Stochastic Orders Induced随机订单的诱导.doc

Stochastic Order Induced by a Measurable PreorderbyDavid C. NachmanDepartment of FinanceJ. Mack Robinson College of BusinessGeorgia State UniversityAtlanta, Georgia 30303-3083Phone: 404-651-1696Fax: 404-651-2630E-mail: November, 2005Abstract. Kamae, et. al. 8, Theorem 1 presents a general characterization of the partial ordering of probability measures induced by a closed partial ordering on the underlying Polish state space. A preorder is a reflexive and transitive, but not necessarily antisymetric relation. This paper presents a similar characterization of the preordering of probability measures induced by a measurable preordering on the underlying Polish space. We then apply this result to obtain a characterization of stochastic majorization, the preorder induced by the widely applied majorization preorder on Euclidean space. We show that a multifunction associated with this preorder is compact and convex valued and continuous, and hence satisfies the hypotheses of our characterization. The continuity properties of the majorization ordering and the induced stochastic majorization ordering have not been widely recognized and are of interest in their own right.Kamae, et. al. 8, Theorem 1 presents a general characterization of the partial ordering of probability measures induced by a closed partial ordering on the underlying Polish state space. A preorder is a reflexive and transitive, but not necessarily antisymetric relation, and so is a weaker kind of relation than a partial order. The intention here is to present a similar characterization of the preordering of probability measures induced by a measurable preordering on the underlying Polish space. Marshall and Olkin 9, Ch. 17.C already hint that Kamae, et. al. 8, Proposition 1 is “. . . a much more general version involving general preorders.” Specifically, we exploit the properties of the preorder and theorems of Strassen 11 and Himmelberg and Van Vleck 6 to provide a Kamae, et. al. like characterization of the induced stochastic preorder. We then apply this result to obtain a characterization of stochastic majorization, Marshall and Olkin 9, Ch. 11, the preorder induced by the widely applied majorization preorder on Euclidean space. We show that a multifunction associated with this preorder is compact and convex valued and continuous, and hence satisfies the hypotheses of our characterization. The continuity properties of the majorization ordering and the induced stochastic majorization ordering have not been widely recognized and are of interest in their own right.A Borel space is a Borel subset a Polish space, a complete separable metric space. A multifunction from to a set is a function with domain with value a nonempty subset of , for each . It is Borel measurable if is a Borel set in for each closed subset of . Let be a Polish space. Throughout is assumed to be endowed with a preorder, denoted by , a reflexive and transitive relation on . For each , let . By reflexivity, , and by transitivity, implies that . Thus is a multifunction from to . We say the preorder is a measurable preorder if the multifunction is Borel measurable. We assume that is a compact valued and Borel measurable. For compact valued multifunctions, Borel measurability has many equivalent definitions. See 7, Theorem 3.Let , denote the set of probability measures on , respectively on , endowed with the topology of weak convergence. In this case, are both Polish spaces (in the Prohorov metric, 2, Theorem 6.8). For denote by the support of , the smallest closed subset of with measure one, 3, p. 18. For each , let and let be the probability measure such that . Theorem 1. For each , and implies that . is a compact and convex valued Borel measurable multifunction from to . Proof. The first statement follows from the same properties of . Thus is a multifunction. For each , is convex, closed and tight. By Prohorovs theorem, 2, Theorem 5.1, is compact. That is Borel measurable follows from Himmelberg and Van Vlect 6, Theorem 3 (ii)Let denote the collection of real valued Borel measurable functions on that are increasing (nondecreasing) in the preorder on , i. e., for , the real line, Borel measurable, if and only if for and , . We extend the preorder to as follows. For , say that is larger in this extended preorder than , and write , if and only if for every for which both these integrals exist. Then our extension of to is truly an extension since for , in if and only if in . Intuitively, if puts more weight on elements that are less extreme in the relation on than does . This intuition is formalized by the characterization of on given below (Theorem 2).This definition of on is typical in the literature on stochastic orderings (8 and 9, 17.A.3, but there are many others. See 10, Chs. 1, 4. There are many definitions given in terms of valued random quantities say and . For example, one version of stochastic majorization of interest here is the relation in 9, Ch. 11. This relation, denoted , is stated as , stochastically majorizes