The FKG inequality is an effective device when the requisite assumptions can be verified. Sometimes these have to be approached circuitously. This is discussed with reference to past uses and suggestions for work on the range of applicability. New areas of potential application are also presented

Joag-Dev, Kumar

Shepp, Larry A

Vitale, Richard A

English

Crossref

Remarks and open problems in the area of the FKG inequality

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsStatistics Papers Wharton Faculty Research1984Remarks and Open Problems in the Area of theFKG InequalityKumar Joag-DevUniversity of IllinoisLarry A. SheppUniversity of PennsylvaniaRichard A. VitaleClaremont Graduate SchoolFollow this and additional works at: http://repository.upenn.edu/statistics_papersPart of the Mathematics Commons, Other Physical Sciences and Mathematics Commons, andthe Statistics and Probability CommonsThis paper is posted at ScholarlyCommons. http://repository.upenn.edu/statistics_papers/200For more information, please contact repository@pobox.upenn.edu.Recommended CitationJoag-Dev, K., Shepp, L. A., & Vitale, R. A. (1984). Remarks and Open Problems in the Area of the FKG Inequality. Institute ofMathematical Statistics Lecture Notes - Monograph Series, 5 121-126. http://dx.doi.org/10.1214/lnms/1215465638Remarks and Open Problems in the Area of the FKG InequalityAbstractThe FKG inequality is an effective device when the requisite assumptions can be verified. Sometimes thesehave to be approached circuitously. This is discussed with reference to past uses and suggestions for work onthe range of applicability. New areas of potential application are also presented.KeywordsAssociation, distributive lattice, FKG inequality, positive quadrant dependence, total orderingDisciplinesMathematics | Other Physical Sciences and Mathematics | Statistics and ProbabilityThis journal article is available at ScholarlyCommons: http://repository.upenn.edu/statistics_papers/200Inequalities in Statistics and ProbabilityIMS Lecture Notes-Monograph Series Vol. 5 (1984), 121-126REMARKS AND OPEN PROBLEMS IN THE AREA OF THE FKG INEQUALITYBY KUMAR JOAG-DEV,1 L. A. SHEPP and RICHARD A. VITALEUniversity of Illinois, Bell Laboratories and Claremont Graduate SchoolThe FKG inequality is an effective device when the requisite assumptions can be ver-ified. Sometimes these have to be approached circuitously. This is discussed with referenceto past uses and suggestions for work on the range of applicability. New areas of potentialapplication are also presented.1. Sufficiency and Necessity of the Conditions for the FKG Inequality. The FKGinequality in its original form (Fortuin, Ginibre and Kasteleyn (1971)) states that if (a) Γ isa distributive lattice i.e. order isomorphic to an algebra of subsets of a set, (b)/ and g areincreasing on Γ, (c) μ is a positive function on Γ with(1.1) μ(*)μ(y) ^ μ(* Λ y)Φ v y) foraii*,y,then(1.2) %fe)μ.(x)Xg(y)μ(y) ^ ϊf{x)g(x)μ(x)ϊμ(y).A simple example of how the FKG inequality can be used in a combinatorial setting is thefollowing. Suppose A, A, are fixed subsets ofN={l9 ... , n} and k, k> are given integers,i = 1, ... , r. Choose a subset of S of N at random by choosing each element to be in Sindependently with probability/?, fixed. Let A, = |A, Π S\. ThenP[A ^ k \ A i ^ k h i ^ r] = ar^P[A ^k] = aQ.To prove this let Γ be the set of all subsets S of N ordered by inclusion, and let/(S) =χ(Ai > ki9 i < r), g(S) = χ(A > k), and μ(S) = 1. It is easy to verify that (a)-(c), (1.1)hold and this gives the result. The result may not seem surprising until it is realized thatar is not always increasing in r. Indeed with n = 2, A = {1}, Aλ = {1,2}, A2 = {2} withp = xh gives a counterexample since αo = Vi < a\ = % > ai = V2. This class of problemswas posed by Frank Hwang and will be further developed elsewhere.We will see that FKG is often hard to apply even when one feels it should apply. Thismay also be illustrated by Hwang's example: It can be shown by a direct argument thatP[Ai^ jfcf, i^r\A ^k]^P[Ai^ku i^r\λ = jfc],But Shepp does not see just now how to give an FKG proof. The obvious choice g(S) =χ(A ^ k), μ(S) = χ(A ^ k), and/as before yields the desired conclusion but (1.1) fails.Is there a reordering of Γ to make an FKG proof?FKG themselves point out that (1.1) is not necessary and one could assume the alternatecondition(1.1')Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command underGrant No. 82-K-0007. The United States Government is authorized to reproduce and distribute reprints forgovernment purposes not withstanding any copyright notation thereon.AMS 1980 subject classifications. Primary 60E15, Secondary 62H20, 06D99Key words and phrases: Association, distributive lattice, FKG inequality, positive quadrant dependence, totalordering.121122 KUMAR JOAG-DEV, L. A. SHEPP, and RICHARD A. VITALEwhich may hold without (1.1), where / and O are the extreme elements of Γ and the sumis over all other elements.Shepp (1982) shows by an example the slack in (a)-(c), (1.1), where by merelyredefining < in the lattice one gets to satisfy (a)-(c), (1.1) and hence obtain (1.2) where the"natural" ordering fails to satisfy (a)-(c), (1.1). Note (1.2) does not depend on " < " .Ahlswede and Daykin (1978) and others (see survey on FKG by Graham (1983)) give moregeneral versions, also not necessary.The question then arises as to whether it is possible to find conditions which are necessaryor at least closer to being necessary. In this regard, it may be interesting to note that fora distributive lattice of length 2,(1.1) and (1. Γ) are equivalent and necessary. In this case,there are two possible structures:I = x VyFIGURE 1. Structures for a 2-and x <f y y length lattice.O = x ΛyThe first is easily seen. For the second, take two increasing functions/ and g with/(0)= f(y) = g(P) = g(x) = O andf(x) =/(/) = g(y) = g(I) =/. Then/and g are positivelycorrelated <=> (1.1) <=> (1.1').Neither (1.1) nor (1. Γ) is necessary for lattices of length greater than 2, but it is interest-ing to see heuristically why they are sufficient. The clue lies in the assumption of a distribu-tive lattice. Among other things, this means that certain sublattice structures do not occurand, in fact, that locally the lattice looks like the pictured length 2 cases. Thus (1.1) andthe distributivity assumption are paired to ensure that things work locally.As mentioned, this approach is generally too strong. One point of departure for an alter-nate approach is to hold the distributivity assumption in abeyance. This allows previouslyforbidden sublattice structures, which we can view as lattices in their own right:FIGURE 2. Structures for a(nondistributive) lattice.Consider, for the first structure, increasing/and g v/ith f(O) =f(z) = g(O) = g(x) = g(y)= O and/U) =/(y ) =/(/ ) = g(z) = g(I) = 1. If lattice elements are equiprobable, thenCov(/g) = — 1/sl A similar example can be given for the second structure. This shows that,for nondistributive lattices, increasing functions are not always positively correlated—atleast for equiprobable lattice elements. It would be interesting to see if any general resultis possible for the nondistributive case. Subsequent to setting down these remarks, Kempe r-man's paper (1977, Theorem 7) wich treats the necessity of distributivity was noticed.The FKG inequality is a powerful device, but it may not be straightforward to use. Infact, as we said, Shepp (1982) treats a problem in which a ' 'natural'' ordering fails to satisfyFKG INEQUALITY 123the requirements but another ordering does work. This raises the question of whether thereis a systematic way to use the FKG inequality.One way of formalizing this is as follows:(Existence) Let a finite set Ω, /, g: Ω -> R, and a probability measure μ on Ω be given.It is desired to find a distributive lattice structure on Ω such that the FKG hypotheses ((a)-(c), (1.1)) hold. When can this be done? Is Cov(f,g) ^ 0 close to a condition?A second question is how long would it take to find a compatible lattice structure.(Multiplicity) Given the previous set-up (Ω,/, g, μ): what fraction of distributive latticestructures on Ω satisfy the FKG requirements ((a)-(c), (1.1))?Of course in both of these questions, compatibility of the lattice structure with μ andwith the pair (f,g) can be considered separately.We next discuss two problems to which the FKG inequality seems likely to be able tocontribute some insight.2. A Possible Application to a Partition Problem.Problem 1. Let ait i = 1, ... , m be a random sample drawn (without replacement)from 1,2, ... , m+n and suppose the remaining numbers are denoted by bjf j = 1, ... ,n. The sumS = Σα, has two interpretations.