Almost indiscernible Sequences and convergence of Canonical Bases

We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes & Rosenthal [BR85]. 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 which admit almost indiscernible sub-sequences. • We apply these tools to ARV , the theory (atomless) random variable spaces. 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 & Rosenthal.