Functional BKR Inequalities, and their Duals, with Applications

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1 Functional BKR Inequalities, and their Duals, with Applications Larry Goldstein and Yosef Rinott University of Southern California, Hebrew University of Jerusalem April 26, 2007 Abbreviated Title: Functional BKR Inequalities Abstract The inequality conjectured by van den Berg and Kesten in [9], and proved by Reimer in [6], states that for A and B events on S, a finite product of finite sets, and P any product measure on S, P (A B) P (A)P (B), where the set A B consists of the elementary events which lie in both A and B for disjoint reasons. This inequality on events is the special case, for indicator functions, of the inequality having the following formulation. Let X be a random vector with n independent components, each in some space S i (such as R d ), and set S = n i=1 S i. Say that the function f : S R depends on K 1,..., n if f(x) = f(y) whenever x i = y i for all i K. Then for any given finite or countable collections of non-negative real valued functions f A, g B on S, depending on K, A and L, B respectively, sup f (X)g (X) K L = sup f (X) sup g (X) Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth [4], are also considered. Applications include order statistics, assignment problems, and paths in random graphs. Key words and phrases: graphs and paths, positive dependence, order statistics. 1 Introduction For x = (x 1,..., x n ) S, where S = n i=1 S i any product space, and K = k 1,..., k m n := 1,..., n with k 1 <... < k m, define x K = (x k1,..., x km ) and [x] K = y S : y K = x K, 0 AMS 2000 subject classifications. Primary 6015 Research supported by Israel Science Foundation Grant No. 473/04. 1

2 the restriction of x to the indicated coordinates, and the collection of all elements in S which agree with x in those coordinates, respectively. For A, B S we say that x A, y B disjointly if there exist and denote K, L n, K L = such that [x] K A and [y] L B, (1) A B = x : x A, x B disjointly. (2) The operation A B corresponds to elementary events which are in both A and B for disjoint reasons in the sense that inclusion in A and B is determined on disjoint sets of coordinates. Theorem 1.1 was conjectured in van den Berg and Kesten [9]. It was proved in [9] for A and B increasing sets and S = 0, 1 n, and it was also demonstrated there that Theorem 1.1 follows from its special case S = 0, 1 n. Using the latter fact, the conjecture was established in general by Reimer [6]. Theorem 1.1 For P = n i=1 P i any product measure on S = n i=1 S i, S i finite, P (A B) P (A)P (B). (3) Many useful formulations can be found in van den Berg and Fiebig [8], in addition to the following motivating example which appeared earlier in [9]. Independently assign a random direction to each edge e = v i, v j of a finite graph, with p e (v i, v j ) = 1 p e (v j, v i ) the probability of the edge e being directed from vertex v i to v j. With V 1, V 2, W 1, W 2 sets of vertices, Theorem 1.1 yields that the product of the probabilities that there exist directed paths from V 1 to V 2 (event A) and from W 1 to W 2 (event B) is an upper bound to the probability that there exist two disjoint directed paths, one from V 1 to V 2 and another from W 1 to W 2 (event A B). The main thrust of this paper is to show how Theorem 1.1 implies inequalities in terms of functions, of which (3) is the special case of indicators, and similarly for the dual inequality of [4]. These functional inequalities, and their duals, are stated in Theorems 1.2 and 1.5, and their proofs can be found in Section 3. Applications to order statistics, allocation problems, and random graphs are given in Section 2. Specializing to monotone functions, we derive related inequalities and stochastic orderings in Section 4; these latter results are connected to those of Alexander [1]. For each i = 1,..., n, let (S i, S i ) be measurable spaces, and set S = n i=1 S i and S = n i=1 S i, the product sigma algebra. Henceforth, all given real valued functions on S, such as f, g, A, B are assumed to be (S, B) measurable where B denotes the Borel sigma algebra of R, and functions on S with values in 2 n, such as K(x) in inequality (d) of Theorem 1.2 below, are assumed to be (S, 2 2n ) measurable. Measurability issues arise in definitions (10), (4), and (17), and are settled in Section 5. We also show in Section 5 that Theorem 1.2 applies to the completion of the measure space (S, S) with respect to the measure P appearing in the theorem; similarly for Theorem 1.5. For K n we say that a function f defined on S depends on K if x K = y K implies f(x) = f(y). The inequalities in Theorems 1.2 and 1.5 require one of two frameworks, the first of which is the following. 2

