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  "body": "\n## Statement\n\nLet $(X_k)_{k = 0}^n$ be a [[martingale|0D0GBNDKDH]] with respect to a filtration $(\\mathcal{F}_k)_{k = 0}^n$.\nAssume that there are constants $c_1, \\dots, c_n \u003e 0$ with $\\abs{X_k - X_{k-1}} \\le c_k$ almost surely for $k = 1, \\dots, n$.\nThen for every $t \u003e 0$,\n$$P(X_n - X_0 \\ge t) \\le \\exp\\left(-\\frac{t^2}{2 \\sum_{k=1}^n c_k^2}\\right),$$\nand hence $P(\\abs{X_n - X_0} \\ge t) \\le 2 \\exp\\left(-\\frac{t^2}{2 \\sum_{k=1}^n c_k^2}\\right)$.\n\n## Remarks\n\nProof idea: let $D_k = X_k - X_{k-1}$. Since $E[D_k \\mid \\mathcal{F}_{k-1}] = 0$ and $D_k \\in [-c_k, c_k]$, Hoeffding's lemma applied to the [[conditional expectation|0TTDGTEFMN]] gives $E[e^{\\lambda D_k} \\mid \\mathcal{F}_{k-1}] \\le e^{\\lambda^2 c_k^2 / 2}$.\nBy induction $E[e^{\\lambda (X_n - X_0)}] \\le e^{\\lambda^2 \\sum_k c_k^2 / 2}$, and Markov's inequality with $\\lambda = t / \\sum_k c_k^2$ gives the bound.\nFor independent centred $D_k$ the result is a form of [[Hoeffding's inequality|0WPDNXGSSN]], and it is also called the Azuma–Hoeffding inequality.\nExample: for a simple random walk $X_n = D_1 + \\dots + D_n$ with independent fair $\\pm 1$ steps, $P(X_n \\ge t) \\le e^{-t^2 / (2n)}$.\n\n## Explanations\n\n### L1 human\n\nA martingale is a fair game: on average each step adds nothing.\nSuppose each step can change the fortune by at most a fixed amount.\nThen the total after $n$ steps is very unlikely to be far from the start.\nWhen each step is at most $1$, the chance of being more than $t$ away drops like $e^{-t^2 / (2n)}$.\n\n### L2 human\n\nAzuma's inequality is applied to [[Doob martingales|Z57CG6W9S6]] $X_k = E[f \\mid \\mathcal{F}_k]$ of a function $f$ of independent variables, revealed one at a time.\nThis gives concentration of quantities such as the chromatic number of a random graph, and it leads to [[McDiarmid's inequality|8APFT1RPM4]].\n",
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