in the sense of , if for all for which these expectations exist. In 9, Ch. 11, is the cone of Borel measurable Schur convex functions. It is easy to see that this is equivalent to the above definition since these expectations are given by integration with respect to the distributions in of these random quantities and given these distributions there are valued random quantities with these distributions. There is a another definition of stochastic majorization that Marshall and Olkin 9, pp. 282-283 call that implies and appears ostensibly to be stronger than . There if for all , where is the typical meaning of stochastically larger, 10, 1.A. Clearly implies since stochastically larger random variables have larger expectations. It turns out that in this particular case we also have implies . See the argument in 9, top of p. 283. We will use this argument to show one part of the characterization of the relation in defined above. Here the orders and are the orderings as defined above, but in the abstract setting of this paper.Let denote the Borel subsets of . A Markov kernel on is a map such that for each set the map is Borel measurable and for fixed . For such a Markov kernel and a probability measure denote by the element of defined by , for measurable rectangles, . We say that the first marginal of is and denote the second marginal . Finally, we say that a set is increasing if its indicator function belongs to (necessarily, then ). These designations are borrowed from 8, pp. 899-900. The following is the desired characterization of on and flushes out the intuition given above.Theorem 2. For the following are equivalent:(i) ;(ii) There exists a Markov kernel on such that and , almost every ;(iii) There exists a probability measure with with first marginal and second marginal ;(iv) There exists a real valued random variable and two measurable functions with (i. e.,) such that the distribution of is and the distribution of is ;(v) There exist valued random variables and such that and the distribution of is and the distribution of is ;(vi) for every increasing set .Proof: The key equivalence is (i) and (ii). The rest follow easily. Let and assume that (i) holds. For every bounded continuous function define . By Theorem 1 and 7, Theorem 2, is Borel measurable in , and for each , since and . Thus is bounded as well, so all integrals below exist. Finally, , for if and then by Theorem 1, , and hence . It follows that , the last inequality from (i). Condition (ii) then follows from 11, Theorem 3.Assume (ii) and let . Since is Borel measurable, its graph is a Borel subset of , 7, Theorem 3. Then , since the -section for every . This gives (iii). Therefore assume (iii). The construction in 8, Theorem 1(iii) goes through here as well and this gives (iv). Assuming (iv) let and . Then clearly and (v) follows from the fact that , since satisfies 9, (3) p. 283. Therefore assume (v). If and the indicator, then , which is (vi), where the inequality follows from the fact that .It remains to show that (vi) imples (i). Assume (vi). For , for all real . It follows from (vi) that . Since , (the equalities here hold by 1, (4), p. 223). Also implies that , and we get that . As a result, (the equalities here again by 1, (4), p. 223). If both the integrals exist, it follows that , which is (i)The crucial implication (i) implies (ii) in Theorem 2 relies on Theorem 3 of Strassen 11. In obtaining essentially the same implication Kamae, et. al. 8, Theorem 1, (i) implies (ii) and 9, 17.B.1 use Theorem 11 of Strassen 11, which requires that the graph of the multifunction be closed. Our assumption of measurability of yields the weaker condition that is a measurable set. Of course if is closed, as it is in the majorization application to follow, then for the measure in Theorem 2(iii), .All that is required to use Theorem 3 of 11 is that the preorder be sufficiently regular to give Borel measurability in of the function , defined above in the first paragraph of the proof of Theorem 2. Some measurability of the multifunction seems essential to this result, but weaker conditions than compact valuedness of may give the result. See for example 5, Proposition 3, p.60. Theorem 2 of 7, however, is very handy. Transitivity of the preorder gives monotonicity of in the preorder. Reflexivity of the preorder gives and is convex whether is or not. This convexity is essential to apply 11, Theorem 3, but it comes at no cost. The result Theorem 2(ii) formalizes our intuition expressed above that if puts more weight on elements that are less extreme in the relation on than does . The Markov kernel of Theorem 2(ii) is such that for almost every , . In this sense shifts weight of to elements less extreme in the relation on . Borrowing from the language suggested in 8, top of p. 900, the kernel might be termed “downward.”Let us now consider the application to majorization. Let and be n-tuples of real numbers and let and denote the vectors and with coordinates rearranged in decreasing order, i. e., and . The vector is majorized by the vector (or