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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

Let (Xn)n∈N be independent random variables with expected values E[Xn] = an and variances var(Xn) = σn2, such that ∑∞n=1an converges in ℝ and ∑∞n=1 σi2 < ∞. Then ∑∞n=1 Xn converges in ℝ almost surely.

Proof

References

  • 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
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