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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$ and $E[X_1^2] = 1$.\nLet $S_n = X_1 + \\cdots + X_n$.\nThen, [[almost surely|0ST21PGDAP]],\n$$\\limsup_{n \\to \\infty} \\frac{S_n}{\\sqrt{2 n \\log \\log n}} = 1 \\quad \\text{and} \\quad \\liminf_{n \\to \\infty} \\frac{S_n}{\\sqrt{2 n \\log \\log n}} = -1,$$\nwhere $\\log$ is the [[natural logarithm|04DXAFN2ES]] and $\\limsup$ is the [[limit superior|0T3RZQXDYD]].\n\n## Remarks\n\nThis version under a finite second moment is the Hartman–Wintner theorem; Khinchin proved the first case, for coin tossing.\nOne proof first treats the [[Wiener process|22735FGJV6]] $W$, for which $\\limsup_{t \\to \\infty} W_t / \\sqrt{2 t \\log \\log t} = 1$ almost surely.\nThe upper bound uses the Gaussian tail and the [[Borel–Cantelli lemma|0T6WS7TT77]] along a geometric sequence of times $t_k = q^k$.\nThe lower bound uses the [[second Borel–Cantelli lemma|0NWXRAMEE7]] for the independent increments between these times.\nThe case of general $X_k$ then follows by a strong approximation of the walk by Brownian motion, such as [[Skorokhod's embedding theorem|XHTYZ7ER4Z]].\n\nExample: take a simple symmetric [[random walk|04VZE3FZXF]], with steps $\\pm 1$, and fix $\\varepsilon \u003e 0$.\nAlmost surely, the walk exceeds $(1 - \\varepsilon)\\sqrt{2 n \\log \\log n}$ for infinitely many $n$.\nAlmost surely, it exceeds $(1 + \\varepsilon)\\sqrt{2 n \\log \\log n}$ for only finitely many $n$.\n\nThe theorem sits between two other limit theorems.\nThe [[strong law of large numbers|0DFWHQ839T]] gives $S_n / n \\to 0$ almost surely.\nThe [[central limit theorem|0EPZC4Z1FC]] gives that $S_n / \\sqrt{n}$ converges in distribution, and it implies that $\\limsup_n S_n / \\sqrt{n} = \\infty$ almost surely.\n\n## Explanations\n\n### L1 human\n\nA random walk with fair steps of size about $1$ drifts away from $0$ by about $\\sqrt{n}$ after $n$ steps.\nOver an infinite time, it reaches slightly further now and then.\nThe law of the iterated logarithm gives the exact size of its record excursions: $\\sqrt{2 n \\log \\log n}$.\n\n### L2 human\n\nThe result describes the almost sure size of the fluctuations of sums, which neither the strong law nor the central limit theorem determines.\nThe normalisation $\\sqrt{2 n \\log \\log n}$ is the boundary between the scale $\\sqrt{n}$ of the central limit theorem and larger scales that the walk exceeds only finitely often.\nStrassen's functional form describes the whole set of limit points of the rescaled path.\n",
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