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VNU JOURNAL OF SCIENCE, Mathematics - Physics T.XIX, Ng3 - 2003 ON WEAK CONVERGENCE IN THE SPACE OF PROBABILITY CAPACITIES IN R‘* Nguyen Nhuy Vietnam National University, Hanoi Le Xuan Son Vinh university Abstract In this paper we are going to extend the weak convergence in the space of probability measures to the one in the space of probability capacities and investigate the relation on weak convergence between probability capacities and their associated probability measures Introduction As well known that integration theory could be founded on order and monotonicity Indeed it turned out that many aspects of integration theory are sustained if additivity is replaced by order and monotonicity It is due to Choquet [2] who was lead to the problem from his research in electrostatics and potential theory Then Choquet’s results had been applied to several research areas, including Artificial Intelligence, Mathematical Economics and Bayesian statistics, (14] and particularly to the areas of upper and lower probabilities, (see, e.g., [15] for an introduction to their use in these areas) In a such way, non-additive set functions, known as capacities, have become uncertainty measures in situations where probability measures not seem to be approriate In the last twenty five years the study of non-additive set functions is useful in interval computations where interval probabilities present uncertainty, and has been carried out by several authors In (10] we have introduced the notion of capacites in R?, where R¢ dimensional Euclidean space with the ordinary metric p(z,y) = 20 denotes d- — ¡)?]Ÿ, and constructed the Choquet integral for these capacities Then the weak topology on the space of probability capacities in R¢ was investigated in 11] We first recall some definitions and facts used in this paper, the details can be found in [10] and [11] Let K(R*), F(R*), G(R*), B(R*) denote the family of all compact sets, closed sets, open sets and Borel sets in R4, respectively T: By a capacity in R¢ we mean a set function R4-+ Rt = (0, +00) satisfying the following conditions: (i) TO) = 0; *This work was supported by the National Science Council of Vietnam Typeset by Ay4s-TEX 46 Nguyen (ii) T is alternating of infinite order: we have Nhuy, For any Borel sets A,, Le Xuan i= 1,2, ,n: Son n> 2, a TI Ads DO (=0#“#7(J4), i=l where Z( Đo (iii) T(A) (iv) si I€7(n) {Ic {1, ,n}, #0} ier and #/ denotes the cardinality of J; = sup{T(C) : C € K(R4), C Cc A} for any Borel set A € B(R4); = inf{T(G): G € G(R*), C Cc G}, for any compact set C € K(R*) A capacity in R¢ is, in fact, a generalization of a measure in R% Clearly any capacity is a non-decreasing set function on Borel sets of R¢ By T(R4\S) support of a capacity = The support T we mean the smallest closed set S C of a capacity T is denoted by supp T We R% such that say that T is a probability capacity in R¢ if T has a compact support and T(supp T) = By C we denote the family of all probability capacities in R¢ Let A€ T be a capacity in R? Then for any measurable function f : R4 > R*+ and B(R®), the function f4 : R > R defined by FA( =T({z€< A: f(x) > t}) forteR is a non-increasing function in t Therefore we can define the Choquet integral Sa faT of ƒ+ with respect to 7' by [ser [ fa(t)dt =ƒ If lạ fdT < œ, we say that f is integrable fsa T({œ€ A: ƒ(œ) > t})dt In particular for A = R#, we write [ ser: Observe that if f is bounded, then « | faT = | T({ee A: f(a) > t)át, A where a = sup{ f(a): x € A} In the general case if f : R¢ > R is a measurable function, then we define [ra= [rare f par, where f*(r) = max{f(x),0} and f~(r) = max{—f(z), 0} In this note we are going to extend the weak convergence in the space of probability measures to the one in the space of probability capacities and study the relation on the weak convergence is organized between as follows probability capacities and probability measures In section we give a version of the Portmanteau The paper Theorem, On weak in which convergence in the space the space of all bounded of continuous aT real valued functions space of all continuous non-negative In Section a generalization of Portmanteau 3, we prove is replayed by the real valued functions with compact Theorem support in R¢ in the case of space of probability capacities in R? The main result of this paper is presented in Section 4, where the relation on weak convergence between probability capacities and their associated measures is investigated A version of portmanteau Let P be a measure on