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 $\left( X_n \right)_^$ be independent random variables with expected values \mathbf \left X_n \right= \mu_n and variances $\mathbf \left( X_n \right) = \sigma_n^2$, such that $\sum_^ \mu_n$ converges in ℝ and $\sum_^ \sigma_n^2$ converges in ℝ. Then $\sum_^ X_n$ converges in ℝ almost surely.

Proof

Assume WLOG $\mu_n = 0$. Set $S_N = \sum_^N X_n$, and we will see that $\limsup_N S_N - \liminf_NS_N = 0$ with probability 1.

For every $m \in \mathbb$,
$\limsup_ S_N - \liminf_ S_N = \limsup_ \left( S_N - S_m \right) - \liminf_ \left( S_N - S_m \right) \leq 2 \max_ \left, \sum_^ X_ \$

Thus, for every $m \in \mathbb$ and $\epsilon > 0$,
$\begin \mathbb \left( \limsup_ \left( S_N - S_m \right) - \liminf_ \left( S_N - S_m \right) \geq \epsilon \right) &\leq \mathbb \left( 2 \max_ \left, \sum_^ X_ \ \geq \epsilon \ \right) \\ &= \mathbb \left( \max_ \left, \sum_^ X_ \ \geq \frac \ \right) \\ &\leq \limsup_ 4\epsilon^ \sum_^ \sigma_i^2 \\ &= 4\epsilon^ \lim_ \sum_^ \sigma_i^2 \end$

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

By the assumption that $\sum_^ \sigma_n^2$ converges, it follows that the last term tends to 0 when $m \to \infty$, for every arbitrary $\epsilon > 0$.

References

{{reflist

* 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

Probability theorems