Choquet weak convergence of capacity functionals of random sets

Information Sciences 177 (2007) 3239–3250

www.elsevier.com/locate/ins

Choquet weak convergence of capacity functionalsof random sets

Ding Feng, Hung T. Nguyen *

Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, USA

Received 13 November 2006; accepted 13 November 2006

Abstract

The results in this paper are about the convergence of capacity functionals of random sets. The motivation stems fromasymptotic aspects in inference and decision-making with coarse data in biostatistics, set-valued observations, as well asconnections between random sets with several emerging uncertainty calculi in intelligent systems such as fuzziness, belieffunctions and possibility theory. Specifically, we study the counter-part of Billingsley’s Portmanteau Theorem for weakconvergence of probability measures, namely, convergence of capacity functionals of random sets in terms of Choquetintegrals.� 2006 Elsevier Inc. All rights reserved.

Keywords: Random sets; Capacity functionals; Weak convergence

1. Introduction

Statistical analysis of coarse data can be carried out in the framework of random sets, see e.g., [5,9,10].Coarse data appear, for example, in statistical inference with missing data in multivariate analysis, censoringdata in survival analysis, and in grouped data in general, see [7,8,16,17]. However, unlike the situations forrandom vectors, developments of distributional results for random sets are not yet at a satisfactory level(see, e.g., [10]). Our present work aims at studying the convergence in distribution of random sets using theChoquet integral. Specifically, via their capacity functionals, we study the counter-part of Billingsley’s Port-manteau Theorem for weak convergence of probability measures, namely, convergence of capacity functionalsof random sets.

Roughly speaking, a random closed set is a random element in the space of all closed sets of the basic set-ting space. In the classical theory of random sets, the setting space is assumed to be a locally compact, second-countable, Hausdorff (LCSCH) space, see e.g., [9,11,12].

0020-0255/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2006.11.011

* Corresponding author. Tel.: +1 505 646 2105; fax: +1 505 646 1064.E-mail addresses: [email protected] (D. Feng), [email protected] (H.T. Nguyen).

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3240 D. Feng, H.T. Nguyen / Information Sciences 177 (2007) 3239–3250

Throughout this paper, U is always considered as an LCSCH space. Especially, the Euclidean space Rm is anice example of LCSCH spaces. Let F ¼ fall closed sets in Ug, K ¼ fall compact sets in Ug,G ¼ fall open sets in Ug, K0 ¼K� f£g, and BðUÞ ¼ fall Borel sets in Ug.

Let BðFÞ be the r-algebra generated by classes FK , K 2K and FG, G 2 G, where

FK ¼ fF 2F : F \ K ¼£g; FG ¼ fF 2F : F \ G 6¼£g:

Define a random closed set as a measurable mapping S : ðX;R; PÞ ! ðF;BðFÞÞ from a probability spaceinto a measurable space. Let Q be the probability measure induced on BðFÞ, that is, for each B 2 BðFÞ,Q(B) = P(S�1(B)).

Note that the distribution of a random set S is uniquely determined by its hitting functional TS on K suchthat

T SðKÞ ¼ P ðfx : SðxÞ \ K 6¼£gÞ; K 2K: ð1Þ

It is easy to prove that TS defined above satisfies the following properties:

(T1) TS is upper semi-continuous (u.s.c) on K, i.e.,

Kn & K in K ) T SðKnÞ & T SðKÞ;
(T2) TS(B) = 0 and 0 6 TS 6 1;(T3) TS is monotone increasing on K and for K1;K2; . . . ;Kn 2K, n P 2,
T S

\ni¼1

Ki

!6

X£6¼I�f1;2;...;ng

ð�1ÞjIjþ1T S

[i2I

Ki

!:

Let C denote the class of all the functionals T on K satisfying (T1)–(T3) above. Such functionals are calledalternating Choquet capacities of infinite order or, for brevity, simply capacity functionals.

In the LCSCH space U, the capacity functional T can be extended on Borel sets [9]) by

T ðBÞ ¼ supfT ðKÞ : K 2K;K � Bg for B 2 BðUÞ: ð2Þ

Then we get that

(T1 0). Upper semi-continuity of T on G means that

Gn % G in G ) T SðGnÞ % T SðGÞ:

We will use the following lemma in our main results.

