##
**Almost indiscernible sequences and convergence of canonical bases.**
*(English)*
Zbl 1338.03060

Summary: We give a model-theoretic account for several results regarding sequences of random variables appearing in [I. Berkes and H. P. Rosenthal, Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 473–507 (1985; Zbl 0554.60044)]. In order to do this, 0.6cm

- \(\bullet\)
- 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 \(\aleph_{0}\)-categorical stable theories in which the last two agree.
- \(\bullet\)
- We characterise sequences that admit almost indiscernible sub-sequences.
- \(\bullet\)
- We apply these tools to the theory of atomless random variables (ARV). 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 [loc. cit.].

### MSC:

03C45 | Classification theory, stability, and related concepts in model theory |

03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |

60G09 | Exchangeability for stochastic processes |

### Keywords:

stable theory.; \(\aleph_{0}\)-categorical theory; beautiful pair; almost indiscernible sequence; random variable; almost exchangeable sequence### Citations:

Zbl 0554.60044
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XMLCite

\textit{I. Ben Yaacov} et al., J. Symb. Log. 79, No. 2, 460--484 (2014; Zbl 1338.03060)

### References:

[1] | Geometric stability theory 32 (1996) |

[2] | Model theory for metric structures 2 pp 315– (2008) |

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