3 Framework 1 f (x) A and g (y) B are given collections of non-negative functions on S, such that f, g depend respectively on subsets of n K, L in K = K A and L = L B, where A and B are finite or countable. The elements of K and L are not assumed to be distinct; we may have, say, K = K γ for some γ and f f γ. Note also that if a function depends on K, it depends on any subset of n containing K. For notational brevity we may write for K ; for example, we may use as an abbreviation for K L, and also x for x K. The second framework is Framework 2 f and g are two given non-negative functions, and K and L any collections of subsets of n. With P a probability measure on (S, S) define for K K, L L, f K (x) = ess inf y [x] K f(y), and g L (x) = ess inf y [x] L g(y). (4) Our functional extension of the BKR inequality (3) is Theorem 1.2 Let X = (X 1,..., X n ) S be a random vector and P a probability measure on (S, S) under which X 1,..., X n are independent. 1. Under framework 1, sup f (X)g (X) = sup f (X) sup g (X). (a) 2. Under framework 2, max f K L= K (X)g L (X) f(x) g(x). (b) K K,L L The special case of (b) where K = L = 2 n, the collection of all subsets of n, clearly implies the inequality in general. Theorem 1.2 applies to a vector X having independent coordinates taking values in any measure space. In previous work the operation was defined only on discrete finite product spaces, that is, a finite product of finite sets. Note, however, that definition (2) can be applied on any product space. Further, with f(x) and g(x) the indicator functions of A and B respectively, we can equivalently express A B as the set whose indicator function is given by 1 A B (x) = max f (x)g (x), (5) K L= K L where f K, g L are given in (4). Therefore, inequality (b) of Theorem 1.2 specialized to the case that K = L = 2 n and f and g are indicators says that the original BKR inequality for events in finite spaces extends to more general product spaces. The following is a straightforward generalization of Theorem 1.2, stated here only for inequality (a). Note that in (6) below, as m increases the pairwise constraints i j = become more restrictive, and the inequality less sharp. 3

4 Theorem 1.3 Let X S be a random vector with independent coordinates. Then for given finite or countable collections of non-negative functions f i, Ai depending on K i, Ai, i = 1,..., m, sup ( 1,...,m) A 1 Am k l =, k l m m f i,i (X) i=1 i=1 sup f i, (X). (6) A i Next we describe an inequality of Kahn, Saks, and Smyth [4], which may be considered dual to the BKR inequality (3), and then provide a function version. We use a notation compatible with (3). With disjointly defined in (1), denote A B = (x, y) : x A, y B disjointly. The following, which we call the KSS inequality, is dual to Theorem 1.1 and is given in [4]. Theorem 1.4 If P denotes the uniform measure over 0, 1 n, then for any (A, B) 0, 1 n 0, 1 n, (P P )(A B) P (A B). (7) Our functional extension of the KSS inequality is as follows. Theorem 1.5 Let X = (X 1,..., X n ) S be a random vector, P any probability measure on (S, S) such that X 1,..., X n are independent, and Y an independent copy of X. 1. Under framework 1, 2. Under framework 2, sup f (X)g (Y) = max K L= K K,L L sup f (X)g (X),. (a ) f K (X)g L (Y) f(x)g(x). (b ) The operation in Theorem 1.4 on 0, 1 n 0, 1 n was defined implicitly in [4], and the inequality was extended there to product measure on discrete finite product spaces. With f(x) and g(x) the indicator functions of sets A and B respectively, we can equivalently express A B as the set whose indicator function is given by 1 A B (x, y) = max f (x)g (y), (8) K L= K L where f K, g L are given in (4). Therefore, inequality (b ) of Theorem 1.5 specialized to the case where K = L = 2 n and f and g are indicators says that the original KSS inequality for events in discrete finite product spaces extends to vectors having independent coordinates taking values in any measure space. 4

5 Note by (5) and (8) that 1 A B (x) = 1 A B (x, x). We next discuss further formulations of Theorems 1.2 and 1.5 which are of independent interest, and will be used in the proof. Under Framework 1, for any K and L subsets of n, define f K (x) = sup :K K f (x) and g L (x) = sup :L L g (x). (9) For any given functions K(x) and L(x) defined on S and taking values in 2 n, under Framework 1, extend (9) to f K(x) (x) = and under Framework 2, extend (4) to sup f (x) and g L(x) (x) = sup g (x), (10) :K K(x) :L L(x) f K(x) (x) = ess inf f(y), and g y [x] L(x) (x) = ess inf g(y). (11) K(x) y [x] L(x) Proposition 3.1 shows that parts (c) and (d) of Proposition 1.1 are reformulations of (a) of Theorem 1.2, and likewise (e) a reformulation of (b). Proposition 1.1 Let the hypotheses of Theorem 1.2 hold. In Framework 1 (c) and (d) below obtain. max f K (X) g L (X) K L= sup f (X) sup g (X). (c) K K,L L fk(x) (X) g L(X) (X) sup f (X) sup g (X) holding for any given K(x) K and L(x) L such that K(x) L(x) =. In Framework 2 we have f K(X) (X)g L(X) (X) f(x) g(x), holding for any given K(x) K and L(x) L such that K(x) L(x) =., (d) As in Theorem 1.2 the special cases of (c) and (d) where K and L both equal 2 n implies the inequality in general. Similarly, the special case of inequality (e) with K = L = 2 n and L(x) = K c (x), where K c denotes the complement of K, yields the inequality in general, that is, (e) is equivalent to the statement that for any given K(x), f K(X) (X)g (X) f(x) g(x). Kc(X) Parallel to the claims of Proposition 1.1, parts (c ) and (d ) below are reformulations of (a ) of Theorem 1.5, and (e ) a reformulation of (b ). For given f and a function K(x, y) taking values in 2 n, define f K(X,Y) and f K(X,Y) by replacing K(x) by K(x, y) in (10) and (11) respectively. 5 (e)

