dy þ
ðF 2en Þc 0
hðyÞEmn þg ðrðy; XÞr 1U 0 Þ dy R1 .
are the restrictions of mn onto F 0 and F c0 , respectively, then
Emn þg ðrðy; XÞr 1U 0 Þ ¼ Em0n þg ðrðy; XÞr 1U 0 Þ;
y 2 F 0;2en ,
Emn þg ðrðy; XÞr 1U 0 Þ ¼ Em00n þg ðrðy; XÞr 1U 0 Þ;
n c y 2 ðF 2e 0 Þ .
Therefore,
Z
Eðh; mn ÞX F 0;2en
where
Z R2 ¼ p
Z
F 0;2en
F 0;2en
Z ðF 2en Þc
hðyÞEm00n þg rðy; XÞr dy R1 R2 ,
0
hðyÞEm0n þg ðrðy; XÞr 1U c Þ dy þ 0
Z ðF 2en Þc 0
hðyÞEm00n þg ðrðy; XÞr 1U c Þ dy 0
hðyÞðEm0n þg ðrðy; XÞ2r Þ1=2 dy Pm0n þg ðU c0 Þ1=2
Z
þ
hðyÞEm0n þg rðy; XÞr dy þ
ðF 2en Þc 0
hðyÞðEm00n þg ðrðy; XÞ2r Þ1=2 dy Pm00n þg ðU c0 Þ1=2
pc1 ðnb þ nbðd2Þ naðd1Þ Þen
baðd1Þ
for a certain constant c1 . Thus, nr=d R2 ! 0 as n ! 1. Let n1 ¼ m0n ðRd Þ and n2 ¼ m00n ðRd Þ and k be the total intensity of Pg . Since k=n ! 0 as n ! 1, r=d Z n ðn1 þ kÞr=d hðyÞEm0n þg rðy; XÞr dy lim nr=d Eðh; mn ÞX lim n!1 n!1 n1 þ k F0 r=d Z n ðn2 þ kÞr=d hðyÞEm0 0n þg rðy; XÞr dy. þ lim n!1 n2 þ k Fc 0
Since ðni þ kÞ=n ! ni 2 ð0; 1 for i ¼ 1; 2, lim E n ðhÞX lim nr=d Eðh; mn Þ
n!1
n!1 r=d r=d Xn1 lðF 0 ÞI þ n2 lðF c0 ÞI r=d r=d ¼ ðn1 LðF 0 ÞðdþrÞ=d þ n2 LðF c0 ÞðdþrÞ=d ÞlðIÞI XðLðF 0 Þ þ LðF c0 ÞÞlðIÞI ¼ lim E n ðhÞ. n!1
Therefore, n1 ¼ LðF 0 Þ and n2 ¼ LðF c0 Þ.
&
5. Conclusions An interesting open problem is to find explicitly the constant I from Theorem 4, i.e. the limit of I n ðIÞ as n ! 1. Note that I depends on r, the power of rðy; XÞ in (4). We conjecture that I ¼ lim nr=d IðI; nlÞ, n!1
(13)
ARTICLE IN PRESS 1132
I. Molchanov, N. Tontchev / Statistics & Probability Letters 77 (2007) 1123–1132
where l is the Lebesgue measure on I. Note that munif ¼ nl does not necessarily minimise nr=d IðI; mÞ among all n r=d measures of total mass n and therefore Iplimn!1 n IðI; nlÞ. Theorem 6 provides an argument in favour of (13), since the intensity mn minimising nr=d IðI; mÞ weakly converges to the uniform distribution. Numerical computations of the optimal Poisson measures using the steepest descent algorithm described in Molchanov and Zuyev (2002) (on the unit interval with r ¼ 2) have shown that the difference between n2 Ið½0; 1; munif n Þ and I n ð½0; 1Þ decreases to zero as n ! 1. It is also interesting to generalise (5) by replacing the distance function rðy; XÞ with a more general nonnegative function Zðy; XÞ, as it has been done in Molchanov and Tontchev (2007) for the case of deterministic X. For instance, Zðy; XÞ may be defined to be the volume of the Voronoi cell generated by X and containing y, or it may represent the error of approximation of a convex surface by its convex downward triangulation. The main difficulty here is caused by possible bad (or exceptional) configurations X which result in the infinite values for Zðy; XÞ and the resulting expectation IðK; mÞ. For instance if Zðy; XÞ is the volume of the Voronoi cell and K ¼ I, then with a positive probability there appear configurations of points with at least one unbounded Voronoi cell. Recall that in case of the quantisation error given by (4) these exceptional configurations are only those which result from the empty X, and it is much easier to control the probability of their appearance as it has been done in Lemma 2. References Cohort, P., 2004. Limit theorems for random normalized distortion. Ann. Appl. Probab. 14, 118–143. Daley, D.J., Vere-Jones, D., 1988. An Introduction to the Theory of Point Processes. Springer, New York. Delattre, S., Graf, S., Luschgy, H., Page`s, G., 2004. Quantization of probability distributions under norm-based distortion measures. Statist. Decision 22, 261–282. Ellis, S.P., 1986. A limit theorem for spatial point processes. Adv. Appl. Probab. 18, 646–659. Graf, S., Luschgy, H., 2000. Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730, Springer, Berlin. Gruber, P.M., 2004. Optimum quantization and its applications. Adv. Math. 186, 456–497. Hall, P., 1985. On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13, 991–1002. Molchanov, I., Tontchev, N., 2007. Optimal approximation and quantisation. J. Math. Anal. Appl. 325, 1410–1429. Molchanov, I.S., Zuyev, S.A., 2000. Variational analysis of functionals of a Poisson process. Math. Oper. Res. 25, 485–508. Molchanov, I.S., Zuyev, S.A., 2002. Steepest descent algorithms in space of measures. Statist. Comput. 12, 115–123. Penrose, M.D., Yukich, J.E., 2003. Weak laws in geometric probability. Ann. Appl. Probab. 13, 277–303.