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ALMOST INDISCERNIBLE SEQUENCES AND CONVERGENCE OF CANONICAL BASES

Published online by Cambridge University Press:  25 June 2014

ITAÏ BEN YAACOV
Affiliation:
UNIVERSITÉ CLAUDE BERNARD – LYON 1, INSTITUT CAMILLE JORDAN, CNRS UMR 5208, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCEhttp://math.univ-lyon1.fr/~begnac/
ALEXANDER BERENSTEIN
Affiliation:
UNIVERSIDAD DE LOS ANDES, CARRERA 1 NO 18A-10, BOGOTÀ, COLOMBIAhttp://matematicas.uniandes.edu.co/∼aberenst
C. WARD HENSON
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, 1409 WEST GREEN STREET, URBANA, ILLINOIS 61801, USAhttp://www.math.uiuc.edu/∼henson

Abstract

We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. In order to do this,

  • We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise א0-categorical stable theories in which the last two agree.

  • We characterise sequences that admit almost indiscernible sub-sequences.

  • We apply these tools to the theory of atomless random variables (ARV). We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes and Rosenthal.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

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