B(R‘) P(R®) theorem Then the measure P is called the probability measure if = By slight modification of Billingsley’s definition {1] of weak convergence, we say that a sequence of probability measures P, converges weakly to a probability measure P if f fdP, + f fdP for every f € Cf (R*), where Cy (R*) denotes all non-negative real valued continuous functions with compact support in R¢ From the definition of a weakly convergent sequence and the property of a convergent sequence, it follows immediately the following proposition 2.1 Proposition P, convergences weakly to P if and only if each subsequence {Py} contains a further subsequence {Py} such that Py» converges weakly to P Following [1], we say that a set A C R¢ is called T-continuity set if its boundary 0A satisfies T(0A) = In this section we prove the following theorem, which is a version of the Portmanteau Theorem in situation where every bounded continuous real valued function is replayed by non-negative real valued continuous function with compact support and provides useful conditions equivalent to weak convergence for probability measures 2.2 Theorem Let P,P, be probability measures on B(R*) Then the following state- ments are equivalent (i) Pa converges weakly to P; (ii) limsup,, Pn(F) < P(F) for all closed F; () liminfa Pa(G) > P(G) for all open G; (iv) limnoo Pn(A) = P(A) for all P-continuity set A We prove Theorem 2.1 by establishing the implications in the following diagram (i) + (4) © (iii) and (i) © (iv) Proof of (i) (ii) Let F € F(R‘) and ¢ > Then there exists G € G(R*),G > F such that P(G) < P(F) +e (2.1) 48 Nguyen Nhuy, Le Xuan Son Suppose that frg : R¢ > (0, 1] is a continuos function such that fre(z) = { ifreF ifr¢G Thus function exists by Urysohn-Tietze Theorem Then by (2.1) we get Fy(Œ)= Í feedF, < | lardP, ¬ [ IeedP F = [ fredP < P(G) < PIF) +6, G which implies lim sup Pn(F) < P(F) +c Since e was arbitrary, the assertion follows Prooƒ oƒ (ñä)—> (0) Suppose (ii) holds We first show that for any real-valued continuous f with compact support, we have timsup f sar, < [ sar By addition in f a constant if necessary, we may assume (2.2) that f(x) > for all z e R4, Then we have there exists Go € G(R4),Go D K such that T(Go) < T(K) +€ We put ifreK fx(z) = { ifagK and suppose that fx,¢) : R* > [0,1] is a continuos function such that ifz€K Fi.Go(2) = { ife¢Go Thus function fxg, exists by the Urysohn-Tietze Theorem Clearly ƒwœạ Cg (R*) We have T„(K) = J Jv4T, < J fxGotTn > J Saget < [ T(Go)dt = (Go) Hence lim sup T,(K) a < Mm, Í /.euđT, = [nem nà T(G) -« Let ƒwạ,ø : R# — [0,1] be a continuos function such that ifz€K Jxee(#) = { ife¢Go Then fog € Co (R) We have T,(G) > | fxuodf, ¬ | fr.odT Hence : lim inf T,(G) > jim, | fr.oaTs = [em >T(Ko) > T(G) -€ Since € was arbitrary, we get lim inf 7, (G) > T(G) n (iii) Suppose A is a bounded T-continuity in R¢ Let int A and A denote the interior of A and the closure of A, respectively Note that A is compact By Lemma 3.1 we have T(A) =T(int ALJAA) = Tint A) If T,, converges weakly to T, then (i) and (ii) hold set functions, we have Since T,, and T ' T(A) > limsup T,(A) > lim sup Tn(A) > lina T,(A) > lim inf Tint A) > T(int A), which, together with (3.1) implies tim Ta(A) = T(A) (3.1) are increasing On weak convergence 3.3.Problem in the space of 53 Docs cach of three statements (i), (ii), and (iii) of Theorem 3.2 imply the weak convergence of capacities? 4, Weak convergence of capacities and of their associated Let E be a locally compact separable Hausdorff space Let K,F,G measures denote the classeses of all compact, closed and open subsets of E, respectively Following Matheron Fa={F |9], we topologize F as follows EF: F(\AÂ0} The miss-and-hit topology on F(Rđ) {FE where is the topology with the base ¢,:K €K and G}, ,Gn € 9}, FE It was shown For every A C E we denote and FA = {Fe F: F()A=0} ø, =F) Fe () - in {9} that for a locally compact Faun en separable Hausdorff space E, the space F with the miss-and-hit topology is compact, Hausdorff and separable family of all Borel sets of F in the miss-and-hit By Choquet Theorem, Let B(F) denote the topology there exists a bijection between probability measures P on B(F) and probability capacities T : € — |0, +ee) satisfying the equality P(Fx) = T(K) for every K € K In this case we say that the probability measure P is associated with the probability capacity T Now T,Tn,n we take E to be a compact = 1,2, , let P,Pa,n set K in R¢ For given probability capacities = 1,2, denote their associated probability measures To prove the main result in this section, we need