Lemma 1.1 (Choquet’s Theorem). Every probability measure Q on ðF;BðFÞÞ determines a Choquet capacity

functional T on K through the correspondences

(C1) T ðKÞ ¼ QðFKÞ; 8K 2K, and

(C2) T ðGÞ ¼ QðFGÞ; 8G 2 G.

Conversely, every Choquet capacity functional T on K determines a unique probability measure Q on (F, B(F))that satisfies above conditions (C1) and (C2).

A compact set K 2K is called a continuity set for the T iff T(K) = T(Int(K)). Thus a compact set K is T-continuous if and only if FK is Q-continuous. Let ST ¼ fK 2K : T ðKÞ ¼ T ðIntðKÞÞg.

Results on the weak convergence of random closed sets in U have been discussed in many papers, see, e.g.,[6,10,13–15]. The problem we are going to address is this. Since the analysis of random sets is much simpler interms of their capacity functionals Tn, T, it is desirable to establish the convergence in distribution in terms ofthe Tn’s and T, rather than the Qn’s and Q. The following lemma is given by [10].

D. Feng, H.T. Nguyen / Information Sciences 177 (2007) 3239–3250 3241

Lemma 1.2. Let U be an LCSCH space. Then the following statements are equivalent:

(a) Qn)W

Q;

(b) limnT nðAÞ ¼ T ðAÞ; 8A 2ST .

In order to study the counter-part of Billingsley’s Portmanteau Theorem [1,2], this will first require anappropriate definition of weak convergence for capacity functions (which are generalizations of probabilitymeasures on ðU ;BðUÞÞ).

Let Cb(U) be the space of all bounded real-valued continuous functions on U. For f 2 Cb(U), the Choquet

integral of f with respect to T is

Zf dT :¼

Z þ1

0

T ðff P tgÞdt þZ 0

�1½T ðff P tgÞ � T ðUÞ�dt:

We say that the sequence of capacity functionals Tn converges, in the Choquet weak sense, to the capacity T in

U, ifR

f dT n !R

f dT ; 8f 2 CbðUÞ, in symbols, T n )C�W

T as n!1.

2. Preliminaries

The capacity functional T on the setting space U has the following properties (see [9]):

(a) T ðGÞ ¼ supfT ðKÞ : K 2K;K � Gg; G 2 G;(b) T ðKÞ ¼ inffT ðGÞ : G 2 G;G � Kg; K 2K.

Let Jf denote the hit-or-miss topology on F generated by two classes FK , K 2K, and FG, G 2 G. Nowobserve that F, with the hit-or-miss topology, is compact and metrizable, see e.g., [3,4,9]. Thus, the conver-gence in distribution of a sequence of random closed sets Sn to S can be studied in the weak topology on thismetric space.

A subset B � U is called T-continuous functional closed set if B = {x : f(x) P a} for some a 2 R, f 2 Cb(U)such that

T ðfx : f ðxÞP agÞ ¼ T ðfx : f ðxÞ > agÞ:

For a given capacity T, a family of capacity functionals A � C is said to be T-tight if for any T-continuous

functional closed set B and for each e > 0, there exists a compact set KB � U such that

supR2A½RðBÞ � RðB \ KBÞ� < e:

Especially, since U is T-continuous functional closed set for any given T, there exists a compact set KU � U suchthat

supR2A½RðUÞ � RðKU Þ� < e:

Lemma 2.1. Let T 2 C and f 2 Cb(U). Then
Z b
aT ðff P tgÞdt ¼

Z b

aT ðff > tgÞdt 8a; b 2 R:

Moreover, we have
Zf dT ¼
Z kf k

�kf kT ðff P tgÞdt � T ðUÞkf k ¼

Z kf k

�kf kT ðff > tgÞdt � T ðUÞkf k:


Proof. To see this, simply note that

Z b

aT ðff P tgÞdt ¼ lim

n!1

Z bþ1n

aþ1n

T ðff P tgÞdt ¼ limn!1

Z b

aT f P t þ 1

n

� �� dt 6

Z b

aT ðff > tgÞdt

6

Z b

aT ðff P tgÞdt:

ThusR b

a T ðff P tgÞdt ¼R b

a T ðff > tgÞdt. h

From this lemma, we know that {f P t} is a T-continuous functional closed set almost everywhere ont 2 (�kfk � 1,kfk + 1).

Put W ¼ fB 2 BðUÞ : B 2Kg. Say that a class L �W is separating if for any K 2K and G 2 G withK � G, there exists some A 2L such that K � A � G. The following proposition is a conclusion of Theorem2.1 in Norberg [13].