6 Proposition 1.2 Let the hypotheses of Theorem 1.5 hold. In Framework 1 (c ) and (d ) below obtain. max f K (X) g L (Y) sup f (X)g (X). (c ) K K,L L = fk(x,y) (X) g L(X,Y) (Y) sup f (X)g (X) =, (d ) holding for any given K(x, y) K and L(x, y) L replacing K(x) and L(x) in (10), respectively, and satisfying K(x, y) L(x, y) =. In Framework 2 we have f K(X,Y) (X)g L(X,Y) (Y) f(x)g(x), (e ) for given K(x, y) K and L(x, y) L replacing K(x) and L(x) in (11), respectively. 2 Applications xample 2.1 Order Statistics Type Inequalities Let X = (X 1,..., X n ) be a vector of independent non-negative random variables with associated order statistics X [n] X [1]. Let A = B be the collection of all the singletons n and f (x) = g (x) = x. Then max f (X) = X [1], max f (X)g (X) = X [1] X [2], = and inequality (a) of Theorem 1.2 provides the middle inequality in the string X [1] X [2] X [1] X [2] (X [1] ) 2 X 2 [1]. The leftmost inequality is true since order statistics are always positively correlated (moreover they are associated as defined by sary et al [2], and even MTP 2, see Karlin and Rinott [5]); the rightmost inequality follows from Jensen. Theorem 1.2 allows a large variety of extensions of this basic order statistics inequality. For example, taking A and B to be all k and l subsets of n respectively, then with f (x) = j x j and g (x) = j x j (12) we derive ( k+l ) ( k ) ( l ) X [j] X [j] X [j]. j=1 j=1 j=1 Dropping the non-negativity assumption on X 1,..., X n, we have for all t > 0, e t(x [1]+X [2] ) [e tx [1] ] 2 = e tx [1] e ty [1] = e t(x [1]+Y [1] ), with Y i s being independent copies of the X i s. Likewise, for all t > 0, e t(x [n]+x [n 1] ) [e tx [n] ] 2 = e tx [n] e ty [n] = e t(x [n]+y [n] ). 6

7 Moment generating function and Laplace orders are discussed in Shaked and Shanthikumar [7]. Returning to non-negative variables, a variation of (12) follows by replacing products with sums, that is, f (x) = j x j and g (x) = j x j, (13) which for, k = l = 2 say, yields max (X [i] + X [j] )(X [k] + X [l] ) [(X [1] + X [2] )] 2. i,j,k,l=1,2,3,4 Though the maximizing indices on the left hand side will be 1, 2, 3, 4 as indicated, the choice is not fixed and depends on the X s; note, for example, that (X [1] + X [2] )(X [3] + X [4] ) is never maximal apart from degenerate cases. Definition (12) and (13) are special cases where f and g are increasing non-negative functions of k and l variables and when f and g are symmetric, f (x) = f(x ) and g (x) = g(x ); (14) max f(x [i1 ],..., X [ik ])g(x [j1 ],..., X [jl ]) f(x [1],..., X [k] )g(x [1],..., X [l] ). i 1,...,i k,j 1,...,j l =1,...,k+l We now give an example which demonstrates that these order statistics type inequalities can be considered in higher dimensions. Let X 1,..., X n be independent vectors in R m, and for, n with = = 3 let f and g be given as in (14), where f(x 1, x 2, x 3 ) = g(x 1, x 2, x 3 ) is, say, the area of the triangle formed by the given three vectors. Theorem 1.2 gives that the expected greatest product of the areas of two triangles with distinct vertices is bounded above by the square of the expectation of the largest triangular area. To explore the dual inequality in these settings, let X be a vector of independent variables with support contained in [0, 1], and Y an independent copy. With A = B the collections of all singletons in n, and f (x) = x, g (x) = 1 x, inequality (a ) of Theorem 1.5 gives max X (1 Y ) X [1] (1 X [n] ). (15) Note that max X (1 Y ) X [1] (1 Y [n] ); the right hand side might be larger because of the restriction. Removing the restriction reverses (15), that is, X [1] (1 X [n] ) X [1] (1 X [n] ) = X [1] (1 Y [n] ) = X [1] (1 Y [n] ) = max X (1 Y ),, where the inequality follows by the negative association of X [1] and 1 X [n]. Following our treatment of applications of Theorem 1.2 we can extend (15) as follows: with A and B the collection of all k and l subsets of n respectively, and f (x) = j x j and g (x) = j (1 x j ), 7

