Kolmogorov's two-series theorem

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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[edit]

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


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 .


  • 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