Summary
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].
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References
Chung, K.L.: A course in probability theory, 2nd ed. New York-London: Academic Press 1974
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 4th ed. Oxford: Clarendon 1960
Padgett, W.G., Taylor, R.L.: Lecture notes in Mathematics360. Berlin-Heidelberg-New York: Springer 1973
Smythe, R.T.: Strong laws of large numbers forr-dimensional arrays of random variables. Ann. of Probab.1, 1, 164–170 (1973)
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Etemadi, N. An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheorie verw Gebiete 55, 119–122 (1981). https://doi.org/10.1007/BF01013465
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DOI: https://doi.org/10.1007/BF01013465