Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T21:29:39.427Z Has data issue: false hasContentIssue false
  • English
  • Français

ALMOST INDISCERNIBLE SEQUENCES AND CONVERGENCE OF CANONICAL BASES

Published online by Cambridge University Press:  25 June 2014

ITAÏ BEN YAACOV ,
ALEXANDER BERENSTEIN  and
C. WARD HENSON
ITAÏ BEN YAACOV
Affiliation:
UNIVERSITÉ CLAUDE BERNARD – LYON 1, INSTITUT CAMILLE JORDAN, CNRS UMR 5208, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCEhttp://math.univ-lyon1.fr/~begnac/
ALEXANDER BERENSTEIN
Affiliation:
UNIVERSIDAD DE LOS ANDES, CARRERA 1 NO 18A-10, BOGOTÀ, COLOMBIAhttp://matematicas.uniandes.edu.co/∼aberenst
C. WARD HENSON
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, 1409 WEST GREEN STREET, URBANA, ILLINOIS 61801, USAhttp://www.math.uiuc.edu/∼henson
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. 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 that admit almost indiscernible sub-sequences.

  • • 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 Berkes and Rosenthal.

Keywords

03C4503C9060G09Stable theory.א0-categorical theorybeautiful pairalmost indiscernible sequencerandom variablealmost exchangeable sequence
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 460 - 484
Copyright
Copyright © The Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Yaacov, Itaї Ben, Berenstein, Alexander, and Henson, C. Ward, Model-theoretic independence in the Banach lattices L p (μ), Israel Journal of Mathematics, vol. 183 (2011), pp. 285–320, doi:10.1007/s11856-011-0050-4, arXiv:0907.5273.CrossRefGoogle Scholar
Yaacov, Itaї Ben, Berenstein, Alexander, Henson, C. Ward, and Usvyatsov, Alexander, Model theory for metric structures, Model theory with Applications to Algebra and Analysis, vol. 2 (Chatzidakis, Zoé, Macpherson, Dugald, Pillay, Anand, and Wilkie, Alex, editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.Google Scholar
Yaacov, Itaї Ben, Lovely pairs of models: the non first order case, this Journal, vol 69 (2004), no. 3, pp. 641–662, doi:10.2178/jsl/1096901759, arXiv:0902.0119.Google Scholar
Yaacov, Itaї Ben. On supersimplicity and lovely pairs of cats, this Journal, vol. 71 (2006), no. 3, pp. 763–776, doi:10.2178/jsl/1154698575, arXiv:0902.0118.Google Scholar
Yaacov, Itaї Ben. Schrödinger’s cat, Israel Journal of Mathematics, vol. 153 (2006), pp. 157–191, doi:10.1007/BF02771782.CrossRefGoogle Scholar
Yaacov, Itaї Ben. On perturbations of continuous structures, Journal of Mathematical Logic, vol. 8 (2008), no. 2, pp. 225–249, doi:10.1142/S0219061308000762, arXiv:0802.4388.Google Scholar
Yaacov, Itaї Ben. Topometric spaces and perturbations of metric structures, Logic and Analysis, vol. 1 (2008), no. 3–4, pp. 235–272, doi:10.1007/s11813-008-0009-x, arXiv:0802.4458.CrossRefGoogle Scholar
Yaacov, Itaї Ben. Definability of groups in א0-stable metric structures, this Journal, vol. 75 (2010), no. 3, pp. 817–840, doi:10.2178/jsl/1278682202, arXiv:0802.4286.Google Scholar
Yaacov, Itaї Ben. Stability and stable groups in continuous logic, this Journal, vol. 75 (2010), no. 3, pp. 1111–1136, doi:10.2178/jsl/1278682220, arXiv:0810.4087.Google Scholar
Yaacov, Itaї Ben. On uniform canonical bases in L plattices and other metric structures, Journal of Logic and Analysis, vol. 4 (2012), p. 12, 30, doi:10.4115/jla.2012.4.12, arXiv:1008.5031.Google Scholar
Yaacov, Itaї Ben. On theories of random variables, Israel Journal of Mathematics, vol. 194 (2013), no. 2, pp. 957–1012, doi:10.1007/s11856-012-0155-4, arXiv:0901.1584.Google Scholar
Berkes, Istvàn and Rosenthal, Haskell P., Almost exchangeable sequences of random variables, Zeitschri für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 70 (1985), no. 4, pp. 473–507, doi:10.1007/BF00531863.CrossRefGoogle Scholar
Yaacov, Itaї Ben and Usvyatsov, Alexander, On d-finiteness in continuous structures, Fundamenta Mathematicae, vol. 194 (2007), pp. 67–88, doi:10.4064/fm194-1-4.Google Scholar
Yaacov, Itaї Ben and Usvyatsov, Alexander. Continuous first order logic and local stability, Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, 5213–5259, doi:10.1090/S0002-9947-10-04837-3, arXiv:0801.4303.Google Scholar
Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996, Oxford Science Publications.Google Scholar
Poizat, Bruno, Paires de structures stables, this Journal, vol. 48 (1983), no. 2, pp. 239–249, doi:10.2307/2273543.Google Scholar