Almost indiscernible sequences and convergence of canonical bases

TL;DR

This paper provides a model-theoretic framework for understanding convergence of sequences of random variables, linking logical, metric, and canonical base convergence, and applies it to prove a key theorem in probability theory.

Contribution

It introduces a new model-theoretic perspective on convergence of types, characterizes theories where different notions of convergence coincide, and applies these results to random variable spaces.

Findings

Characterizes $eth_0$-categorical stable theories where convergence notions agree.

Identifies conditions for sequences to have almost indiscernible subsequences.

Proves the Main Theorem of Berkes & Rosenthal using model theory.

Abstract

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}