Finitely additive mixtures of probability measures

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1984

An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a a-field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions. Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.

International Journal of Approximate Reasoning, 2008

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions.

Journal of Theoretical Probability, 2013

Let L be a linear space of real bounded random variables on the probability space (Ω, A, P0). There is a finitely additive probability P on A, such that P ∼ P0 and EP (X) = 0 for all X ∈ L, if and only if c EQ(X) ≤ ess sup(−X), X ∈ L, for some constant c > 0 and (countably additive) probability Q on A such that Q ∼ P0. A necessary condition for such a P to exist is L − L + ∞ ∩ L + ∞ = {0}, where the closure is in the norm-topology. If P0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on A, such that P ≪ P0 and EP (X) = 0 for all X ∈ L, if and only if ess sup(X) ≥ 0 for all X ∈ L.

Eprint Arxiv Math 0511406, 2005

Let f 1 , f 2 ,. .. , f n be a family of independent copies of a given random variable f in a probability space (Ω, F , µ). Then, the following equivalence of norms holds whenever 1 ≤ q ≤ p < ∞

Journal of Theoretical Probability, 2007

We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions. Key words and phrases. Hausdorff moment problem, coherent lower prevision, lower distribution function, complete monotonicity. 1 This is because the set of polynomials on [0, 1] is uniformly dense in the set of continuous functions on [0, 1], so a Hausdorff moment sequence corresponds to a unique positive linear functional on the set of continuous functions on [0, 1]. The F. Riesz Representation Theorem in its more modern form (see, for instance, [13, Section V.1]) allows us to extend this functional uniquely (under σ -additivity) to all Borel-measurable functions. More details are also given in Section 3.

The Review of Symbolic Logic

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

Mathematische Zeitschrift, 1979

We show that there is a probability space X and a bounded scalarly measurable function from X to #~ which has no Pettis integral (Theorem 2B). Our method relies on a new decomposition theorem for additive functionals defined on power sets (Theorem 1H). As another corollary we prove the existence of an indefinite Pettis integral with non-totally-bounded range (Example 2D).

2020

We present a new approach to infinitesimal freeness that reduces infinitesimal freeness to a special case of operator valued freeness, thus answering a question of F{\&#39;e}vrier and Nica. Moreover, our approach easily extends to the operator-valued case. We then construct operator-valued infinitesimal cumulants. We show that the operator valued infinitesimal freeness of Curran and Speicher is equivalent to the vanishing of our mixed cumulants. We then show how to find the free additive convolution, in the context of operator valued infinitesimal freeness, using the subordination functions in two different ways. The first using our new characterization and the second using differentiable paths.

Journal of Mathematical Psychology, 1974

Axioms for additive conjoint measurement and qualitative probability are given. Representation theorems and uniqueness theorems are proved for structures that satisfy these axioms. Both Archimedean and nonarchimedean cases are considered. Approximations of infinite structures by sequences of finite structures are also considered. At the present time, there is one set of techniques for proving representation theorems for finite measurement structures an another set for infinite structures. Techniques for finite structures were developed in Scott (1964) and basically consist of solving finite sets of inequalities; techniques for infinite structures in one way or another resemble those used in Holder (1901) an d consist of the construction of fundamental sequences. Although finite structures often admit good axiomatizations in the sense that necessary and sufficient conditions for their representations can be given, they do not admit good uniqueness results. Infinite structures, however, often have uniqueness results for their representations but assume structural (nonnecessary) conditions in their axiomatizations. In this paper, new techniques are developed which allow infinite structures to be represented in terms of their finite substructures and thus simultaneously achieve good axiomatizations and representation theorems. These new techniques use the compactness theorem of mathematical logic in a way similar to Abraham Robinson's use in his Nomtandurd Analysis (Robinson, 1966). However, to avoid the introduction of a large amount of mathematical logic into this paper, algebraic constructions are given for the various uses of the compactness theorem. This makes the paper relatively self-contained. These new techniques also allow a bridge to be built from finite to infinite structures. Thus, in Section 7 it is shown that certain infinite structures with unique representations are limits of sequences of finite structures. In terms of representations this means that as more elements are included into the qualitative structure the more "unique" the representation becomes. These new techniques also avoid the use of Archimedean axioms.

We give an extension of de Finetti's concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to unbounded random variables that we believe has advantages over Lebesgue-style integrals in the finitely additive setting. We also give a general version of the Fundamental Theorem of Prevision to deal with conditional previsions and unbounded random variables.