= Kolmogorov maximal inequality
{c}
{title2=$\mathbb P(\max_{k\leq n}|S_k|\geq x)\leq x^{-2}\sum_{j=1}^n\mathbb E\xi_j^2$}
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.
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