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

Let (X_n)_n∈N be independent random variables with expected values E[X_n] = a_n and variances var(X_n) = σ_n², such that ∑^∞_n=1a_n converges in ℝ and ∑^∞_n=1 σ_i² < ∞. Then ∑^∞_n=1 X_n converges in ℝ almost surely.

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