8 we obtain max = i,j X i (1 Y j ) 1 i k,1 j l X [i] (1 X [n j+1] ). We now consider resource allocation problems of the following type. Suppose that two projects A and B have to be completed using n available resources represented by the components of a vector x. ach resource can be used for at most one project, and an allocation is given by a specification of disjoint subsets of resources. For any given subsets, n, let f (x) and g (x) count the number of ways that projects A and B can be completed using the resources x and x respectively. The exact definitions of the projects and the counts are immaterial; in particular larger sets do not necessarily imply more ways to carry out a project. For an allocation,, =, the total number of ways to carry out the two projects together is the product f (x)g (x). When the resources are independent variables, inequality (a) of Theorem 1.2 bounds the expected maximal number of ways of completing A and B together, by the product of the expectations of the maximal number of ways of completing each project alone. The bound is simple in the sense that it does not require understanding of the relation between the two projects. In particular, it can be computed without knowledge of the optimal allocation of resources. xample 2.2 With J a list of tasks, consider fulfilling the set of tasks on (not necessarily disjoint) lists A J and B J, in two distant cities using disjoint sets of workers chosen from 1, 2,..., n. ach worker may be sent to one of the cities and assigned a single task or a set of tasks which he can perform. A worker may be qualified to fill more than one set of tasks. For i n, let x i 2 J be the collection of possible assignments of tasks for worker i, (that is, the sets of tasks worker i is qualified to fulfill.); For, n and x = (x 1,..., x n ), let f (x) equal the number of ways the collection of workers can complete A, and g (x) the number of ways the collection can complete B. When the qualifications X i, i n are independent, Theorem 1.2 bounds the expectation of the maximal number of ways of fulfilling the task requirements in both cities, by the product of the expectations of the maximal numbers of ways that the requirements in each collection can be separately satisfied. xample 2.3 Paths on Graphs Consider a graph G with an arbitrary fixed vertex set V = v 1,..., v n, where for each pair of vertices the existence of the edge v i, v j is determined independently using a probability rule based on v i, v j, perhaps depending only on d(v i, v j ) for some function d. Let X = X i,j where X i,j is the indicator that there exists an edge between v i and v j. For instance, with V R m and Z i,j, 1 i, j n independent non-negative variables, we may take for v i, v j V, X i,j = 1(d(v i, v j ) < Z i,j ) where d(v i, v j ) = v i v j (uclidean distance). Note that since the variables Z i,j do not have to be identically distributed, we can set Z i,i = 0 and avoid self loops should we wish to do so. Let a path in the graph G from u to w be any ordered tuple of vertices v i1,..., v ip with v i1 = u, v ip = w and X ik,i k+1 = 1 for k = 1,..., p 1, and having all edges v ik, v ik+1 distinct. For u, v and w in V and, i, j : 1 i, j n, let f (X) be the number of paths in the graph from u to v which use only edges v i, v j for i, j ; in the same 8

