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

Proof

Assume WLOG ${\mu }_n=0$. Set $S_N=\sum _{n=1}^N{X}_n$, and we will see that $\underset{N}{\mathrm{lim\; sup}}{S}_N-\underset{N}{\mathrm{lim\; inf}}{S}_N=0$ with probability 1.

For every $m\in \mathbb{N}$, $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\; sup}}{S}_N-\underset{N\to \mathrm{\infty }}{\mathrm{lim\; inf}}{S}_N=\underset{N\to \mathrm{\infty }}{\mathrm{lim\; sup}}(S_N-S_m)-\underset{N\to \mathrm{\infty }}{\mathrm{lim\; inf}}(S_N-S_m)\le 2\underset{k\in \mathbb{N}}{\max }\left|\sum _{i=1}^k{X}_{m+i}\right|$$

Thus, for every $m\in \mathbb{N}$ and $ϵ>0$, $$\begin{array}{cc}\mathbb{P}(\underset{N\to \mathrm{\infty }}{\mathrm{lim\; sup}}(S_N-S_m)-\underset{N\to \mathrm{\infty }}{\mathrm{lim\; inf}}(S_N-S_m)\ge ϵ)& \le \mathbb{P}(2\underset{k\in \mathbb{N}}{\max }\left|\sum _{i=1}^k{X}_{m+i}\right|\ge ϵ)\\ & =\mathbb{P}(\underset{k\in \mathbb{N}}{\max }\left|\sum _{i=1}^k{X}_{m+i}\right|\ge \frac{ϵ}{2})\\ & \le \underset{N\to \mathrm{\infty }}{\mathrm{lim\; sup}}4{ϵ}^{-2}\sum _{i=m+1}^{m+N}{\sigma }_i^2\\ & =4{ϵ}^{-2}\underset{N\to \mathrm{\infty }}{\lim }\sum _{i=m+1}^{m+N}{\sigma }_i^2\end{array}$$

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

By the assumption that $\sum _{n=1}^{\mathrm{\infty }}{\sigma }_n^2$ converges, it follows that the last term tends to 0 when $m\to \mathrm{\infty }$, for every arbitrary $ϵ>0$.

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