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  "body": "\n## Statement\n\nLet $(S, \\mathcal{S})$ be a measurable space, and let $X_1, X_2, \\dots$ be [[independent and identically distributed|00MRN0FARV]] random variables with values in $S$.\nLet $B \\subseteq S^{\\N}$ be a measurable set for the product σ-algebra.\nAssume that $B$ is symmetric: for every [[permutation|05RWFKVTJ7]] $\\pi$ of $\\N$ that moves only finitely many indices, $(x_n)_n \\in B$ if and only if $(x_{\\pi(n)})_n \\in B$.\nThen the [[event|031TPJVS74]] $A = \\set{(X_1, X_2, \\dots) \\in B}$ has $P(A) \\in \\set{0, 1}$.\n\n## Remarks\n\nProof idea: approximate $A$ by an event $A_n$ that depends only on $X_1, \\dots, X_n$.\nLet $A_n'$ be the image of $A_n$ under the permutation that swaps the blocks $1, \\dots, n$ and $n+1, \\dots, 2n$.\nBy symmetry $A_n'$ approximates $A$ as well, and $A_n, A_n'$ are independent with equal probability.\nSo $P(A) = \\lim P(A_n \\cap A_n') = \\lim P(A_n)^2 = P(A)^2$.\nEvery tail event is symmetric, so the theorem extends [[Kolmogorov's zero–one law|073Q8Z94KG]] in the i.i.d. case.\nExample: let $X_i$ be real and $S_n = X_1 + \\dots + X_n$. The event $\\set{S_n \\in I \\text{ for infinitely many } n}$, for an interval $I$, is symmetric, because a finite permutation leaves $S_n$ unchanged for all large $n$. It is not a tail event in general, since changing $X_1$ shifts all $S_n$.\nThe identical distribution is needed: if $X_1$ is a fair coin and $X_n = 0$ for $n \\ge 2$, then $\\set{\\text{some } X_n = 1}$ is symmetric and has probability $\\frac{1}{2}$.\n\n## Explanations\n\n### L1 human\n\nTake an infinite sequence of independent draws from one distribution.\nAny event that does not care about the order of finitely many draws is either almost certain or almost impossible.\nFor instance, whether a random walk returns to its start infinitely often has probability $0$ or $1$.\n\n### L2 human\n\nThe theorem is used to show that quantities such as $\\limsup_n S_n$ for a random walk are almost surely constant, and to obtain the recurrence dichotomy for random walks.\nCompared with [[Kolmogorov's zero–one law|073Q8Z94KG]], it needs identical distribution but covers more events, namely all events of the exchangeable σ-algebra.\nIn the language of [[de Finetti's theorem|W19129PVKQ]], it says that an i.i.d. sequence is an exchangeable sequence whose mixing measure is a point mass.\n",
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