Kolmogorov's two-series theorem
# 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 ℝ and $\sum _{n=1}^{\infty }\sigma _{n}^{2}$ converges in ℝ. Then $\sum _{n=1}^{\infty }X_{n}$ 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 $\limsup _{N}S_{N}-\liminf _{N}S_{N}=0$ 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 $\epsilon >0$,

${\begin{aligned}\mathbb {P} \left(\limsup _{N\to \infty }\left(S_{N}-S_{m}\right)-\liminf _{N\to \infty }\left(S_{N}-S_{m}\right)\geq \epsilon \right)&\leq \mathbb {P} \left(2\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|\geq \epsilon \ \right)\\&=\mathbb {P} \left(\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|\geq {\frac {\epsilon }{2}}\ \right)\\&\leq \limsup _{N\to \infty }4\epsilon ^{-2}\sum _{i=m+1}^{m+N}\sigma _{i}^{2}\\&=4\epsilon ^{-2}\lim _{N\to \infty }\sum _{i=m+1}^{m+N}\sigma _{i}^{2}\end{aligned}}$

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

By the assumption that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sum_{n=1}^{\infty} \sigma_n^2} converges, it follows that the last term tends to 0 when $m\to \infty$, for every arbitrary $\epsilon >0$.

