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 $\left( X_n \right)_{n=1}^{\infty}$ be independent random variables with expected values $\mathbf{E} \left[ X_n \right] = \mu_n$ and variances $\mathbf{Var} \left( X_n \right) = \sigma_n^2$ , such that $\sum_{n=1}^{\infty} \mu_n$ converges in $\mathbb{R}$ and $\sum_{n=1}^{\infty} \sigma_n^2$ converges in $\mathbb{R}$ . Then $\sum_{n=1}^{\infty} X_n$ converges in $\mathbb{R}$ almost surely.

Proof

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

For every $m \in \mathbb{N}$ ,

$\limsup_{N \to \infty} S_N - \liminf_{N \to \infty} S_N = \limsup_{N \to \infty} \left( S_N - S_m \right) - \liminf_{N \to \infty} \left( S_N - S_m \right) \leq 2 \max_{k \in \mathbb{N} } \left| \sum_{i=1}^{k} X_{m+i} \right|$

Thus, for every $m \in \mathbb{N}$ and $\epsilon > 0$ ,

$\begin{align} \mathbb{P} \left( \limsup_{N \to \infty} \left( S_N - S_m \right) - \liminf_{N \to \infty} \left( S_N - S_m \right) \geq \epsilon \right) &\leq \mathbb{P} \left( 2 \max_{k \in \mathbb{N} } \left| \sum_{i=1}^{k} X_{m+i} \right| \geq \epsilon \ \right) \\ &= \mathbb{P} \left( \max_{k \in \mathbb{N} } \left| \sum_{i=1}^{k} X_{m+i} \right| \geq \frac{\epsilon}{2} \ \right) \\ &\leq \limsup_{N \to \infty} 4\epsilon^{-2} \sum_{i=m+1}^{m+N} \sigma_i^2 \\ &= 4\epsilon^{-2} \lim_{N \to \infty} \sum_{i=m+1}^{m+N} \sigma_i^2 \end{align}$

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

By the assumption that $\sum_{n=1}^{\infty} \sigma_n^2$ converges, it follows that the last term tends to 0 when $m \to \infty$ , 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

Category:Theorems in probability theory