THE NATURAL VECTOR BUNDLE OF THE SET OF MULTIVARIATE DENSITY FUNCTIONS

We find a description of the set of multivariate density functions with given marginals and introduce an associated vector bundle. The interest for the probability theory is restricted to the nonnegative elements in the sets of the derived vector bundle. The fiber is the space of all correlation measures among a multivariate density function and its unidimensional marginals.


Introduction
An important branch in probability and statitics is the study of distributions with given marginals.This study was initialized by Wassily Hoeffding (1940,1941) and Maurice Fréchet (1951).Since then, its theoretical and applied development have been remarkable (for an account of the first steps in this development see Dall'Aglio (1991)).When the marginals are unidimensional, the most important tool in this study are copulas.Copulas were introduced by Abe Sklar (1959) in response to a query of Fréchet.Sklar proved that if H is the joint distribution function of n random variables X 1 , ..., X n , and F 1 , ..., F n are the distribution functions of X 1 , ..., X n , respectively, then there exists an n-copula C such that H(x 1 , ..., x n ) = C(F 1 (x 1 ), ..., F n (x n )).The n-copula is uniquely determined on range(F 1 ) × ... × range(F n ), so that C can be thought of as a description of the way in which a joint distribution function is related to its one-dimensional marginals.Excellent references about copulas are Joe (1997) and Nelsen (1999).Independently of the above mentioned development and using a geometrical approach, Ezio Marchi (1969Marchi ( , 1972) ) studied the set of discrete multivariate density functions with given unidimensional marginals which it motivates a great variety of new concepts of cooperative equilibrium in game theory (see Marchi (1976Marchi ( , 1986aMarchi ( , 1986bMarchi ( , 1988Marchi ( , 1992Marchi ( , 1996) ) and Marchi, García Jurado and Prada (1991)).He first studied the bivariate case concerning to some problems about bimatrix zero sum games.Then, he studied the multivariate case introducing a natural vector bundle.
In particular, Marchi (1972) showed by considering finite sets X i , i = 1, ..., n, that the Cartesian product e X N of the sets of density functions f X i defined on X i , i = 1, ..., n, is naturally embedded in the set g X N of the density functions defined on the Cartesian product X N of the sets X i .Moreover, he derived g X N as a vector bundle having as the underlined manifold the image of the sets f X i under the natural embedding.The interest for the probability theory is restricted to the nonnegative elements in the sets of the derived vector bundle.Specifically, the base of the vector bundle contains the set e X N and the space contains the set of tensor products of elements in f X i , i = 1, ..., n.If g i ∈ f X i , i = 1, ..., n, then the set of functions in g X N with unidimensional marginals g i consists of the nonnegative elements of the fiber in the vector bundle at the tensor product ⊗ n i=1 g i .Moreover, any element of this set can be written as the sum of ⊗ n i=1 g i and an element of the type fiber of the vector bundle.So the fiber plays an important role, namely that of being the space of all correlation measures among a multivariate density function and its unidimensional marginals.
In this paper we extend the results obtained by Marchi (1972) considering σ-finite measure spaces (X i , µ i ), i = 1, ..., n.It is worth noting that in the papers just mentioned, Marchi, in order to prove the results, considered extreme points and introduced basis which are not available in the general setting of arbitrary measure spaces.Here we use decompositions for the different involved spaces simplifying considerably the proofs of the different results.
The rest of the paper is organized as follows.In section 2 we obtain a decomposition of L 1 (X N , µ N ) where (X N , µ N ) denotes the product measure space corresponding to the measure spaces (X i , µ i ), i = 1, ..., n.We express L 1 (X N , µ N ) as a topological sum of tensor products of some subspaces of L 1 (X i , µ i ) , i = 1, ..., n.In section 3 we obtain a description of the set of density functions in L 1 (X N , µ N ) with given unidimensional marginals and, in section 4, we introduce the vector bundle.Finally, some concluding remarks are given in section 5.