Proposition 2.2. Suppose U is an LCSCH space. Let S1,S2, . . . be random closed sets in U,T1,T2, . . . the

corresponding hitting functions, and Q1,Q2, . . . the corresponding probability distributions. If there exists a

separating class L �W and a capacity functional R such that

RðIntðAÞÞ 6 lim infn!1

T nðAÞ 6 lim supn!1

T nðAÞ 6 RðAÞ 8A 2L

then there exists a random set S in U with the probability distribution Q and hitting functional T satisfying

Qn)W

Q and T = R on K.

3. Choquet weak convergence

The main result presented here is that, for the case where U is LCSCH, T n )C�W

T is equivalent to the con-vergence in distribution of corresponding random sets with T-tightness of the sequence fT ng1n¼1. In thesequence Qn, Q denote probability measures on BðFÞ, and Tn, T denote corresponding hitting capacity func-tionals. Now we give the following example first.

Example 3.1. Let U ¼ R1. Take F n : R1 ! ½0; 1� defined by

F nðxÞ ¼x

1þx

� �nif x P 0;

0 if x < 0:

(

Then Fn(x) is the distribution of a random variable Xn. Since Fn(x) N 0 for each x 2 R1, Xn is not weak con-vergent. Let Tn(K) be the capacity functional of the random closed set Sn = {Xn}. It is easy to seeT nðKÞ ! 0; 8K 2K by

0 6 T nðKÞ 6 F nðmaxðKÞÞ ! 0:

Let Qn denote probability measures on BðFÞ determined by Tn and Q be the probability measure on BðFÞdetermined by the capacity 0. Since T nðKÞ ! 0; 8K 2K, we get Qn)

WQ by the above Lemma 1.2. But T n )

C�W0

does not hold. Indeed, let f(x) = 1. Then for 0 < t < 1, we have

T nðff P tgÞ ¼ 1:

So we obtain

Zf dT n ¼

Z 1

0

T nðff P tgÞdt ¼ 1


and
Zf d0 ¼
Z 1

0

0ðff P tgÞdt ¼ 0:

Thus T n )C�W

0 does not hold.

Then we will give following two lemmas.

Lemma 3.2. If T n )C�W

T , then

lim supn!1

T nðBÞ 6 T ðBÞ 8B 2F:

Proof. Let £ 6¼ B 2F. By the definition of the capacity T (e.g., see [9, p. 30]), there exists a decreasingsequence fGmg � G such that Gm& B and T(Gm)& T(B). Hence B and UnGm are disjoint closed sets in U.By Urysohn’s lemma, there exists a continuous function fm : U! [0, 1] such that fm(x) = 0 for x 2 UnGm,and fm(x) = 1 for x 2 B. Note that

ffm > tg ¼£ for t P 1; ffm > tg ¼ U for t < 0:

Moreover for 0 6 t < 1, it implies

B � ffm > tg � ffm > 0g � Gm: ð3Þ
By Lemma 2.1, we know Z
fm dT n ¼Z 1

0

T nðffm > tgÞdt; n ¼ 1; 2; . . .

and
Zfm dT ¼
Z 1

0

T ðffm > tgÞdt:

Since T n )C�W

T1, we obtain

lim supn!1

T nðBÞ ¼ lim supn!1

Z 1

0

T nðBÞdt 6 lim supn!1

Z 1

0

T nðffm > tgÞdt ¼ lim supn!1

Zfm dT n ¼

Zfm dT

¼Z 1

0

T ðffm > tgÞdt:

By using (3), we get

lim supn!1

T nðBÞ 6 lim infm!1

Z 1

0

T ðffm > tgÞdt 6 limm!1

Z 1

0

T ðGmÞdt ¼ T ðBÞ;

which completes the proof. h


T , then

lim infn!1

T nðGÞP T ðGÞ 8G 2 G:

Proof. Let £ 6¼ G 2 G. By the property of the capacity T (e.g., see [9]), there exists a increasing sequencefKmg �K such that Km% G and T(Km)% T(G). Hence Km and UnG are disjoint closed sets in U. By Ury-sohn’s lemma, there exists a continuous function gm : U! [0,1] such that gm(x) = 0 for x 2 UnG, andgm(x) = 1 for x 2 Km. Simply note that {gm P t} = B for t > 1, {gm P t} = U for t 6 0. Moreover, for0 < t 6 1, it implies