the following Lemma 4.1 Lemma Let T,5 be the probability capacities in C such that [tarThen S = T From on B(R“), [tas for allf € Cf (R*) i.e., T(A) = S(A) for all A € B(R?) this follows that {Tn} C Ê can not converge weakly to two different limits at the same time Proof Because of the equality T(A) = sup{T(K) : K € K(R‘),K c A} 54 Nguyen Nhuy, Le Xuan Son for any Borel set A € B(R*), it suffices to show that T(K) = S(K) for all K e K(R*) If it is not the case, then T(K) # S(K) for some compact set K € K(IR4) Assume that T(K) — S(K) = > Then there exists an open set G D K such that S(G) < S(K) + ; (4.1) Let fx,g(x) € Cf (R42) such that ifeeKk fale) = { ifx¢G Then by (4.1) we have [ tras < 5(G) < S(K)) + 56 This contradiction completes the proof QO In this section we prove the folollowing theorem 4.2 Theorem The sequence of probability capacities T,, converges weakly to T if and only if the associated sequence of probability measures P,, converges weakly to P Proof Assume that P, converges weakly to P Let f € Co (IR4) We put œ =sup{ƒ(z):z € R“} < œ Note that if fị,fạ € (0œ) are distinct, then ÔZ{/>¡) and ØZ(/»/„ÿj hence at most countably many of them can have positve P-measure are disjoiat, a P-continuity set almost everywhere on (0.q), i.e., [ ` P|Ø(Fqyxa)jwt =0 Hence, by Theorem 2.2 and Lebesgue’s bounded convergence Theorem, we have din f fats = f sar) = jim [erates > 9) - TU > t)je im, [ PalZu>a) — PFiseni és and Therefore, Z,/>¿) is =ƒ a lim [Pa(#(y>e)) = P(#g xa) lát =0 On weak convergence in the space of 55 Conversely, assume that 7;, converges weakly to T We will show that P,, converges weakly to P Since the space F(K) by Theorem with the miss-and-hit topology is a compact 6.4 [13] for each subsequence {Py} such that metric space, there exists a futher subsequence {Py} j fdPa» > J JdP for every ƒ € C,(R2) and for some probability measure P! on Z(), where Cạ(R#) denotes the space of all bounded continuos real valued functions on R# In particular Tham ice.,Pav converges weakly > [tar to P’, Hence Tn» for everyf € Cf (R*), converges weakly to T’ By Lemma 4.1 we have T=T"' Therefore, by Choquet Theorem we get P=P That means J fdPa J fdP for every fe Ct(R4) Consequently, P, converges weakly to P by Proposition 2.1 oO References P Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York Chichester Brisbane ww G Choquet, S Graf, Toronto, 1968 Theory of Capacities, Ann A Radon - Nikodym Inst Fourier (1953 - 1954), 131- 295 Theorem for Capacities, Mathematik, 320 (1980) 192 - 214 Measure P J Huber and V Strassen, Minimar Capacities, Ann und Angewandte Theory, Springer-Verlag New York Inc, 1974 — P R Halmos, J Reine Test and The Neyman-Pearson Lemma for Statist., Vol.1, No.2, 251-263 ° Hung T Nguyen, Nhu T Nguyen and Tonghui Wang, On Capacity Functionals in Interval Probabilities To appear in International Journal of Uncertainty, Fuzziness and Knowledge-Based _ Hung T Nguyen Systems ,2001 and Tonghui Wang, On Convergence of Possibility Measures, To appear in International Journal of Uncertaity, Fuzziness and Knowlege-Based Systems,2002 Nguyen œ J B Kadane and Wasserman, Symmetric, Coherent, Nhuy, Choquet Le Xuan Capacities, Son Ann Statist Vol.24, No 3, 1996, 1250-1264 oo Matheron, G Random Sets and Integral Geometry, J Wiley, 1975 10 Nguyen Nhuy and Le Xuan Son, Probability Capacities in R¢ and Choquet Integral for Capacities,To appear in Acta Math Viet (2003) bơi 11 Nguyen Nhuy and Le Xuan Son, The weak Topology Capacities in R¢ , To appear in Viet J Math 12 T Norberg, Random Capacities and on the Space of Probability (2003) their Distributions, Prob Theory Relat Fields, 73 (1986), 281-297 13 K R Parthasarathy, Probability Measures on Metric Spaces ,Academic Press New York and London(1967) 14 D, Schmeidler, Sujective Probability and Ecpected Utility without Additivity, Econo- metrica, 57 (1989), 571-587 P Walley, Statistical Reasoning with Imprecise New York, 1991 Probabilities, Chapman and Hall, ... version of the Portmanteau The paper Theorem, On weak in which convergence in the space the space of all bounded of continuous aT real valued functions space of all continuous non-negative In. .. measure on B(R‘) P(R®) theorem Then the measure P is called the probability measure if = By slight modification of Billingsley’s definition {1] of weak convergence, we say that a sequence of probability. .. completes the proof QO In this section we prove the folollowing theorem 4.2 Theorem The sequence of probability capacities T,, converges weakly to T if and only if the associated sequence of probability