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 \cite{Berkes-Rosenthal:AlmostExchangeableSequences}. In order to do this, \begin{itemize} \item 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 $\aleph_0$-categorical stable theories in which the last two agree. \item We characterise sequences which admit almost indiscernible sub-sequences. \item 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. \end{itemize}