Kolmogorov maximal inequality ($\mathbb{P}({\max }_{k\le n}|S_k|\ge x)\le x^{-2}{\sum }_{j=1}^n\mathbb{E}{\xi }_j^2$)

For independent zero-mean square-integrable increments, their partial sums form a martingale, and their squares form a nonnegative submartingale. Apply the Doob maximal inequality to the squared process at threshold $x^2$. Independence and zero means make its terminal second moment the sum of the increment variances. Identical distributions are not required.