(i) Let Xλ, ... , Xm; Yx, ... , Yn be two independent random samples drawn from a singlepopulation with a continuous distribution function so that the probability of one or moreties among the observations is zero. Let Ru ... , Rm be the ranks of the X observationsamong the m+n observations. Then Σ/?, has the same distribution as S, in fact, /?, couldbe identified with ar In this case S is known as the Wilcoxon statistic (with the "nulldistribution").(ii) Suppose α f' s are arranged so that ax < a2 < ... < am. Then a. -i represents the number ofbj's smaller than α, and it is easy to verify thatT = S - m(m + l)/2 = Σ7 (α, -/)has a distribution symmetric about mn/2 and assumes values, 0,1, ... , mn.Further, for an integer k, 0 < k < mn,where (k;m,ri) is the number of partitions of k into m parts, with each part ^ n. In otherwords, (k;m,n) represents the total number of distinct sequences of non-negative integers^ n and of length m such that each sequence is nondecreasing and the sum of the integersin the sequence is A:.There has been a wealth of literature on the theory of partitions. For the most recent source,see Andrews (1976, Section 3.2).The problem we are concerned with here is the unimodality of the distribution of T, thatis(k;m,n) - (k-l;m,n) ^ 0 , 0 ^ k^ mn/2,which is directly connected with exceedances of at over b/s. Dynkin (1950) proved theunimodality of T; however, the proof is based on the representation theory of Lie algebras.Recently, Hughes (1977) and Stanley (1980,1981) discuss several problems regarding uni-124 KUMAR JOAG-DEV, L. A. SHEPP, and RICHARD A. VΓΓALEmodality of sequences arising from Lie algebras. So far, no direct combinatorial or prob-abilistic proof is available. One possible approach is to compare the conditional prob-abilities of Γ, governed by the order structure of α, and b/s while one or more /?, are fixed.Such probability comparisons are similar to those appearing in Shepp (1982). The latticestructure and possible partial orderings are the same as in that paper. Could the FKG in-equality again succeed to give a simple proof?3. A Restriction to Linear Functions.Problem 2. Let (X\,Xi) be a pair of real random variables. The conclusion of the FKGinequality for the measure generated by {X\yXi) is that(3.1) cov[/(Xi,X2),£(Xi,X2)] > 0,for every pair of (co-ordinatewise) nondecreasing functions/,g. This property of positivedependence for (XiJί?) was termed as "association" by Esary, Proschan and Walkup(1967). A weaker notion of positive dependence called "positive quadrant dependence"(PQD) is defined by requiring(3.2) FXίχ2(u,v) ^ FX](u)Fχ2(v)9 for all (ιι, v)where F with the appropriate subscripts denotes the distribution function. Lehmann (1966)studied PQD and showed that (3.2) is equivalent to(3.3) cov(/ii(Xi),/i2(X2))>0,for every pair of nondecreasing functions hi ,h2. Although it is easy to show that (3.2) (or(3.3)) does not imply (3.1), if ffg are restricted to linear nondecreasing functions then theimplication does hold. This was proved by Shaked (1982). We give a very simple proof.Note that, we want to prove the following:P[Xλ >xx ,X 2>JC 2] ^ P{Xλ >JC,]P[X 2>Λ: 2] for all (JC, ,x2)impliesP&iaft > c, Σibfr >d\> PpSfc^ > ctf^bft > d\.for ait hi nonnegative and arbitrary constants c, d.FIGURE 3. Quadrants.Proof. Let Q, aif βf be the probabilities of the regions as shown, created by interesect-ing lines ίλ, /2 representing Σ ? ^ = c and XfbpCi = d respectively. We have to show that(3.4)given that(3.5)However, from (3.5) it is easy to check thatwhich implies (3.4).FKG INEQUALITY 125Remark. Notice that in the above proof the quadrant could very easily be replaced bya region defined by an intersection of half planes other than Xi > JCI and Xi > xi, the onlyrequirement being that it is contained in the intersection of Σ Λ A > c and ΣftjX/ > d.We will set up an analogy for the measures on lattices. Let Γ be a lattice. Suppose Tλ,T2 are total ordering relations. Consider a