Startseite Mathematik Rate of convergence of empirical measures for exchangeable sequences
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Rate of convergence of empirical measures for exchangeable sequences

  • Patrizia Berti EMAIL logo , Luca Pratelli und Pietro Rigo
Veröffentlicht/Copyright: 30. November 2017
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 67 Heft 6

Abstract

Let S be a finite set, (Xn) an exchangeable sequence of S-valued random variables, and μn = (1/n) ∑i=1nδXi the empirical measure. Then, μn(B)⟶a.s.μ(B) for all B ⊂ S and some (essentially unique) random probability measure μ. Denote by 𝓛(Z) the probability distribution of any random variable Z. Under some assumptions on 𝓛(μ), it is shown that

an≤ρ[L(μn),L(μ)]≤bnandρ[L(μn),L(an)]≤cnu

where ρ is the bounded Lipschitz metric and an(⋅) = P(Xn+1 ∈ ⋅ | X1, …, Xn) is the predictive measure. The constants a, b, c > 0 and u ∈ (12, 1] depend on 𝓛(μ) and card (S) only.

MSC 2010: Primary 60G09; 60B10; 60A10; 60G57; 62F15
Keywords: empirical measure; exchangeability; predictive measure; random probability measure; rate of convergence
Dedicated to Professor Paolo de Lucia on his 80th birthday

Communicated by Anna De Simone

References

[1] Berti, P.—Pratelli, L.—Rigo, P.: Limit theorems for a class of identically distributed random variables, Ann. Probab. 32 (2004), 2029–2052.10.1214/009117904000000676Suche in Google Scholar

[2] Berti, P.—Crimaldi, I.—Pratelli, L.—Rigo, P.: Rate of convergence of predictive distributions for dependent data, Bernoulli 15 (2009), 1351–1367.10.3150/09-BEJ191Suche in Google Scholar

[3] Berti, P.—Pratelli, L.—Rigo, P.: Limit theorems for empirical processes based on dependent data, Electron. J. Probab. 17 (2012), 1–18.10.1214/EJP.v17-1765Suche in Google Scholar

[4] Berti, P.—Pratelli, L.—Rigo, P.: Exchangeable sequences driven by an absolutely continuous random measure, Ann. Probab. 41 (2013), 2090–2102.10.1214/12-AOP786Suche in Google Scholar

[5] Berti, P.—Pratelli, L.—Rigo, P.: Asymptotic predictive inference with exchangeable data, submitted (2016), currently available at http://www-dimat.unipv.it/rigo/smps2016.pdfSuche in Google Scholar

[6] Cifarelli, D. M.—Regazzini, E.: De Finetti’s contribution to probability and statistics, Statist. Science 11 (1996), 253–282.10.1214/ss/1032280303Suche in Google Scholar

[7] Cifarelli, D. M.—Dolera, E.—Regazzini, E.: Frequentistic approximations to Bayesian prevision of exchangeable random elements, submitted (2016), arXiv:1602.01269v1.10.1016/j.ijar.2016.06.007Suche in Google Scholar

[8] Fortini, S.—Ladelli, L.—Regazzini, E.: Exchangeability, predictive distributions and parametric models, Sankhya A 62 (2000), 86–109.Suche in Google Scholar

[9] Goldstein, L.—Reinert, G.: Stein’s method for the Beta distribution and the Polya-Eggenberger urn, J. Appl. Probab. 50 (2013), 1187–1205.10.1017/S0021900200013875Suche in Google Scholar

[10] Mijoule, G.—Peccati, G.—Swan, Y.: On the rate of convergence in de Finetti’s representation theorem, submitted (2016), arXiv:1601.06606v1.10.30757/ALEA.v13-42Suche in Google Scholar

Received: 2016-4-28
Accepted: 2016-9-26
Published Online: 2017-11-30
Published in Print: 2017-11-27

© 2017 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
https://doi.org/10.1515/ms-2017-0070
Schlagwörter für diesen Artikel
empirical measure; exchangeability; predictive measure; random probability measure; rate of convergence

Artikel in diesem Heft

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0070/pdf
Button zum nach oben scrollen