An elementary proof of the strong law of large numbers

@article{Etemadi1981AnEP,
  title={An elementary proof of the strong law of large numbers},
  author={N. Etemadi},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete},
  year={1981},
  volume={55},
  pages={119-122}
}
  • N. Etemadi
  • Published 1 February 1981
  • Mathematics
  • Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
SummaryIn 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… 
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References

SHOWING 1-5 OF 5 REFERENCES
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,…
A Course in Probability Theory
TLDR
This edition of A Course in Probability Theory includes an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.
Strong Laws of Large Numbers for $r$-Dimensional Arrays of Random Variables
Lecture notes in Mathematics 360
  • Berlin-Heidelberg-NewYork: Springer
  • 1973
A course in probability theory, 2nd ed
  • New York-London: Academic Press
  • 1974