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  "body": "\n## Definition\n\nLet $X_1, X_2, \\dots$ be [[independent and identically distributed|00MRN0FARV]] real random variables with [[cumulative distribution function|0H8H7ZTHW2]] $F$.\nFor $n \\ge 1$, the *empirical distribution function* is\n$$F_n(x) = \\frac{1}{n} \\sum_{i=1}^{n} \\mathbf{1}\\{X_i \\le x\\}, \\qquad x \\in \\R.$$\nThe *empirical process* is the [[stochastic process|0QH5ENKZM1]] indexed by $x \\in \\R$ given by\n$$G_n(x) = \\sqrt{n}\\,\\bigl(F_n(x) - F(x)\\bigr).$$\n\n## Remarks\n\nFor each fixed $x$, $n F_n(x)$ is binomial with parameters $n$ and $F(x)$.\nSo $E[G_n(x)] = 0$ and $\\operatorname{Var}(G_n(x)) = F(x)(1 - F(x))$, and by the central limit theorem $G_n(x)$ converges in distribution to a normal law with these parameters.\nFor the uniform law on $[0, 1]$, [[Donsker's theorem|818832PMSV]] says that $G_n$ converges in distribution, as a random function, to a [[Brownian bridge|3MCVMBAGD1]].\nA non-example: if $X_i = X_1$ for all $i$, the variables are not independent, $F_n(x) = \\mathbf{1}\\{X_1 \\le x\\}$ for all $n$, and $\\sqrt{n}(F_n - F)$ does not have the behaviour above.\nMore generally, for a probability measure $P$ and a class $\\mathcal{F}$ of functions, the empirical process indexed by $\\mathcal{F}$ is $f \\mapsto \\sqrt{n}\\bigl(\\frac{1}{n}\\sum_{i=1}^n f(X_i) - \\int f \\, dP\\bigr)$; the case above takes the indicator functions $\\mathbf{1}_{(-\\infty, x]}$.\n",
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