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ON THEORIES OF RANDOM VARIABLES

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30 pages
ON THEORIES OF RANDOM VARIABLES ITAÏ BEN YAACOV Abstract. Nous étudions des théories d'espaces de variables aléatoires : en un premier temps, nous considérons les variables aléatoires à valeurs dans l'intervalle [0, 1], puis à valeur dans des structures métriques quelconques, généralisant la procédure d'aléatoirisation de structures classiques due à Keisler. Nous démontrons des résultats de préservation et de non-préservation de propriétés modèle-théoriques par cette construction : (i) L'aléatoirisée d'une structure ou théorie stable est stable. (ii) L'aléatoirisée d'une structure ou théorie simple instable n'est pas simple. Nous démontrons également que dans la structure aléatoirisée, tout type est un type de Lascar. We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: (i) The randomisation of a stable structure is stable. (ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type. Introduction Mathematical structures arising in the theory of probabilities are among the most natural examples for metric structures which admit a model theoretic treatment, albeit not in the strict setting of classical first order logic.

  • logic

  • continuous logic

  • valued random

  • since ?

  • ?¯ ?

  • then ?

  • p? ?

  • theoretic independence


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