Kolmogorov's two-series theorem

In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers.

Statement of the theorem

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Let   be independent random variables with expected values   and variances  , such that   converges in   and   converges in  . Then   converges in   almost surely.

Proof

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Assume WLOG  . Set  , and we will see that   with probability 1.

For every  ,  

Thus, for every   and  ,  

While the second inequality is due to Kolmogorov's inequality.

By the assumption that   converges, it follows that the last term tends to 0 when  , for every arbitrary  .

References

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  • Durrett, Rick. Probability: Theory and Examples. Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 1.8, pp. 60–69.
  • M. Loève, Probability theory, Princeton Univ. Press (1963) pp. Sect. 16.3
  • W. Feller, An introduction to probability theory and its applications, 2, Wiley (1971) pp. Sect. IX.9