Kolmogorov's two-series theorem explained

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

\left( X _{n} \right) ^{inf ty} _{n =1} be independent random variables with expected values E \left[ X _{n} \right] = \mu _{n} and variances Var \left( X _{n} \right) = \sigma ^{2} _{n} , such that \sum ^{inf ty} _{n =1} \mu _{n} converges in R and \sum ^{inf ty} _{n =1} \sigma ^{2} _{n} converges in R . Then \sum ^{inf ty} _{n =1} X _{n} converges in R almost surely.

Proof

\mu _{n} = 0 . Set S _{N} = \sum ^{N} _{n =1} X _{n} , and we will see that \limsup _{N} S _{N} - \liminf _{NS} _{N} = 0 with probability 1.

For every

m \in N ,$$\backslash limsup\_\; S\_N\; -\; \backslash liminf\_\; S\_N\; =\; \backslash limsup\_\; \backslash left(S\_N\; -\; S\_m\; \backslash right)\; -\; \backslash liminf\_\; \backslash left(S\_N\; -\; S\_m\; \backslash right)\; \backslash leq\; 2\; \backslash max\_\; \backslash left|\; \backslash sum\_^\; X\_\; \backslash right|$$

Thus, for every

m \in N and \epsilon > 0 ,

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

By the assumption that

\sum ^{inf ty} _{n =1} \sigma ^{2} _{n} converges, it follows that the last term tends to 0 when m \to inf ty , for every arbitrary \epsilon > 0 .

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