Tail fields of partially exchangeable arrays

AbstractWe give an elementary, direct proof that if an array of random variables {(Xij, α, ξi, ηj); i, j ∈ N} is separately exchangeable, then X = {Xij; i, j ∈ N} and {(α, ξi, ηj); i, j ∈ N} are conditionially independent given the shell σ-field SX of X. We show further that if (X, Y) = {(Xij, Yij); i, j ∈ N} is separately exchangeable, then X and SX, Y are conditionally independent given SX

Optimisation of measures on a hyperfinite adapted probability space

Albeverio S, Herzberg F. Optimisation of measures on a hyperfinite adapted probability space. ACTA APPLICANDAE MATHEMATICAE. 2008;100(1):1-14.The minimisation problem for a functional P bar right arrow u(Gamma,P,(g) over tilde) is considered, where (g) over tilde is an R-n-valued stochastic process, defined on some filtered probability space Gamma = (Gamma(G(t))(t is an element of[0,1]), P), and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure P on Gamma, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function phi of ((g) over tilde (t1),..., (g) over tilde (tk)), for t(1),..., t(k) is an element of [0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of vertical bar phi vertical bar being uniformly bounded from below and some condition on the dimension of phi ((g) over tilde (t1),..., (g) over tilde (tk)) the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a "standard result": It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis

Algorithmic Aspects of the Intersection and Overlap Numbers of a Graph

The intersection number of a graph G is the minimum size of a ground set S such that G is an intersection graph of some family of subsets F ⊆ 2 S. The overlap number of G is defined similarly, except that G is required to be an overlap graph of F. Computing the overlap number of a graph has been stated as an open proble

On purification of measure-valued maps

Games, Purification, Measure-valued maps, C60, C70,