G � fgm > 0g � fgm P tg � Km: ð4Þ


All this means that
Zgm dT n ¼
Z 1

0

T nðfgm P tgÞdt; n ¼ 1; 2; . . .

and
Zgm dT ¼
Z 1

0

T ðfgm P tgÞdt:

By T n )C�W

T and the monotone property of the capacity functionals, we obtain

lim infn!1

T ðGÞ ¼ lim infn!1

Z 1

0

T nðGÞdt P lim infn!1

Z 1

0

T nðfgm P tgÞdt ¼ lim infn!1

Zgm dT n ¼

Zgm dT

¼Z 1

0

T ðfgm P tgÞdt:

By (4), we imply

lim infn!1

T nðGÞP lim supm!1

Z 1

0

T ðfgm P tgÞdt P lim supm!1

Z 1

0

T ðKmÞdt ¼ T ðGÞ

yielding the desired result. h

It is easy to show that any family of finitely many capacity functionals is T-tight for any given T. By Lem-mas 3.2 and 3.3, however, we conclude the following result:

Theorem 3.4. If T n )C�W

T , then the sequence {Tn} is T-tight.

Proof. Since U is LCSCH, there exists a sequence of open sets fCqg1q¼1 such that Cq 2K; 8q P 1, andCq% U. Let B = {x : f(x) P a} be a T-continuous functional closed set with

T ðfx : f ðxÞP agÞ ¼ T ðfx : f ðxÞ > agÞ:

By the upper semi-continuity of T, [Cq \ {x : f(x) > a}]% {x : f(x) > a} implies

T ðCq \ fx : f ðxÞ > agÞ % T ðfx : f ðxÞ > agÞ:

For each e > 0, there exists a q0 such that

0 6 T ðfx : f ðxÞ > agÞ � T ðCq \ fx : f ðxÞ > agÞ < e28q P q0:

By Lemmas 3.2 and 3.3, we have

lim supn!1½T nðBÞ � T nðCq \ BÞ� 6 lim sup

n!1½T nðBÞ � T nðCq \ fx : f ðxÞ > agÞ�

6 lim supn!1

T nðBÞ � lim infn!1

T nðCq \ fx : f ðxÞ > agÞ 6 T ðBÞ � T ðCq \ fx

: f ðxÞ > agÞ ¼ T ðfx : f ðxÞ > agÞ � T ðCq \ fx : f ðxÞ > agÞ < e2:

Since any family of finitely many capacity functionals is T-tight, we obtain that the sequence {Tn} is T-tight. h

Lemma 3.5. If Qn)W

Q and the sequence fT ng1n¼1 is T-tight, then

lim infn!1

Zf dT n P

Zf dT 8f 2 CbðUÞ:


Proof. For any given f 2 Cb(U), Lemma 2.1 implies

Zf dT n ¼

Z kf k

�kf kT nðff > tgÞdt � T nðUÞkf k 8n P 1;

Zf dT ¼

Z kf k

�kf kT ðff > tgÞdt � T ðUÞkf k:

Note that Fff>tg is open in the F, with the hit-or-miss topology. Since Qn)W

Q, Billingsley’s Portmanteau The-orem (e.g., see [1]) implies

lim infn!1

QnðFff>tgÞP QðFff>tgÞ 8t 2 ð�kf k; kf kÞ:

On the other hand, for each e > 0, by the T-tightness of the sequence fT ng1n¼1, there exists a compact K � U

such that

supn½T nðUÞ � T nðKÞ� < e:

Simply note that FK is closed in the F, with hit-or-miss topology. For Qn)W

Q, by Billingsley’s PortmanteauTheorem, we obtain lim supn!1QnðFKÞ 6 QðFKÞ. By using Choquet’s Theorem and Fatou’s Lemma, weobtain

lim infn!1

Zf dT n P lim inf

n!1

Z kf k

�kf kT nðff > tgÞdt � lim sup

n!1T nðUÞkf k

P lim infn!1

Z kf k

�kf kQnðFff>tgÞdt � lim sup

n!1½T nðKÞ þ e�kf k

PZ kf k

�kf klim inf

n!1QnðFff>tgÞdt � lim sup

n!1QnðFKÞkf k � ekf k

PZ kf k

�kf kQðFff>tgÞdt � QðFKÞkf k � ekf k;

i.e.,

lim infn!1

Zf dT n P

Z kf k

�kf kQðFff>tgÞdt � QðFU Þkf k � ekf k ¼

Z kf k

�kf kT ðff > tgÞdt � T ðUÞkf k � ekf k

¼Z

f dT � ekf k:

Since e is arbitrary, it implies

lim infn!1

Zf dT n P

Zf dT : �

Lemma 3.6. If Qn)W

Q and the sequence fT ng1n¼1 is T-tight, then

lim supn!1

Zf dT n 6

Zf dT 8f 2 CbðUÞ:

Proof. For any fixed f 2 Cb(U), the decreasing function T({f P t}) is Riemann integrable on [�kfk,kfk]. ByLemma 2.1, {f P t} is T-continuous functional closed set almost everywhere on t 2 (�kfk�1,kfk+1). For eache > 0, there exists a subdivision of [�kfk,kfk]

�kf k ¼ t0 < t1 < t2 < � � � < tj < tjþ1 < � � � < tm ¼ kf k


such that

(a) {f P tj} is T-continuous functional closed set for j = 0,1, . . . ,m � 1;(b) the following inequality holds

Xm�1

j¼0

T ðff P tjgÞðtjþ1 � tjÞ � T ðUÞkf k <Z

f dT þ e:

Since the sequence fT ng1n¼1 is T-tight, there exists a compact set K � U such that

supn½T nðff P tjgÞ � T nðff P tjg \ KÞ� < e; j ¼ 0; 1; . . . ;m� 1:

Note that FU is open and Fðff Ptjg\KÞ is closed in the F, with hit-or-miss topology. For Qn)W

Q, by Billingsley’sPortmanteau Theorem, we obtain

lim infn!1

QnðFUÞP QðFU Þ; lim supn!1

QnðFðff Ptjg\KÞÞ 6 QðFðff Ptjg\KÞÞ:

Hence, monotone increasing of each Tn and Choquet’s Theorem imply

lim supn!1

Zf dT n ¼ lim sup

n!1

Z kf k

�kf kT nðff > tgÞdt � T nðUÞkf k

" #

6 lim supn!1

Xm�1

j¼0

Z tjþ1

tj

T nðff P tgÞdt

" #� lim inf

n!1Qn FUð Þkf k

6 lim supn!1

Xm�1

j¼0

T nðff P tjgÞðtjþ1 � tjÞ � QðFU Þkf k

6 lim supn!1

Xm�1

j¼0

½T nðff P tjg \ KÞ þ e�ðtjþ1 � tjÞ � T ðUÞkf k

6 lim supn!1

Xm�1

j¼0

½QnðFðff Ptjg\KÞÞ�ðtjþ1 � tjÞ þ 2ekf k � T ðUÞkf k

6

Xm�1

j¼0

QðFðff Ptjg\KÞÞðtjþ1 � tjÞ þ 2ekf k � T ðUÞkf k

¼Xm�1

j¼0

T ðff P tjg \ KÞðtjþ1 � tjÞ þ 2ekf k � T ðUÞkf k:

So we obtain

lim supn!1

Zf dT n 6

Xm�1

j¼0

T ðff P tjg \ KÞðtjþ1 � tjÞ þ 2ekf k � T ðUÞkf k

6

Xm�1

j¼0

T ðff P tjgÞðtjþ1 � tjÞ � T ðUÞkf k þ 2ekf k <Z

f dT þ eþ 2ekf k:

Since e is arbitrary, we obtain lim supn!1R

f dT n 6R

f dT . h

By summary of Lemmas 3.5 and 3.6, we obtain

Theorem 3.7. If Qn)W

Q and the sequence fT ng1n¼1 is T-tight, then T n )C�W

T .

It turns out that the converse of Theorem 3.7 holds, since


T , then Qn)W

Q.


Proof. We will, at first, prove that the class ST �W is separating. In fact, let K 2K and G 2 G with K � G.Since U is an LCSCH space, by the compactness of K, there exists an open set O � U such that O 2K andK � O � G. Note that K and UnO are disjoint closed sets in U. By Urysohn’s lemma, there exists a continuousfunction f : U! [0, 1] such that f(x) = 0 for x 2 UnO, and f(x) = 1 for x 2 K. Simply note that

