An elementary proof of the strong law of large numbers

  • N. Etemadi


In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].


Stochastic Process Probability Theory Random Vector Mathematical Biology Independent Random Variable 
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.
    Chung, K.L.: A course in probability theory, 2nd ed. New York-London: Academic Press 1974Google Scholar
  2. 2.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 4th ed. Oxford: Clarendon 1960Google Scholar
  3. 3.
    Padgett, W.G., Taylor, R.L.: Lecture notes in Mathematics360. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  4. 4.
    Smythe, R.T.: Strong laws of large numbers forr-dimensional arrays of random variables. Ann. of Probab.1, 1, 164–170 (1973)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • N. Etemadi
    • 1
  1. 1.Mathematics DepartmentUniversity of Illinois at Chicago CircleChicagoUSA

Personalised recommendations