9 manner, let g (X) be the number of paths in the graph from v to w which use only edges v i, v j for i, j. The projects A and B in this framework are to create paths from u to v using, and from v to w using, respectively, which combine together, when =, to give the overall project of creating a path from u to w passing through v. As the product f (X)g (X) for = is the number of paths from u to w via v for the given allocation, Theorem 1.2 provides a bound on the expected maximal number of such paths over all allocations in terms of the product of the expectations of the maximal number of paths from u to w and from w to v when the paths are created separately. Though finding the optimal allocation may be demanding, the upper bound can be computed simply, for this case in particular by monotonicity of f (x), g (x) in and for fixed x, implying that the maximal number of paths created separately is attained when using all available edges, i.e. at = = n. However, the result and the upper bound hold even in constrained situations where the existence of more edges does not lead to more paths, that is, in cases where the functions f, g are not monotone in and. One such case would be where the existence of a particular edge mandates that all paths from u to v use it. More specifically, for some fixed i 0, j 0 suppose that if i 0, j 0 and x i0,j 0 = 0 then f (x) counts the number of paths from u to v. On the other hand if x i0,j 0 = 1 then f (x) counts the number of paths from u to v which use the edge v i0, v j0. In general such f will not be monotone. This example easily generalizes to paths with multiple waypoints. We may also consider directed graphs where for 1 i j n the directed edge (v i, v j ) from v i to v j exists when X ij = 1, the directed edge (v j, v i ) from v j to v i exists when X ij = 1 and X ij = 0 when no edge exists. Returning to the graph example following the statement of Theorem 1.1, when the signed edge indicators X ij 1 i<j n are independent, inequality (a) of Theorem 1.2 provides a bound on the expected maximal number of paths from vertices v 1 to v 2 and w 1 to w 2 using disjoint edges. Another possible extension is to consider paths between subsets of vertices. For application of the dual inequality, consider for example two directed graphs on the same vertex set, determined by equally distributed and independent collections of signed edge indicators X and Y, each having independent (but not necessarily identically distributed) components. Let, (i, j) : 1 i j n, and f (X) be the number of directed paths in the graph from vertices u to v which use only X edges (v i, v j ) with (i, j) ; in the same manner, let g (Y) be the number of directed paths in the graph from v back to u which use only Y edges (v i, v j ) with (i, j). Consider the expected maximal number of paths, over all and with =, that go from u to v using the X edges and return to u from v using the Y edges. Then Theorem 1.5 implies that this expectation is bounded by the expected maximal number of paths, over all and, to move from u to v using, and then returning to u using, all with X edges, but where edges used on the forward trip may now also be used for the return. 3 Proofs 3.1 Proofs of Proposition 1.1 and Theorem 1.2 We first reduce the problem by proving the following implications between the parts of Theorems 1.2 and Proposition

10 Proposition 3.1 (a) (c) (d) (a) and (b) (e). Proof: (a) (c): Apply inequality (a) to the finite collections f K K K, g L L L and use ( ) = max K L= K K,L L sup f (x)g (x) = = ( max K L= K K,L L sup f (x) sup g (x) K K L L ) sup f (x)g (x) K K,L L = max K L= K K,L L (c) (d): Apply f K(x) (x) g L(x) (x) max K L=,K K,L L fk (x) g L (x). f K (x) g L (x). (16) (d) (a): Note that the right hand side of (16) equals f K(x) (x) g L(x) (x) for some K(x) K and L(x) L with K(x) L(x) =. (b) (e): Apply f K(x) (x)g L(x) (x) max K L=,K K,L L f K (x)g L (x). (e) (b): Use the fact that there exist some disjoint K(x) K, L(x) L such that max f K L= K (x)g L (x) = f K(x) (x)g L(x) (x). (17) K K,L L Our next objective is proving the inequalities of Framework 1, to be accomplished in Lemma 3.6. We start with a simple extension of inequality (3), expressed in terms of indicator functions, from finite spaces to spaces endowed with finite sigma algebras. Lemma 3.1 Let Q be any probability product measure on H, a finite product sigma algebra. Then, inequality (a) holds when expectations are taken with respect to Q, and f A, g B are H measurable indicator functions. Proof: Since the sigma algebra H is finite, we may assume that the space on which Q is defined is finite by identifying the elements of H with elements of some finite set; hence, (3) implies Then Q(A B) Q(A)Q(B). (18) Now let events A and B be defined by their indicator functions 1 A (x) = max f (x), 1 B (x) = max g (x). (19) max f (x)g (x) max 1 A = = K (x)1 BL (x) max 1 A K L= K (x)1 BL (x) = 1 A B (x), (20) where the first inequality above holds since f (x) 1 A (x) and therefore f (x) = f K (x) 1 AK (x), and the last equality is (5). Now (20) gives the first inequality below, (18) the second inequality, and (19) the last equality in Q max = f (X)g (X) Q(A B) Q(A)Q(B) = Q 10 max f (X) Q max g (X).

