I am writing a paper, and I need to cite the law of large numbers.

Precisely, I want to use the statement that if $\{x_{1},...,x_{n}\}$ are independent random samples generated from a distribution with expectation $\mu $, then $$ \lim_{n\to\infty} \sum_{i=1}^n \frac{x_{i}}{n} = \mu. $$

Which is the standard reference for this theorem?

  • 1
    $\begingroup$ That there should be a standard reference is questionable. That there are myriad accessible references is evident from a Web search for weak law of large numbers $\endgroup$
    – whuber
    Apr 10, 2018 at 20:03
  • $\begingroup$ Which law of large numbers? Strong or weak? $\endgroup$
    – AdamO
    Apr 10, 2018 at 21:20

1 Answer 1


In my reference statistical book, this theorem is listed on page 200 as Theorem 4.3.1 (Strong law of large numbers). [Probability & Statistics - The Science of Uncertainty, M.J. Evans & J.S.Rosenthal, W.H.Freeman and Company, New York, 2004].

For a more extensive proof of the 'Strong law of large numbers', see for example the reference work: J.S. Rosenthal. A First Look at Rigorous Probability Theory 2nd Edition, World Scientific, 2006.


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