K � ff P tg � O � G 8t 2 ð0; 1Þ: ð5Þ
By Lemma 2.1, T({f P t}) = T({f > t}) is almost everywhere on t 2 (0,1). Choose B = {f P t0} for somet0 2 (0, 1) with T({f P t0}) = T({f > t0}). Keeping in mind that {f > t0} � Int(B) � B, we getT(B) = T(Int(B)) = T({f > t0}). Moreover, B ¼ B � O 2K implies B 2K and therefore B 2 ST . Thus theclass ST �W is separating by (5).
On the other hand, by Lemmas 3.2 and 3.3, T n )C�W

T implies, for any K 2K,

lim supn!1

T nðKÞ 6 T ðKÞ;

lim infn!1

T nðIntðKÞÞP T ðIntðKÞÞ:

Thus, for K 2ST ,

T ðIntðKÞÞ 6 lim infn!1

T nðIntðKÞÞ 6 lim infn!1

T nðKÞ 6 lim supn!1

T nðKÞ 6 T ðKÞ:

By Proposition 2.2, there exists a random set S 0 with the probability distribution P and the corresponding hit-

ting functional R satisfying Qn)W

P and R = T on K. By Choquet’s Theorem, we have Q = P, i.e. Qn)W

Q. h

Theorem 3.9. Let U be an LCSCH space. Then the following statements are equivalent:

(a) T n )C�W

T ;

(b) Qn)W

Q and the sequence fT ng1n¼1 is T-tight;

(c) limnT nðKÞ 6 T ðKÞ for every compact set K in U, limnTn(G) P T(G) for every open set G in U and the

sequence fT ng1n¼1 is T-tight;

(d) limnTn(A) = T(A) for each A 2ST , and the sequence fT ng1n¼1 is T-tight.

Corollary 3.10. Let (U,d) be a compact metric space. Then the following statements are equivalent:

(a) T n )C�W

T ;

(b) Qn)W

Q;

(c) limnT nðKÞ 6 T ðKÞ for every compact set K in U and limn Tn(G) P T(G) for every open set G in U;(d) limn Tn(A) = T(A) for each A 2ST .

4. Equivalent conditions to the weak convergence in distribution of random sets

Let CþS ðUÞ denote the space of all nonnegative real-valued, continuous functions on U with compactsupports.

Lemma 4.1. Let U be an LCSCH space. If Qn)W

Q, then

limn!1

Zf dT n ¼

Zf dT 8f 2 CþS ðUÞ:

Proof. For any given f 2 CþS ðUÞ, by definition, we know that {x 2 U : f(x) 5 0} � K for some K 2K. Simplynote that Lemma 2.1 implies

Zf dT n ¼

Z kf k

0

T nðff P tgÞdt ¼Z kf k

0

T nðff > tgÞdt;

Zf dT ¼

Z kf k

0

T ðff P tgÞdt ¼Z kf k

0

T ðff > tgÞdt:


For each t 2 (0,kfk), the closed set {f P t} � K is compact in the LCSCH space U, so the setFff Ptg ¼F�Fff Ptg is closed in the space F with the hit-or-miss topology. Moreover Fff>tg is open inthe space F. On the other hand, we know

oðFff PtgÞ ¼Fff Ptg � IntðFff PtgÞ �Fff Ptg �Fff>tg:

Then Choquet’s Theorem implies

0 6

Z kf k

0

Q½oðFff PtgÞ�dt 6Z kf k

0

QðFff Ptg �Fff>tgÞdt ¼Z kf k

0

½T ðff P tgÞ � T ðff > tgÞ�dt ¼ 0:

All this means that Fff Ptg is a Q-continuous set almost everywhere on t 2 (0,kfk). Note the facts that jQnj 6 1,jQj 6 1, and the given condition Qn)

WQ. By using the Lebesgue dominated convergence theorem and Billings-

ley’s Portmanteau theorem (see [1]), we imply

limn!1

Zf dT n �

Zf dT

� ¼ lim

n!1

Z kf k

0

½T nðff P tgÞ � T ðff P tgÞ�dt

¼ limn!1

Z M

0

½QnðFff PtgÞ � QðFff PtgÞ�dt

¼Z M

0

limn!1½QnðFff PtgÞ � QðFff PtgÞ�dt ¼ 0: �

Let U ¼ ðRm; dÞ be the Euclidean space. For any K 2K0, j = 1,2, . . ., define WjK : Rm ! ½0; 1� by

WjKðxÞ :¼ ½1� jdðx;KÞ� _ 0. Then Wj

K 2 CþS ðRmÞ.