11 Let P denote the collection of all product sets of the form C = n i=1 S i, S i C i where C i S i are finite for all i = 1,..., n, and F C be the product sub-sigma algebra of S generated by C. Then S = F J with J = C P F C, (21) where F J denotes the sigma algebra generated by J. We say a collection of functions is FP if it generates a finite product sub-sigma algebra of S; note that a finite union of FP collections is FP. Lemma 3.2 Inequality (a) is true for P any probability product measure on (S, S), and f A, g B, any finite collections of FP indicator functions. Proof: Let H be the sigma algebra generated by f A, g B, and Q := P H, the restriction of P to the finite product sigma algebra H. For h an H measurable indicator function, that is, for h(x) = 1 A (x) for some A H, we have Q h = Q(A) = P (A) = P h. (22) Since the product of H measurable indicators is an H measurable indicator, and the same is true for the maximum, we have by Lemma 3.1 and (22), P max f (X)g (X) = = Q max f (X)g (X) = Q Q max f (X) = P max f (X) P max max g (X) g (X) Lemma 3.3 generalizes the inequality from FP indicator functions to S measurable indicator functions. Lemma 3.3 Inequality (a) is true for any probability product measure P and finite collections f A and g B of S measurable indicator functions. Proof: For R, S subsets of A B satisfying R S = and R S = A B, we proceed by induction on the cardinality of the set S in the statement I(R, S): inequality (a) is true when f R A, g R B are finite FP collections of indicator functions, and f S A, g S B are any finite collections of S measurable indicators. Lemma 3.2 shows that I(A B, ) is true, and the conclusion of the present lemma is I(, A B). Assume for some such R, S with S = A B, that I(R, S) is true. For γ R with, say γ A, let M be the collection of all sets A S such that (a) holds for f γ = 1 A, when f R A\γ, g R B is a finite FP collections of indicators, and f S A, g S B any collection of S measurable indicators. The singleton collection f γ is FP for any A J given by (21). Therefore, for any A J, the union f γ, f R A\γ, g R B is FP. By the induction hypothesis, J M. Since M is a monotone class and J is an algebra which generates S, the monotone class theorem implies S M. This completes the induction. At this point we have proved the BKR inequality (3) on (S, S, P ) for indicator functions. To move to more general functional inequalities we now relax the requirement that the functions be indicators. 11.

12 Lemma 3.4 Inequality (d) is true for any product measure P and finite collections of S measurable functions f A and g B which assume finitely many non-negative values. Proof: We prove (d) by induction on m and l, the number of values taken on by the collections f A, g B respectively. By Lemma 3.3 inequality (a) is true for finite collections of measurable indicators, and hence by Proposition 1.1, so is inequality (d). Now the base case m = 2, l = 2 follows readily by extending from indicators to two valued functions by linear transformation. Assume the result is true for some m and l at least 2, and consider a collection f A assuming the values 0 a 1 <... < a m+1 ; a similar argument applies to induct on l. For some k, 2 k m, define A,k = x : f (x) = a k, and for a k 1 a a k+1, let h a (x) = f (x) + (a a k )1 A,k (x), the function f with the value of a k replaced by a. We shall prove that for all a [a k 1, a k+1 ] inequality (d) holds with h a A replacing f A. By the induction hypothesis we know it holds at the endpoints, that is, for a a k 1, a k+1, since then the collection h a A takes on m values; clearly, the case a = a k suffices to prove the lemma. Given Γ(x), a function with values in 2 A, with some abuse of notation denote f Γ(x) (x) = sup f (x). (23) : Γ(x) Note that f K(x) (x) in (10) corresponds to Γ(x) = : K K(x), and similarly for g L(x) (x); for measurability issues see Section 5. For any given function Γ(x) with values in 2 A, we have for all a [a k 1, a k+1 ], C Γ := x : h a Γ(x)(x) = a, fγ(x) (x) a k 1, a k+1 = x : f Γ(x) (x) = a k, showing that C Γ does not depend on a. Let D = C Γ for Γ(x) = A, and note that sup h a (x) = a1 D + sup f (x)1 D c. Then the right hand side of (d), with h a A replacing f A, equals aδ + λ, where δ = P (D) sup g (x)dp (x) and λ = sup f (x)dp (x) sup g (x)dp (x) D c do not depend on a. Now, let = C Γ for Γ(x) = : K = K(x) and note that h a K(x)(x) = a1 + f K(x) (x)1 c. Similarly, the left hand side of (d), with h a A replacing f A, equals aθ + η, where θ = g L(x) (x)dp (x) and η = fk(x) (x) g L(x) (x)dp (x) c 12

