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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|00MRN0FARV]] real random variables.\nAssume that the logarithmic [[moment-generating function|0WQVXP7PDB]] $\\Lambda(\\lambda) = \\log E[e^{\\lambda X_1}]$ is finite for every $\\lambda \\in \\R$.\nLet $\\Lambda^*(x) = \\sup_{\\lambda \\in \\R} (\\lambda x - \\Lambda(\\lambda))$ be its convex conjugate, and let $\\bar X_n = (X_1 + \\cdots + X_n)/n$.\nThen the laws of $\\bar X_n$ satisfy the [[large deviation principle|MNPWBBNCAD]] on $\\R$ with the good [[rate function|140N2PQSTE]] $\\Lambda^*$.\nIn particular, for every $x \u003e E[X_1]$,\n$$\\lim_{n \\to \\infty} \\frac{1}{n} \\log P(\\bar X_n \\ge x) = -\\Lambda^*(x).$$\n\n## Remarks\n\nThe upper bound is the Chernoff bound.\nFor $\\lambda \\ge 0$, Markov's inequality gives $P(\\bar X_n \\ge x) \\le e^{-n \\lambda x} E[e^{\\lambda S_n}] = e^{-n(\\lambda x - \\Lambda(\\lambda))}$, with $S_n = n \\bar X_n$, and one optimises over $\\lambda$.\nThe lower bound uses a change of measure, the tilted law $e^{\\lambda y - \\Lambda(\\lambda)} \\, P(X_1 \\in dy)$ with $\\lambda$ chosen so that its mean is $x$.\nUnder the tilted law the event $\\bar X_n \\approx x$ is typical, and the cost of the change of measure is $\\Lambda^*(x)$.\n\nThe function $\\Lambda^*$ is convex, non-negative, and equal to $0$ exactly at the mean $E[X_1]$.\n\nExample: for standard normal $X_k$, $\\Lambda(\\lambda) = \\lambda^2/2$ and $\\Lambda^*(x) = x^2/2$.\nExample: for fair coin tosses with values $0$ and $1$, $\\Lambda^*(x) = x \\log x + (1 - x) \\log(1 - x) + \\log 2$ on $[0, 1]$, and $\\Lambda^* = \\infty$ outside $[0, 1]$.\n\nOn $\\R$ the large deviation principle holds for every i.i.d. sequence, with rate function $\\Lambda^*$, but without the finiteness assumption $\\Lambda^*$ need not be good.\n\n## Explanations\n\n### L1 human\n\nThe average of many independent copies of a random number is very likely close to the mean.\nCramér's theorem says how unlikely a given deviation is: the probability falls exponentially in the number of copies.\nThe exponent is computed from the moment-generating function by a Legendre transform.\n\n### L2 human\n\nThe theorem is the first large deviation principle and the model for the others.\nIts proof introduces the two main tools of the subject: exponential Chebyshev bounds for the upper bound and exponential tilting for the lower bound.\nIn $\\R^d$ the same result holds with $\\lambda x$ replaced by an inner product, and the Gärtner–Ellis theorem extends it to dependent sequences.\n[[Sanov's theorem|QWEF2WAXJM]] is the corresponding result for empirical measures; Cramér's theorem for finitely valued variables can be deduced from it by the contraction principle.\n",
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