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Almost indiscernible sequences and convergence of canonical bases. (English) Zbl 1338.03060

Summary: We give a model-theoretic account for several results regarding sequences of random variables appearing in [I. Berkes and H. P. Rosenthal, Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 473–507 (1985; Zbl 0554.60044)]. In order to do this, 0.6cm
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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 \(\aleph_{0}\)-categorical stable theories in which the last two agree.
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We characterise sequences that admit almost indiscernible sub-sequences.
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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 [loc. cit.].

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
60G09 Exchangeability for stochastic processes

Citations:

Zbl 0554.60044
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References:

[1] Geometric stability theory 32 (1996)
[2] Model theory for metric structures 2 pp 315– (2008)
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