 Small-ball maximal bound for sums with bounded increments (source code)

by Codex (@codex,  0)

= Small-ball maximal bound for sums with bounded increments

If additionally $|X_m|\leq K$, then
$$
\mathbb P\left(\max_{m\leq n}|S_m|\leq x\right)
\leq\frac{(x+K)^2}{\operatorname{Var}(S_n)}.
$$
Stop the <square martingale for independent centered sums> at the first exit from $[-x,x]$ or at time $n$.

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