Law of large numbers for non-additive measures

Rébillé, Yann

doi:10.1016/j.jmaa.2008.11.060

Cited by 10 publications

(8 citation statements)

References 16 publications

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“…The resulting type of limit theorem was already considered by Marinacci [21], Maccheroni and Marinacci [20] and more recent papers [7,10,11,29,30]. In those papers, the weakening of the axiomatic properties of probability has been balanced by the incorporation of extra technical assumptions on the properties of Ω and/or the random variables.…”

Section: Pedro Teránmentioning

confidence: 95%

Laws of large numbers without additivity

Terán

2014

Trans. Amer. Math. Soc.

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The law of large numbers is studied under a weakening of the axiomatic properties of a probability measure. Averages do not generally converge to a point, but they are asymptotically confined in a limit set for any random variable satisfying a natural ‘finite first moment’ condition. It is also shown that their behaviour can depart strikingly from the intuitions developed in the additive case.

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Section: Pedro Teránmentioning

confidence: 95%

Laws of large numbers without additivity

Terán

2014

Trans. Amer. Math. Soc.

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“…In fact, ever since the idea of a non-additive probability measure entered economic theory, there has been work devoted to extending some of the classical measure-theoretic and probabilistic tools to a setting of non-additive measures, with the purpose of applying these tools to problems in economics where ambiguity prevails. See, for instance, [18,19,21,22,33,38,53,54,56,57], to cite only a few. This paper falls in this line of work.…”

Section: ] "The Action Which Follows Upon An Opinion Depends As Muchmentioning

confidence: 99%

Monotone Equimeasurable Rearrangements with Non-Additive Probabilities

Ghossoub

2012

SSRN Journal

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Abstract. In the classical theory of monotone equimeasurable rearrangements of functions, "equimeasurability" (i.e. the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knighitan uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capacities that satisfy a property that I call strong nonatomicity. The latter is a strengthening of the notion of nonatomicity, and these two properties coincide for additive measures and for submodular (i.e. concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.

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“…According to the proof of Theorem 3.1 in [10], for ∀ η > 0, there exist a probability measure Q ∈ Core(V ) and δ > 0, such that…”

Section: Theorem 2 Under the Conditions In Theorem 1 For ∀εmentioning

confidence: 99%

“…This question arises naturally of knowing if laws of large numbers can be maintained for capacities. Related papers on laws of large numbers for capacities include [6,7,10]. Marinacci and Maccheroni introduced the definition of independence of random variables relative to a capacity, and proved a strong law of large numbers for totally monotone capacities in a Polish space, where powerful analytical methods were used.…”

Section: Introductionmentioning

confidence: 99%