Theorem 4.2. Let U ¼ ðRm; dÞ. Then the following statements are equivalent:

(a) Qn)W

Q;

(b) limn!1R

f dT n ¼R

f dT ; 8f 2 CþS ðUÞ;(c) limn!1

RWj

K dT n ¼R

WjK dT ; 8K 2K0; j ¼ 1; 2; . . .;

(d) limn!1R

WjK dT n ¼

RWj

K dT ; 8K 2 ST n f£g; j ¼ 1; 2; . . .;(e) limnT nðKÞ 6 T ðKÞ for every compact set K in U and limnTn(G) P T(G) for every open set G in U;

(f) limnT nðKÞ ¼ T ðKÞ; 8K 2 ST .

Proof. From Lemma 4.1, we obtain (a)! (b). The proofs of (b)! (c), (c)! (d) and (e)! (f) are trivial. ByBillingsley’s Portmanteau Theorem (e.g., see [1]), we can directly get (a)! (e). The proof of (f)! (a) is byLemma 1.2. Now we only need to show (d)! (a).

In fact, suppose that limn!1R

WjK dT n ¼

RWj

K dT ; 8K 2ST n f£g; j ¼ 1; 2; . . . Since the space F, withthe hit-or-miss topology, is a compact metric space, the space of all Borel probability measures is weaklycompact. Hence, for each subsequence fQn0 g

1n0¼1 of the sequence fQng

1n¼1, there exists a further subsequence

fQn0ig1i¼1 weakly converging to some Borel probability measure P on F. Let R be the corresponding hitting

capacity functional on K determined by P. By Lemma 4.1, we know that Qn0i)W

P implies limi!1R

WjK dT n0i

¼RWj

K dR; 8K 2 SR n f£g; j ¼ 1; 2; . . ., i.e., we have the equalityR

WjK dR ¼

RWj

K dT ; 8K 2 ST\SR n f£g; j ¼ 1; 2; . . . By Lemma 4.3 below, we imply T = R on K. So, Choquet’s Theorem impliesQ = P, i.e. Qn0i

)W

Q. Thus we obtain Qn)W

Q by Theorem 2.3 in Billingsley [1]. h

Lemma 4.3. Let U ¼ ðRm; dÞ, and T, R be two capacity functionals on K. IfR

WjK dR ¼

RWj

K dT ; 8K 2ST\SR n f£g; j ¼ 1; 2; . . ., then T = R on K.

Proof. Firstly, we will prove that T = R on ST \SR. In fact, if £ 6¼ K 2ST \SR n f£g, then

K � fx : WjKðxÞP tg ¼ x : qðx;KÞ 6 1� t

j

� �8t 2 ð0; 1�: ð6Þ


This means that fx : WjKðxÞP tg is compact in U and fx : Wj

KðxÞP tg & K as j!1. Hence the Lebesguedominated convergence theorem and the upper semi-continuity of R and T, with

RWj

K dR ¼R

WjK dT , imply

RðKÞ ¼Z 1

0

limj!1

RðfWjK P tgÞdt ¼ lim

j!1

ZWj

K dR ¼ limj!1

ZWj

K dT ¼Z 1

0

limj!1

T ðfWjK P tgÞdt ¼ T ðKÞ:

Thus T = R on ST \SR. It remains to show that T = R on K.Indeed, for any £ 6¼ K 2K, take f : Rm ! ½0; 1� defined by f(x) = [1 � d(x,K)] _ 0, x 2 Rm. Then Lemma

2.1 implies
Z 1
0

T ðfx : f ðxÞP tgÞdt ¼Z 1

0

T ðfx : f ðxÞ > tgÞdt:

This means that T({x : f(x) P t}) = T({x : f(x) > t}) almost everywhere in t 2 (0,1). Thus

0 6 T ðfx : f ðxÞP tgÞ � T ðIntfx : f ðxÞP tgÞ 6 T ðfx : f ðxÞP tgÞ � T ðfx : f ðxÞ > tgÞ:
Since {x : f(x) P t} = {x : d(x,K) 6 1 � t} is compact in U, we imply fx : f ðxÞP tg 2ST almost everywherein t 2 (0, 1). Similarly, we obtain fx : f ðxÞP tg 2SR almost everywhere in t 2 (0,1). Hencefx : f ðxÞP tg 2ST \SR almost everywhere in t 2 (0, 1). Choose a sequence ftig1i¼1 � ð0; 1Þ such thatfx : f ðxÞP tig 2ST \SR, i P 1 and ti " 1 as i!1. Therefore we have {x : f(x) P ti}& K as i!1. Bythe previous part of the proof, we obtain that R({x : f(x) P ti}) = T({x : f(x) P ti}), i P 1. Thus the uppersemi-continuity of R and T implies
RðKÞ ¼ limi!1