majorizes ), written , if for each , with equality holding for , 9, A.1, p. 7.In words, is majorized by if the components of are more evenly spread out than the components of or the components of are more concentrated than the components of . This intuition is reinforced by noting the following. Let , the n-tuple whose coordinates are all equal to one. Then for a vector the inner product is the sum of the components of . Let and let , where appears in the component. The vectors , , and all have the same total sum of components, but the components of are more evenly spread out than those of . Clearly concentrates this sum in one component. In this sense, is the most evenly spread of this sum of components and is the most concentrated of this sum. Indeed, we have that , . We note that the majorization relation is reflexive and transitive (established below in Lemma 5) and hence is a preordering. It is not a partial ordering, however, since it is not antisymmetric. Indeed it is symmetric in that , where is any permutation of .Let denote n-dimensional Euclidean space. All topological properties in the sequel will be with respect to the usual metric on . Let denote the set of permutation matrices and let denote the set of doubly stochastic matrices. Then if and only if there is one one in each row and each column of and all other entries are zero. Similarly, if and only if the entries in are nonnegative and each row and each column sum to one.Theorem 3. For , the following are equivalent:(i) ;(ii) , some ;(iii) , for some , , and some .Proof: The equivalence of (i) and (ii) is due to Hardy, Littlewood and Polya. See 9, Theorem 2.B.2. The equivalence of (ii) and (iii) is due to Birkhoff. See 9, Theorem 2.A.2 For each let , the set of n-tuples that are majorized by . For a picture of this set in the case n = 3 see 9, Figure 3, p. 9. Let , the graph of the multifunction . In the following, the notions of upper and lower semicontinuity in 6 are the same as the notions of upper and lower hemi-continuity in 5, which gives convenient characterizations of these notions in terms of sequences in . We will use the terminology of 6, but refer to these convenient characterizations in 5. A multifunction is continuous if it is both upper and lower semicontinuous. The following are properties of .Theorem 4. is a compact convex valued continuous multifunction in . Consequently is closed in .Proof: Clearly for each , , so is a multifunction. is convex, by Theorem 3(ii) (convex combinations of doubly stochastic matrices are doubly stochastic) and compact since, by Theorem 3(iii), it is the convex polyhedron generated by the finite number of permutations of .Suppose and with . If , by Theorem 3(ii), , some . Then again by Theorem 3(ii), and , so is lower semicontinuous, 5, Theorem 2, p. 27. Let with arbitrary. By Birkhoffs theorem is the convex polyhedron generated by the permutation matrices and hence is compact in . Thus there is a subsequence of the that converges to an element . For this subsequence indexed by , . Thus is upper semicontinuous, 5, Theorem 1, p. 24. The closure of then follows by the same resultAs obvious as these properties of are, except for convexity of , they appear nowhere prominently in the literature on majorization to this authors knowledge. Eaton and Perlman 4 do exploit the upper semicontinuity of in the proof of their Lemma 4.1. The following result establishes the transitivity of majorization and is needed later for the characterization of stochastic majorization.Lemma 5. For , if , then .Proof: For , suppose and . Then by Theorem 3(ii) and some . But then and , 9, 2.A.3, p. 20. Again by Theorem 3(ii), As in the abstract case, for each , let . Let , the graph of the multifunction .Theorem 6. is a compact and convex valued continuous multifunction from to . The graph is closed in .Proof: For , . This result is then Theorem 1 above, but specialized to the example here. By 6, Theorem 3(i), inherits the continuity of established in Theorem 4. The closure of follows from 5, Theorem 1, p. 24The functions that are increasing (non-decreasing) in the majorization relation are called Schur-convex. See 9, Ch. 1.D, Ch. 3 for the origins of this terminology and the characterizations of this class of functions. Denote by the class of Borel measurable Schur-convex functions. The measurability requirement is a restriction, 9, 3.C.4, p. 70. We can extend the relation in to a relation in as we did above by taking . This relation in is the version of stochastic majorization studied in 9, Ch. 11. Here the relations and are the relations in 9, Ch. 11. Let denote the Borel sets in and call Schur convex if . The following characterization of in results.Theorem 7. For the following are equivalent:(i) ;(ii) There exists a Markov kernel on such that and , almost every ;(iii) There exists a probability measure with with first marginal and second marginal ;(iv) There exists a real valued random variable and two measurable functions with (i. e.,) such that the distribution of is and the distribution of is ;(v) T