0.1 Kolmogorov’s Maximal Inequality for Sums
Let \(S_{n}= \xi_{1}+\dots + \xi_{n}\) for independent \(\xi_{i}\)s with \(\mathbb{E}\xi_{i}^2<\infty\). Then for \(\lambda>0\) \[ \mathbb{P}\left(\max_{1\leq k \leq n}(\lvert S_{k}-\mathbb{E}S_{k} \rvert )\geq \lambda\right)\leq \frac{\text{Var}(S_{n})}{\lambda^2}. \]
\begin{proof} Applying Doob-Kolmogorov Inequality to the martingale \(S_{n}-\mathbb{E}S_{n}\) we have \[
\mathbb{P}\left( \max_{1 \leq k \leq n}(\lvert S_{k}-\mathbb{E}S_{k} \rvert )\geq \lambda \right) \leq \frac{{\mathbb{E}\lvert S_{n}-\mathbb{E}S_{n} \rvert^2 }}{\lambda^2}= \frac{\text{Var}(S_{n})}{\lambda^2}.
\] \end{proof}