  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,
{itemize} 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. 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. {itemize}