13 do not depend on a. When a a k 1, a k+1 the collection h, A takes on m values, so by the induction hypotheses (d) holds with h a A replacing f A and we obtain aθ + η aδ + λ, for a a k 1, a k+1. (24) By taking a convex combination, we see that inequality (24) holds for all a [a k 1, a k+1 ], so in particular for a k, completing the induction. Lemma 3.5 Inequality (d) is true for any probability product measure P and finite collections of non-negative S measurable functions f A and g B. Proof: Lemma 3.4 shows that the result is true for simple functions. By approximating the functions f, g below by simple functions, f,k f, g,k g as k, and applying the monotone convergence theorem, we have the result for arbitrary non-negative functions. Lemma 3.6 Inequality (d) is true for countable collections of non-negative S measurable functions f A and g B. Proof: For K, L 2 n, let recalling definition (9). Noting and f K(x) (x) = ϕ K (x) = f K (x) and φ L (x) = g K (x), sup f (x) = :K K(x) sup sup ϕ K (x) = sup K K sup K K(x) :K K f (x) = f K (x) = sup f (x), sup K K(x) f K (x) = ϕ K(x) (x), and similarly for g B, the result follows immediately upon applying Lemma 3.5 to the finite collections ϕ K K 2 n and φ L L 2 n. By Proposition 3.1, at this point we have completed proving all inequalities pertaining to Framework 1. The next proposition connects the two frameworks and completes the proof of Theorem 1.2, and again applying Proposition 3.1, that of Proposition 1.1. Proposition 3.2 Inequality (a) holds in Framework 1 for all collections f A, g B of given functions, if and only if inequality (b) holds in Framework 2 for all given functions f and g and collections K and L. Proof: (a) (b). Given f, g, K and L apply inequality (a) to the collections f K (x) K K and g L (x) L L as in (4), and on the right hand side of the inequality, use max f (x) f(x) and max g (x) g(x) a.e. K K K L L L (b) (a): Given collections of functions f, g depending on K, L, define f(x) = sup f (x) and g(x) = sup g (x). (25) 13

14 Now letting f K, g L be as in (4), we have f (x) = f K (x) f K (x) and likewise g (x) g L (x). Now, for, disjoint, f (x)g (x) f K (x)g L (x) max K L= K 2 n,l 2 n f K (x)g L (x). Taking supremum on the left hand side over all disjoint, and then expectation, the result now follows by applying inequality (b) and (25). 3.2 The Dual Inequality As observed in [4], the techniques in [8] extend the dual inequality (7) from uniform measure on 0, 1 n 0, 1 n to any product measure on a discrete finite product space S. Specifically, Lemmas 3.2(iii), 3.4, and 3.5 of [8] carry over with minimal changes, essentially by replacing by and by appropriately; for example, the dual version of Lemma 3.4 would begin with the identity (f f) 1 (A B) = C 1,C 2 (f f) 1 (C 1 C 2 ) where the union is over all C 1, C 2 such that C 1 is a maximal cylinder of A, C 2 is a maximal cylinder of B, and C 1 C 2 ; see Sections 3 and 2 of [8] for the formal definitions of maximal cylinder, and perpendicularity, respectively. Now the proof of Theorem 1.5 and Proposition 1.2 follow in a nearly identical manner to that of Theorem 1.2 and Proposition 1.1. For instance, to prove (d ), consider C Γ = (x, y) : h a Γ(x,y)(x) = a, fγ(x,y) (x) a k 1, a k+1 = (x, y) : f Γ(x,y) (x) = a k. Setting D = C Γ for Γ(x, y) = : K = K(x, y) we can write the left hand side of (d ) as aθ + η, with θ = g L(x,y) (y)dp (x)dp (y) and η = fk(x,y) (x) g L(x,y) (y)dp (x)dp (y), D and using = C Γ for Γ = A, the right hand side becomes aδ + λ, where δ = sup g (x)dp (x) and λ = f (x)g (x)dp (x) with θ, η, δ and λ not depending on a. D D c sup D c, 4 A PQD ordering inequality Consider a collection f (x) m =1 of functions which are all increasing or all decreasing in each component of x = (x 1,..., x n ) R n. Let X = (X 1,..., X n ) R n be a vector of 14