partial ordering P induced by Tλ, T2 as follows:Definition.Given a measure μ on Γ one may define "marginal distribution functions" Fi, Fi byFi(x) = μ{y;y^Γjc} and the PQD analog would be: for every x e Γ ,(3.6) μ{y:y ^px} ^ μ{y:y ^r,xYviy-J ^r 2*}In view of the remark above one may ask the following: Suppose (Γf, Γf) is anotherpair of linear ordering on Γ such that the induced partial order P* is weaker than P above,that isUnder what conditions would (3.6) be sufficient for the validity of an analogous inequalityinvolving P* and (Γf, Γf)?A related multivariate question is the following. Supose Xi, ... , X* are such that forarbitrary nonnegative constants aiy bi and an arbitrary proper subset A of {1,2, ... k},ΣJcΛflA and Σ^ft/X, are PQD, where A is the complement of A. This property may be called"disjoint positive linear dependence" (DPLD).QUESTION. Does DPLD =>PLD?Here PLD meansH\apCh Σfy&arePQD. (This prob-lem is related to some concepts discussed in Joag-Dev (1983)).It is interesting to see that k = 3 is the most crucial while k = 2 has already been proved.To see that the case k = 3 yields the general result, considerYλ = xλ, Y2 = Σ ^ * A , Y3 = Σ ω ft A ,where A, B are disjoint and do not containt 1. The triplet YλJ2Ji is DPLD. If PLD, itwill show that the linear combinations containing one common variable would be PQD.Using the same technique successively, the cardinality of A Π B can be increased to k.To see the relation between these covariance inequalities and FKG, suppose that L isa product lattice of two components Lλ and L2 with partial ordering Pi and P2 respectively.Suppose μ defined on L satisfies FKG condition with respect to the partial ordering inducedby (PUP2). Then it follows that for every pair of nondecreasing functions (f,g) defined onLλ and L2 respectively,(3.7) Cov[/,£]>0.However, it is well known that the validity of (3.7) for every pair of nondecreasing functionsdoes not imply FKG inequality. Suppose now we restrict the nondecreasing functions to thosewhich are linear, then the above converse seems to be plausible. In fact, it reduces to havingDPLD and PLD conditions equivalent.126 KUMAR JOAG-DEV, L. A. SHEPP, and RICHARD A. VITALEREFERENCESAHLSWEDE, R. and DAYKIN, D. E. (1978). An inequality for the weights of two families of sets,their unions and intersections. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete43 183-185.ANDREWS, G. (1976). Theory of Partitions. Encyclopedia of Mathematics and Its Applications.Addison-Wesley.DYNKIN, E. B. (1950). Some systems of weights of linear representations of semisimple lie groups.Dokl. Akad. Nauk. SSSR (N.S.) 71 221-224.ESARY, J., PROSCHAN, F. and WALKUP, W. (1967). Associated random variables with applica-tions. Ann. Math. Statist. 38 1466-1474.FORTUIN, C M . , GINIBRE, J, and KASTELEYN, P. N. (1971). Correlation inequalities on somepartially ordered sets. Comm. Math Phys. 22 89-103.GRAHAM, R. L. (1983). Applications of the FKG inequality and its relatives. To appear in Mathe-matical Programming Bonn 1982 — The State of the Art, ed. by Bachem, Grόtschel, and Korte.Springer- Verlag.HUGHES, J. W. B. (1977). Lie algebraic proofs of some theorems on partitions. Number Theoryand Algebra. H. Zassenhaus, ed., Academic Press, New York.JOAG-DEV, K. (1983) Characterization of independence via uncorrelatedness. Ann. Probability 111037-1041.KEMPERMAN, J. H. B. (1977). On the FKG inequality for measures on a partially ordered space.lndag.Math.39 313-331.SHAKED, M. (1982). A general theory of some positive dependence notions. J. Multivariate Anal.12 199-218.SHEPP, L. A. (1982). The XYZ conjecture and the FKG inequality. Ann Probability 10 824-827.STANLEY, R. (1980). Unimodal sequences arising from Lie algebras. Young Day Proceedings,127-136. Marcel Dekkar, Inc., N.Y.STANLEY, R. (1981). Two combinatorial applications of the Aleksandrov-Fenchel inequalities. 7.Comb. Theory 31 56-65.

Remarks and Open Problems in the Area of the FKG Inequality

Kosmopolis

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1401&amp;context=statistics_papers

Remarks and Open Problems in the Area of the FKG Inequality

Abstract

Similar works

Full text

Available Versions

Crossref

ScholarlyCommons@Penn

Kosmopolis