Rðfx : f ðxÞP tigÞ ¼ limi!1

T ðfx : f ðxÞP tigÞ ¼ T ðKÞ: �

Example 4.4. Let U ¼ ðRm; dÞ. Suppose {Sn} is a sequence of i.i.d. random closed sets in U with the same dis-tribution as the random closed set S : ðX;R; P Þ ! ðF;BðFÞÞ. Let QS be the probability measure of S and TS

be the corresponding hitting capacity (capacity functional). For estimation of TS, we use the empirical capac-ity functional T x

n ðKÞ, defined by

T xn ðKÞ ¼

1

n

Xn

i¼1

1K\SiðxÞ6¼£; K 2K;

where

1K\SiðxÞ6¼£ ¼1 if K \ SiðxÞ 6¼£;

0 otherwise:

�

Let Qn be the probability measure of S determined by T xn . We claim that Qn)

WQS almost surely (a.s.). To

see this, let B be a countable base of U and L denote the class of sets of all finite unions of the family B. ThenL is countable. By the strong law of large numbers, T x

n ðBÞ ! T SðBÞ a. s. as n!1 for each given B 2L.Hence the countability of L implies

Pfx : T xn ðBÞ ! T SðBÞ 8B 2Lg ¼ 1:

We only need to show that Qn)W

QS if T nðBÞ ! T SðBÞ 8B 2L .Indeed, the local compactness of U implies that L is a separating class of W. To complete the proof, simply

note that limn!1Tn(B) = T(B) for every B 2L. Hence we obtain, for each A 2L,

T SðIntðAÞÞ 6 T SðAÞ ¼ lim infn

T nðAÞ ¼ limn

T nðAÞ ¼ lim supn

T nðAÞ ¼ T SðAÞ 6 T SðA�Þ:

By Proposition 2.2, there exists a random set S 0 with the probability distribution P and the corresponding hit-ting functional T satisfying Qn)

WP and T = TS on K. By Choquet’s Theorem, we have QS = P, i.e. Qn)

WQS .

5. Conclusions

By definition, a random object is characterized by the corresponding probability measure on the set U ofpossible objects, i.e., by a mapping P from a set A � 2U of (measurable) sets to the unit interval [0, 1], a map-ping that describes, for each set A, the probability P(A) that a randomly selected object belongs to this set.


This definition is intuitively clear but its literal implementation is difficult. Indeed, for a set U of n possibleobjects, we have 2n possible sets A � U; thus, to describe a probability measure P(A), we need to store 2n prob-abilities corresponding to 2n possible sets. It is well known that for discrete probability distributions, it is suf-ficient to store n� 2n values – e.g., n probabilities of individual objects.

A random closed set is a particular case of a random object – an object whose possible values are (closed)subsets X � U0 of the basic set U0. For the basic set U0 consisting of n elements, we have 2n possible subsets X,so the set U of possible objects has 2n elements. Thus, there are 22n

subsets A � U, and to describe the corre-sponding probability distribution, we need to list an astronomical number 22n

of probabilities P(A). It isknown that we can drastically decrease the needed number of values if instead of using the original probabil-ities P(A), we instead store, for every (closed) set K � U0, the probability T ðKÞ ¼def PðA \ K 6¼ ;Þ that a randomset A has a non-empty intersection with K. This function is called (Choquet) capacity functional. For a set U0

with n elements, storing the values of T(K) means storing 2n � 22n

values. It is known that if we know T(K),then we can uniquely reconstruct the original probability measure. However, it is not always true that closeT(K) lead to close probability measures, i.e., in precise terms, convergence of capacity functionals is not alwaysequivalent to convergence of the original probability measures. Since the capacity functional representation isextremely important, it is necessary to find conditions under which, e.g., the weak convergence of probabilitymeasures is equivalent to the weak convergence of their capacity functionals (called weak Choquetconvergence).

In this paper, we obtain several sufficient and necessary conditions under which the weak Choquet conver-gence is equivalent to the corresponding weak convergence in distribution of random sets.

Acknowledgements

We would like to thank three referees for their constructive suggestions which led to a significant improve-ment of this paper.

References

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Choquet weak convergence of capacity functionals of random sets

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