Kolmogorov’s inequality

Let $X_1,\mathrm{\dots },X_n$ be independent random variables in a probability space, such that $\mathrm{E}[X_k]=0$ and for$k=1,\mathrm{\dots },n$. Then, for each $\lambda >0$,

$$P(\underset{1\le k\le n}{\max }|S_k|\ge \lambda )\le \frac{1}{{\lambda }^2}\mathrm{Var}[S_n]=\frac{1}{{\lambda }^2}\sum _{k=1}^n\mathrm{Var}[X_k],$$

where $S_k=X_1+\mathrm{\cdots }+X_k$.