A decomposition
If A is a finite set, then |A| denotes the number of elements of A. We denote with N and R the set of natural numbers and the set of real numbers, respectively.If n ∈ N, then N = {1, ..., n}.To avoid ambiguities we assume that n ≥ 2.
If S is a subset of a vectorial space, then hSi is the linear space generated by S. If E is a subset of a normed vectorial space X, then clE is the closure of E in X.If X i , i ∈ N , are normed vectorial spaces with norms k.k i , then X N is equipped with the product topology defined by any of the equivalent norms k( The symbol ⊕ denotes the topological sum.
In this paper we use some notions from measure and integral theory.For Fubini's theorem to be valid, all the measure spaces are considered to be σ-finite.
Proof.Let σ be the continuous function defined by The function σ is one to one and onto.In effect . This shows that σ is onto.Suppose now that Integrating the previous equality on X, and taking into account that By the open map theorem, since L 1 0 (X, µ) and hgi are closed subspaces of L 1 (X, µ), then σ is a topological isomorphism, or which is the same, the subspaces L 1 0 (X, µ) and hgi decompose L 1 (X, µ) in a topological sum. 2 Given n real functions f i : where each X i is an arbitrary set, the tensor product of the functions f i is the function We have the equality In effect, the set of simple integrable functions is dense in L 1 (X N , µ N ) and any simple integrable function can be approximated in L 1 (X N , µ N ) by finite linear combinations of characteristic functions of the type χ this last fact can be deduced from the definition of product measure).
From Theorem 2.1 and (2.2) we obtain a decomposition of L 1 (X N , µ N ) given in the next theorem.
For a detailed proof of the above theorem see Morillas (2003).
For i ∈ N , consider the continuous linear function In this section we are going to give an analytical description of U (g N ).In what follows we will suppose that for i ∈ N , g i , h i ∈ L 1 (X i , µ i ) and Next we will consider some functions introduced by Marchi (1972) for the case in which X i are finite sets.We begin with the continuous linear function Thus, if we consider the functions associated with this topological sum, σ i given by (2.1) and the projection p i given by

Consider now the function
R and and Proof.Suppose that f ∈ M. Using (3.4) we obtain Taking into account that for all i ∈ N , from the previous equality it results that if i ∈ N , then Thus f ∈ M and, by (3.6), On the other hand, since From the previous observations we have, In the sequel we will need the following results which proofs can be found in Morillas (2003).
The following assertions hold: Now are in conditions to give an analytical description of the set by Lemma 3.2.with J = {i} and (3.9) with c = 1, we have , then s i (f ) = g i , and consequently z i = 0. Thus, for i ∈ N , f N\{i} = 0, resulting for f the expression that we like. 2 Remark 3.6.Using (3.8) and a similar argument to that used in the previous proof, we obtain that Consequently, using again Theorem 3.1, . Since (g N ) and (g 0 N ) are arbitrary, this shows that By the previous corollary and (3.10), In what follows we will denote with R the set given by The previous results tell us that the Banach space R (R is a closed subspace of L 1 (X N , µ N )) do not depend of the functions g i nor of the functions h i .
By the results of this section, when each g i ∈ f X i , by Theorem 3.1, U (g N ) consists of the nonnegative elements of M ∩ π −1 h N (Ψ(g N )) whose analytical description is given by Theorem 3.5.Concretely, Then (z N ) = 0. Therefore, for each (g N ) ∈ M N , DΨ(g N )|× n i=1 L 1 0 (X i , µ i ) is one to one.
We also have, from Lemma 3.3 and Theorem 2.2, that for each (g N ) ∈ M N , the set As a consequence of the previous lemma we have Proposition 4.4.4.4.Ψ(M N ) is a submanifold of L 1 (X N , µ N ) and Ψ|M N : In the sequel we will think about Ψ|M N as a function from M N onto Ψ(M N ) and we will consider in M ∩ π −1 h N (Ψ(g N )) the structure of normed vectorial space that makes the map Now we are in condition to state and prove the main result of this section.Theorem 4.5.4.5.(Ψ(M N ), M, π h N |M) is a vector bundle with type fiber R.

Proof. Since s
where p Ψ(M N ) : Ψ(M N ) × R → Ψ(M N ) is the projection.Taking into in account the structure of normed vectorial space defined in M∩π −1 h N (Ψ(g N )) = (π h N |M) −1 (Ψ(g N )), this set is a Banach space and the map By the results of the previous section, if for each i ∈ N , g i ∈ f X i , then U (g N ) consists of the nonnegative elements of the fiber (π h N |M) −1 (Ψ(g N )).Also, any density function f in L 1 (X N , µ N ) with unidimensional marginals g i , can be written as f = Ψ(g N ) + r where r is in the type fiber R.

Concluding remarks.
If the sets X i are finite and we consider on them the cardinal measure, the above results reduce to those in Marchi (1972).As we have mentioned in the introduction, in the discrete case they motivate a great variety of new concepts of cooperative equilibrium in game theory.Now we could use the results obtained in this paper to extend all these concepts for continuous games.On the other hand, there are several properties of the derived fiber bundle that can be studied.For example, problems concerning perturbation of the fibers and cross sections.
It is important to note that the set of density functions with given marginals appears in the study of the optimal transportation problem (see, e. g., Cafarelli, Feldman and McCann (2001), Cafarelli (2003)).More recently, it also appears in conexion with the application of game theory to physics, in particular, to quantum mechanics (see, e.g., Landsburg and references therein).We think that the results obtained here provide new geometrical tools to study all these matters.