Home Rate of convergence of empirical measures for exchangeable sequences
Article
Licensed
Unlicensed Requires Authentication

Rate of convergence of empirical measures for exchangeable sequences

  • Patrizia Berti EMAIL logo , Luca Pratelli and Pietro Rigo
Published/Copyright: November 30, 2017
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 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), 2029–2052.10.1214/009117904000000676Search 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-BEJ191Search 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.v17-1765Search 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-AOP786Search 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.pdfSearch 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/1032280303Search 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.2016.06.007Search in Google Scholar

[8] Fortini, S.—Ladelli, L.—Regazzini, E.: Exchangeability, predictive distributions and parametric models, Sankhya A 62 (2000), 86–109.Search 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/S0021900200013875Search 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.v13-42Search 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

Articles in the same Issue

  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
Keywords for this article
empirical measure; exchangeability; predictive measure; random probability measure; rate of convergence

Articles in the same Issue

  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
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 26.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0070/pdf
Scroll to top button