Almost indiscernible sequences and convergence of canonical bases

Itaï Ben Yaacov (1) , Alexander Berenstein (1, 2) , C. Ward Henson (3)

1 ICJ - Institut Camille Jordan
2 UNIANDES - Departamento de Matematicas [Univ de los Andes Colombia]
3 UIUC - University of Illinois at Urbana-Champaign [Urbana]

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. \begin{enumerate} \item We characterise sequences which admit almost indiscernible sub-sequences. \item We study theories for which metric converge coincides with canonical base convergence (\textit{a priori} weaker). For $\aleph_0$-categorical theories we characterise this property by the $\aleph_0$-categoricity of the associated theory of beautiful pairs. In particular, we show that this is the case for the theory of spaces of random variables. \item Using these tools we give model theoretic proofs for results regarding sequences of random variables appearing in Berkes \& Rosenthal \cite{Berkes-Rosenthal:AlmostExchangeableSequences}. \end{enumerate}

Mots clés: stable theory; $\aleph_0$-categorical theory; beautiful pairs; almost indiscernible sequence; almost exchangeable sequence

Domaines: Logique [math.LO]

Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson. Almost indiscernible sequences and convergence of canonical bases. 2009. ⟨hal-00406952v1⟩