15 independent random variables, Y = (Y 1,..., Y n ) an independent copy of X, and for each = 1,..., m, let H n, and where Z i, = Y i if i H, and Z i, = X i, if i H. Now let Z = (Z 1,,..., Z n, ), (26) U = (f 1 (Z 1 ),..., f m (Z m )) and V = (f 1 (X),..., f m (X)). (27) Inequalities between vectors below are coordinate-wise. When (28) below holds, we say that the components of V are more Positively Quadrant Dependent than those of U, and write U P QD V. Theorem 4.1 For every c = (c 1,..., c m ) R m and H n, = 1,..., m, P (U c) P (V c) and P (U c) P (V c). (28) Proof: Since (28) holds for U, V if and only if it holds for U, V, by replacing the collection f (x) m =1 by f (x) m =1 when the functions are decreasing, it suffices to consider the increasing case. For k 0,..., n let H k = H 0,..., k, and with H replaced by H k, let Z k and U k be defined as in (26) and (27) respectively. We prove the first inequality in (28) by induction on k in P (U k c) P (V c); (29) the second inequality in (28) follows in the same manner. Inequality (29) is trivially true, with equality, when k = 0, since then H k = and Z = X for all m. Now assume inequality (29) is true for 0 k < n and set Then P (U k+1 c) B = : k + 1 H. = P (f 1 (Z k+1 1 ) c 1,..., f m (Z k+1 m ) c m ) = [P (f 1 (Z k+1 1 ) c 1,..., f m (Z k+1 m ) c m X l, Y l, l k + 1)] = [P (f (Z k ) c, B X l, Y l, l k + 1)P (f (Z k+1 ) c, B X l, Y l, l k + 1)] = [P (f (Z k ) c, B X l, Y l, l k + 1)P (f (Z k ) c, B X l, Y l, l k + 1)] [P (f 1 (Z k 1) c 1,..., f m (Z k m) c m ) X l, Y l, l k + 1] = P (U k c) P (V c), where the third equality follows from the independence of X k+1 and Y k+1 and the fourth from the fact that f (Z k ) B has the same conditional distribution when either X k+1 or Y k+1 appears as the k + 1 st coordinate of the Z vector; the first inequality follows from the fact that conditioned on X l, Y l, l k + 1, the functions f (Z k ) are all increasing in X k+1 and are therefore (conditionally) associated, and the second inequality is the induction hypothesis 15

16 (29). In fact, for the first inequality above it suffices to see that the product of the two probabilities conditioned on X l, Y l, l k + 1 is the product of (conditional) expectations of two increasing functions of X k+1, which is smaller than the (conditional) expectation of the product. Taking c = (c,..., c) we immediately have Corollary 4.1 For all c R, P (max f (Z ) c) P (max f (X) c) or equivalently max f (X) ST max f (Z ). Application 1. Consider the framework of Theorem 1.2, with f (x), g (x), A, B all increasing or all decreasing functions which depend on coordinates K, L. Define D to be a collection of functions and for Y = (Y 1,..., Y n ) as above, set D = f (X) + g (X) : K L =, D = f (X) + g (Y) : K L =. By Theorem 4.1 we have Applying Corollary 4.1, D P QD D. max f (X) + g (X) ST max f (X) + g (Y). = = xponentiating the last relation and replacing e f by f, using obvious properties of the max, we obtain max f (X)g (X) ST max f (X)g (Y). (30) = = and therefore max f (X)g (X) max f (X)g (Y) max f(x) max g(x), = = for nonnegative monotone functions f and g. Thus the relation (30) is stronger than the BKR inequality for monotone sets, which was proved in [9]. Alexander [1] presents similar functional versions in this context. As an example we return to order statistics as in Section 2.1. From (30) we derive, for example, that X [1] X [2] ST X [1] Y [2] Y [1] X [2]. Generalizing by using the functions (12), we obtain for any p + q = m, m X [j] ST j=1 max X[iq]Y [jq]. i 1,...,i p j 1,...,j q=1,...,m 16

17 5 Appendix on Measurability In this section we briefly deal with various measurability issues. The measurability of the functions defined in (10) can be seen from f K(x) (x) = K f K (x)1(k(x) = K), since the given function K(x) is assumed measurable. Similarly for (23), f Γ(x) (x) = sup f (x)1(γ(x) = A). A 2 A A We next prove that given a non-negative, (S, B) measurable function f : S R and any K n, the function f K (x) defined in (4) is (S, B) measurable. Letting f r (x) = min(f(x), r) and P L (x) be the marginal of P (x) on the coordinates x L, we have ( lim p (r f r (x)) p dp K c(x)) 1/p = ess sup y [x] K (r f r (y)) = r ess inf y [x] K f r (y). Tonelli s theorem (see e.g. [3]) now implies that ess inf y [x]k f r (y) is measurable. Letting r shows that (4) is measurable. The only complication regarding measurability of the pair (K(x), L(x)) in (17) is that the maximum may not be uniquely attained, since otherwise we would simply have x : K(x) = K, L(x) = L = x : f K (x)g L (x) f K (x)g L (x), K L = a finite intersection of measurable sets, so measurable. To handle the problem of nonuniqueness, let be an arbitrary total order on the finite collection of subsets of n n, so that when the max is not unique we can choose (K(x), L(x)) to be the first disjoint pair that attains the maximum. Then x : K(x) = K, L(x) = L = F G where F = x : f K (x)g L (x) > f K (x)g L (x) and G = (K,L ) (K,L) K L = (K,L ) (K,L) K L = x : f K (x)g L (x) f K (x)g L (x) and again measurability follows. Finally, since all integrals in the inequalities are unchanged by modifications on P null sets, the results hold for the completion (S, S) of (S, S) with respect to P. 17

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