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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]] with $E[X_k] = 0$ and $E[X_k^2] \u003c \\infty$ for each $k$.\nLet $S_k = X_1 + \\cdots + X_k$ for $1 \\le k \\le n$.\nThen for every $\\lambda \u003e 0$,\n$$P\\left(\\max_{1 \\le k \\le n} \\abs{S_k} \\ge \\lambda\\right) \\le \\frac{\\operatorname{Var}(S_n)}{\\lambda^2} = \\frac{1}{\\lambda^2} \\sum_{k=1}^{n} E[X_k^2].$$\n\n## Remarks\n\nLet $\\tau$ be the first $k$ with $\\abs{S_k} \\ge \\lambda$.\nThe event $\\set{\\tau = k}$ depends only on $X_1, \\dots, X_k$.\nBy independence the cross term $E[S_k (S_n - S_k) \\mathbf{1}_{\\tau = k}]$ vanishes, so $E[S_n^2 \\mathbf{1}_{\\tau = k}] \\ge E[S_k^2 \\mathbf{1}_{\\tau = k}] \\ge \\lambda^2 P(\\tau = k)$.\nSumming over $k$ gives $\\lambda^2 P(\\tau \\le n) \\le E[S_n^2]$.\n\nThe bound has the same right side as [[Chebyshev's inequality|02HZ1BK3MW]] for the single variable $S_n$, but it controls the whole path $S_1, \\dots, S_n$.\nIt is a special case of [[Doob's martingale inequality|4G53E7BRXF]], since $(S_k)$ is a [[martingale|0D0GBNDKDH]].\n\nExample: let the $X_k$ be independent signs, equal to $1$ or $-1$ with probability $1/2$ each.\nThen the probability that $\\abs{S_k} \\ge \\lambda$ for some $k \\le n$ is at most $n / \\lambda^2$.\nWith $\\lambda = 10 \\sqrt{n}$ this is at most $1/100$.\n\n## Explanations\n\n### L1 human\n\nA sum of independent centred random steps rarely wanders far from zero.\nChebyshev's inequality says this for the final position.\nKolmogorov's inequality says that the same bound holds for the farthest point reached along the way.\n\n### L2 human\n\nThe inequality is the key step in the proof that a series $\\sum X_k$ of independent centred variables with $\\sum \\operatorname{Var}(X_k) \u003c \\infty$ converges almost surely.\nThat result leads to [[Kolmogorov's three-series theorem|TTTFQ693RJ]] and to the [[strong law of large numbers|0DFWHQ839T]] for independent variables with suitable finite variances.\n",
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