Weak disintegrability as a form of preservation of coherence

Summary Weak disintegrations are investigated from various points of view. Kolmogorov's definition of conditional probability is critically analysed, and it is noted how the notion of disintegrability plays some role in connecting Kolmogorov's definition with the one given in line with de Finetti's coherence principle. Conditions are given, on the domain of a prevision, implying the equivalence between weak disintegrability and conglomerability. Moreover, weak sintegrations are characterized in terms of coherence, in de Finetti's sense, of, a suitable function. This fact enables us to give, an interpretation of weak disintegrability as a form of “preservation of coherence”. The previous results are also applied to a hypothetical inferential problem. In particular, an inference is shown to be coherent, in the sense of Heath and Sudderth, if and only if a suitable function is coherent, in de Finetti's sense. Research partially supported by: M.U.R.S.T. 40% “Problemi di inferenza pura”.     

Strong previsions of random elements

LetC be a class of arbitrary real random elements andP an extended real valued function onC. Two definitions of coherence forP are compared. Both definitions reduce to the classical de Finetti's one whenC includes bounded random elements only. One of the two definitions (called strong coherence) is investigated, and some criteria for checking it are provided. Moreover, conditions are given for the integral representation of a coherentP, possibly with respect to a δ-additive probability. Finally, the two definitions and the integral representation theorems are extended to the case whereC is a class of random elements taking values in a given Banach space.     

A unifying view on some problems in probability and statistics

Statistical Methods & Applications - Let $$L$$ be a linear space of real random variables on the measurable space $$(\varOmega ,\mathcal {A})$$ . Conditions for the existence of a probability...     

A note on zenga concentration index

Summary The Zenga index, , is shown to be a concentration index, in the sense that, ifX andY are non negative random variables with 0<E(X), E(Y)<+∞, then (X)⩾ (Y) whenever the Lorenz curves satisfyL _x(p)≤L _y(p) for all p. Research partially supported by: M.U.R.S.T. 40% ?Inferenza statistica: basi probabilistiche e sviluppi metodologici?.