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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]].\nFor $A \u003e 0$, let $Y_n = X_n \\mathbf{1}_{\\set{\\abs{X_n} \\le A}}$, the variable $X_n$ truncated at level $A$.\nThen the [[series|0NM626FH6Q]] $\\sum_{n} X_n$ converges [[almost surely|0ST21PGDAP]] if and only if, for some $A \u003e 0$, the following three series converge:\n\n1. $\\sum_{n} P(\\abs{X_n} \u003e A)$,\n2. $\\sum_{n} E[Y_n]$,\n3. $\\sum_{n} \\operatorname{Var}(Y_n)$.\n\n## Remarks\n\nIf the three series converge for some $A \u003e 0$, then they converge for every $A \u003e 0$.\nBy [[Kolmogorov's zero–one law|073Q8Z94KG]], the event that $\\sum_n X_n$ converges has probability $0$ or $1$, so when the series do not all converge, $\\sum_n X_n$ diverges almost surely.\n\nIdea of the proof of sufficiency.\nSeries 3 and [[Kolmogorov's inequality|D9B4HE4B4Q]] show that $\\sum_n (Y_n - E[Y_n])$ converges almost surely.\nSeries 2 then gives the convergence of $\\sum_n Y_n$.\nSeries 1 and the [[first Borel–Cantelli lemma|0T6WS7TT77]] show that $X_n = Y_n$ for all but finitely many $n$, almost surely.\nThe proof of necessity uses a symmetrisation argument and a lower bound for the probability that a sum of bounded centred variables stays small.\n\nExample: let $\\varepsilon_n$ be independent signs, equal to $1$ or $-1$ with probability $1/2$ each, and let $X_n = \\varepsilon_n / n^{p}$ with $p \u003e 0$.\nWith $A = 1$, series 1 and 2 are $0$, and series 3 is $\\sum_n n^{-2p}$.\nSo the random series $\\sum_n \\varepsilon_n / n^p$ converges almost surely if $p \u003e 1/2$ and diverges almost surely if $p \\le 1/2$.\n\n## Explanations\n\n### L1 human\n\nWhen you add up infinitely many independent random numbers, the sum may or may not settle down.\nThe theorem gives an exact test with three ordinary series.\nThe large values must be rare, the average sizes of the small values must add up, and their fluctuations must add up.\n\n### L2 human\n\nThe theorem completely describes the almost sure convergence of series of independent variables.\nIt combines [[Kolmogorov's inequality|D9B4HE4B4Q]], the [[Borel–Cantelli lemma|0T6WS7TT77]] and the zero–one law.\nFor independent variables, almost sure convergence, convergence in probability and convergence in distribution of $\\sum_n X_n$ are equivalent, a result of Lévy.\nA common use is to show that random series such as $\\sum_n \\varepsilon_n a_n$ converge almost surely exactly when $\\sum_n a_n^2 \u003c \\infty$.\n",
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