Kolmogorov's Maximal Inequality and the Convergence of Series of Independent Square-Integrable Random Variables

In the setting of The Real Numbers: Standing Notation and Background, let $(\Omega,\mathcal{F},P)$ be a probability space and let $(\xi_{k})_{k\in\mathbb{N}}$ be an independent sequence of square-integrable random variables on it, each with expectation $\mathbb{E}[\xi_{k}]=0$, such that the series $\sum_{k=1}^{\infty}\mathbb{E}[\xi_{k}^{2}]$ converges. Let $T_{0}:\Omega\to\mathbb{R}$ be the zero function and, for $n\in\mathbb{N}$, let $T_{n}=\sum_{k=1}^{n}\xi_{k}$, defined pointwise. Let $C$ be the set of those $\omega\in\Omega$ for which the real sequence $(T_{n}(\omega))_{n\in\mathbb{N}}$ converges, and define $T:\Omega\to\mathbb{R}$ by $T(\omega)=\lim_{n\to\infty}T_{n}(\omega)$ for $\omega\in C$ and $T(\omega)=0$ for $\omega\notin C$. Then the following hold.

1. (Kolmogorov's maximal inequality)¶ Let $n$ be zero or a natural number, let $m\in\mathbb{N}$ with $n<m$, and let $\varepsilon\in\mathbb{R}$ be positive. Then the set $A$ of those $\omega\in\Omega$ with $|T_{j}(\omega)-T_{n}(\omega)|\ge\varepsilon$ for some integer $j$ with $n<j\le m$ is an event, and

$$P(A)\le\frac{1}{\varepsilon^{2}}\sum_{k=n+1}^{m}\mathbb{E}[\xi_{k}^{2}].$$

2. (Almost sure convergence)¶ $C\in\mathcal{F}$ and $P(C)=1$.

3. (Mean-square convergence)¶ $T$ is a square-integrable random variable, and for every $n$ that is zero or a natural number,

$$\mathbb{E}\bigl[(T-T_{n})^{2}\bigr]=\sum_{k=n+1}^{\infty}\mathbb{E}[\xi_{k}^{2}].$$

In particular $\mathbb{E}[T]=0$, $\mathbb{E}[T^{2}]=\sum_{k=1}^{\infty}\mathbb{E}[\xi_{k}^{2}]$, and $\mathbb{E}[(T-T_{n})^{2}]\to0$ as $n\to\infty$.