Archiv der Mathematik

, Volume 111, Issue 1, pp 101–111 | Cite as

A Berry-Esseen type limit theorem for Boolean convolution

  • Octavio ArizmendiEmail author
  • Mauricio Salazar


We give estimates on the rate of convergence on the Boolean central limit theorem for the Lévy distance. In the case of measures with bounded support, we obtain a sharp estimate by giving a qualitative description of this convergence.


Boolean convolution Boolean central limit theorem Lévy distance Berry–Esseen theorem 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Arizmendi and T. Gaxiola, Asymptotic spectral distributions of distance-k graphs of star product graphs, In: XI Symposium on Probability and Stochastic Processes, 47–58, Progr. Probab., 69, Birkhäuser/Springer, Cham, 2015.Google Scholar
  2. 2.
    A.C. Berry, The accuracy of the gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G.P. Chistyakov and F. Götze, Limit theorems in free probability theory. I, Ann. Probab. 36 (2008), 54–90.Google Scholar
  4. 4.
    C.-G. Esseen, On the Liapounoff limit of error in the theory of probability, Ark. Mat. Astr. Fys. 28A (1942), 19 pp.Google Scholar
  5. 5.
    T. Hasebe, Monotone convolution semigroups, Studia Math. 200 (2010), 175–199.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. Hasebe, Free infinite divisibility for beta distributions and related ones, Electron. J. Probab. 19 (2014), 33 pp.Google Scholar
  7. 7.
    H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), 409–438.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), 39–58.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 337–371.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    R. Speicher and R. Woroudi, Boolean convolution, Fields Inst. Commun. 12 (1997), 267–279.MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Voiculescu, Symmetries of some reduced free product \(\text{C}^*\)-algebras, In: Operator Algebras and their Connections with Topology and Ergodic Theory, 556–588, Lecture Notes in Mathematics, 1132, Springer, Berlin, 1985.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMexico

Personalised recommendations