Probability Theory and Related Fields

, Volume 78, Issue 4, pp 523–534 | Cite as

The strong law of large numbers for k-means and best possible nets of Banach valued random variables

  • J. A. Cuesta
  • C. Matran


Let B be a uniformly convex Banach space, X a B-valued random variable and k a given positive integer number. A random sample of X is substituted by the set of k elements which minimizes a criterion. We found conditions to assure that this set converges a.s., as the sample size increases, to the set of k-elements which minimizes the same criterion for X.


Banach Space Positive Integer Assure Random Sample Stochastic Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cuesta, J.A.: Medidas de centralización multidimensionales (ley fuerte de los grandes números). Trab. Estad. e Invest. Oper. 35, 3–16 (1984)Google Scholar
  2. 2.
    Cuesta, J.A., Matran, C.: Strong laws of large numbers in abstract spaces via Skorohod's Representation Theorem. Sankyà, Ser. A, 84, 98–103 (1986)Google Scholar
  3. 3.
    Cuesta, J.A., Matran, C.: Uniform consistency of r-means. Stat. Probab. Lett. to appearGoogle Scholar
  4. 4.
    Diestel, J., Uhl, JJ.: Vector measures. Providence: American Math Society (1977)Google Scholar
  5. 5.
    Garkavi, A.L.: The best possible net and the best possible cross-selection of a set in a normed space. Izv. Akad. Nauk. SSSR Ser. Mat. 26, 87–106 (1962). (English translation in: Am. Math. Soc. Transl. (2), 39, 111–132 (1964))Google Scholar
  6. 6.
    Herrndorf, N.: Approximation of vector-valued random variables by constants. J. Approximation Theory 37, 175–181 (1983)Google Scholar
  7. 7.
    Holmes, R.B.: A course on optimization and best approximation. Berlin Heidelberg New York: Springer 1972Google Scholar
  8. 8.
    IEEE Trans. Inform. Theory 28 (1982)Google Scholar
  9. 9.
    Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967Google Scholar
  10. 10.
    Pollard, D.: Strong consistency of k-means clustering. Ann. Statist. 9, 135–140 (1981)Google Scholar
  11. 11.
    Singer, I.: Best approximation in normed linear spaces by elements of linear subspaces. New York Heidelberg Berlin: Springer 1970Google Scholar
  12. 12.
    Skorohod, A.V.: Limit theorems for stochastic processes. Theor. Probab. Appl. 1, 261–290 (1956)Google Scholar
  13. 13.
    Sverdrup-Thygeson, H.: Strong law of large numbers for measures of central tendency and dispersion in compact metric spaces. Ann. Statist. 9, 141–145 (1981)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • J. A. Cuesta
    • 1
    • 2
  • C. Matran
    • 1
    • 2
  1. 1.Department of StatisticsUniversity of SantanderSantanderSpain
  2. 2.Department of StatisticsUniversity of ValladolidValladolidSpain

Personalised recommendations