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The natural vector bundle of the set of multivariate density functions. (English) Zbl 1095.62067

E. Marchi [Z. Wahrscheinlichkeitstheorie Verw. Geb. 23, 7–17 (1972; Zbl 0278.60010)] studied the set of discrete multivariate density functions with given unidimensional marginals, and in later years used this geometrical approach to develop new concepts of cooperative equilibrium in game theory. In particular, he showed that, for finite sets \(X_i\) \((i = 1,\dots, n)\), the Cartesian product of the sets of density functions defined on \(X_i\) \((i = 1,\dots, n)\) is naturally embedded in the set of density functions defined on the Cartesian product of the sets \(X_i\). Let us consider the following notations, necessary to describe some basic results: for \(X_1, X_2,\dots, X_n\) finite sets, \(X_N\) denotes the Cartesian product \(X_1\times X_2\times\dots\times X_n\), \(D_i\) \( (i = 1,\dots, n)\) is the set of density functions defined on \(X_i\), \(D_N = D_1\times D_2\times\dots\times D_n\) the Cartesian product of \(D_i\) \((i=1\dots n)\), and \(DX_N\) denotes the set of density functions defined on the Cartesian product \(X_N\). With the notations, the previously obtained results of E. Marchi can be announced simpler as follows:
\(D_N\) is proved to be naturally embedded in \(DX_N\). Moreover, \(DX_N\) can be derived as a vector boundle having as the underlying manifold the image of the sets \(D_i\) under the natural embedding. The interest for probability theory is restricted to the nonnegative elements in the sets of the derived vector boundle. More precisely, the base of the vector boundle contains the set \(D_N = D_1\times D_2\times\dots\times D_n\), and the space contains the set of tensor products of elements in \(D_i\) \((i = 1,\dots, n)\). If \(g_i\in D_i\) \((i = 1,\dots, n)\), then the set of functions in \(D_N\) with unidimensional marginals \(g_i\) consists of the nonnegative elements of the fiber in the vector boundle at the tensor product of \(g_i\), \(\bigotimes_{i=1}^{n} g_i\). Moreover, any element of this set can be written as the sum of \(\bigotimes_{i=1}^{n} g_i\) and an element of the fiber type of the vector boundle. Thus, the fiber plays the important role of being the space of all correlation measures among a multivariate density function and its unidimensional marginals.
The main contribution of the present paper is to extend these results of E. Marchi, the finite sets \(X_i\) being replaced with the \(\sigma\)-finite measure spaces \((X_i, \mu_i)\), \((i = 1,\dots, n)\). Since the extreme points and introduced basis are no more available in the novel general setting of arbitrary measure spaces, the authors use various space decompositions, which simplify the proofs considerably. It is worth to mention that the newly obtained results can be used to extend notions of cooperative equilibrium in game theory from the discrete case to the continuous framework. The same is true for the field of quantum mechanics. Several properties are of theoretical interest for the study of the derived fiber boundle, perturbations of the fibers and cross sections.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
55R25 Sphere bundles and vector bundles in algebraic topology
62H20 Measures of association (correlation, canonical correlation, etc.)
53B99 Local differential geometry
91A12 Cooperative games
91A23 Differential games (aspects of game theory)

Citations:

Zbl 0278.60010
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