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  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0PNRKMRC55]].\n$X$ is *sub-Gaussian* if there is a constant $K \u003e 0$ such that\n$$P(\\abs{X} \\ge t) \\le 2 e^{-t^2 / K^2} \\qquad \\text{for all } t \\ge 0.$$\nA distribution on $\\R$ is sub-Gaussian if a random variable with this distribution is sub-Gaussian.\n\n## Remarks\n\nUp to changing $K$ by a constant factor, the condition is equivalent to each of the following: $E[e^{X^2/K^2}] \\le 2$; and $(E[\\abs{X}^p])^{1/p} \\le K \\sqrt{p}$ for all $p \\ge 1$.\nIf $E[X] = 0$, it is also equivalent to a bound on the [[moment-generating function|0WQVXP7PDB]]: there is $\\sigma \u003e 0$ with $E[e^{\\lambda X}] \\le e^{\\lambda^2 \\sigma^2 / 2}$ for all real $\\lambda$.\nIn this last form one says that $X$ is sub-Gaussian with variance proxy $\\sigma^2$.\nExample: if $X$ is [[normal|05AQE0QR0D]] with mean $0$ and variance $\\sigma^2$, then $E[e^{\\lambda X}] = e^{\\lambda^2 \\sigma^2 / 2}$, so $X$ is sub-Gaussian with variance proxy $\\sigma^2$.\nExample: by Hoeffding's lemma, if $a \\le X \\le b$ and $E[X] = 0$, then $E[e^{\\lambda X}] \\le e^{\\lambda^2 (b - a)^2 / 8}$, so $X$ is sub-Gaussian with variance proxy $(b - a)^2 / 4$.\nNon-example: an [[exponential|0E99F51GVB]] variable with rate $1$ has $P(X \\ge t) = e^{-t}$, and $e^{-t} \\le 2 e^{-t^2/K^2}$ fails for large $t$, whatever $K$ is.\nIf $X_1, \\dots, X_n$ are independent and centred with variance proxies $\\sigma_i^2$, then $X_1 + \\dots + X_n$ has variance proxy $\\sigma_1^2 + \\dots + \\sigma_n^2$.\n",
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