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  "body": "\n## Statement\n\nLet $(X_n)_{n \\ge 0}$ be a sequence of real random variables, adapted to a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_n)_{n \\ge 0}$.\nA family $(Y_i)$ of random variables is uniformly integrable if $\\sup_i E\\bigl[\\abs{Y_i} \\mathbf{1}\\{\\abs{Y_i} \u003e K\\}\\bigr] \\to 0$ as $K \\to \\infty$.\n\n1. If $(X_n)$ is a [[submartingale|EQ055QMACW]] and $\\sup_n E[X_n^+] \u003c \\infty$, then $X_n$ converges [[almost surely|0AAXWGT5TD]] to a random variable $X_\\infty$ with $E[\\abs{X_\\infty}] \u003c \\infty$.\n2. If $(X_n)$ is a non-negative [[supermartingale|KYW1PV0SX5]], then $X_n$ converges almost surely to a random variable $X_\\infty$ with $E[X_\\infty] \\le E[X_0]$.\n3. Let $(X_n)$ be a [[martingale|0CDVK9RQBH]].\nThese are equivalent: $(X_n)$ is uniformly integrable; $X_n$ converges almost surely and in $L^1$; $X_n = E[X \\mid \\mathcal{F}_n]$ for an integrable $X$.\nIn this case $X_n = E[X_\\infty \\mid \\mathcal{F}_n]$ for all $n$, where $X_\\infty$ is the limit.\n\n## Remarks\n\nThe proof of part 1 uses [[Doob's upcrossing inequality|10HFW1GRV2]].\nThe bound on $E[X_n^+]$ makes the number of upcrossings of each rational interval finite almost surely, so the limit exists in $[-\\infty, \\infty]$.\nFatou's lemma shows that the limit is finite and integrable.\nPart 2 follows from part 1 applied to $-X_n$, and part 3 uses that uniform integrability upgrades almost sure convergence to $L^1$ convergence.\nExample: for independent $\\xi_i$ with $P(\\xi_i = 0) = P(\\xi_i = 2) = 1/2$, the product $M_n = \\xi_1 \\cdots \\xi_n$ is a non-negative martingale with $E[M_n] = 1$.\nIt converges almost surely to $0$ but not in $L^1$, so it is not uniformly integrable, which shows that part 3 needs its hypothesis.\nThe bound in part 1 is needed: the simple symmetric random walk is a martingale with $E[\\abs{S_n}] \\to \\infty$, and it does not converge.\nThe same statements hold in continuous time for submartingales with right-continuous paths.\n\n## Explanations\n\n### L1 human\n\nA fair or favourable game whose size stays under control must settle down to a limit.\nA gambler who keeps playing fair games with a fortune that cannot go negative ends up with a fortune that stops changing.\nBut the limit may lose mass: the expected value can drop in the limit unless the process is uniformly integrable.\n\n### L2 human\n\nThese theorems are the main reason martingales are useful in analysis and probability.\nThey give short proofs of the strong law of large numbers, of [[Lévy's upward theorem|W01FD2A8NK]] and of Kolmogorov's zero–one law.\nThey also give almost sure convergence of likelihood ratios, of branching process normalisations and of the Radon–Nikodym derivatives on finer and finer partitions.\nThe $L^p$ version, for $p \u003e 1$, says that a martingale bounded in $L^p$ converges almost surely and in $L^p$, by [[Doob's martingale inequality|4G53E7BRXF]].\n",
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