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  "body": "\n## Summary\n\nThe empirical process of a sample is the gap between its empirical distribution function and the true [[cumulative distribution function|0H8H7ZTHW2]], times $\\sqrt{n}$.\nIt is the main object behind uniform laws of large numbers and functional central limit theorems in statistics.\nThe [[Glivenko–Cantelli theorem|E7B78X780X]] and [[Donsker's theorem|818832PMSV]] describe its behaviour as the sample size grows.\n\n## Explanations\n\n### L1 human\n\nDraw $n$ numbers at random from a distribution and plot the fraction of them below each level $x$.\nThis staircase is the empirical distribution function, and it should be close to the true distribution function $F$.\nThe empirical process is the error between the two curves, multiplied by $\\sqrt{n}$ so that it stays of order one.\nFor one fixed $x$ it is just the rescaled error of a proportion, which the central limit theorem says is roughly normal.\nSeen as a random curve in $x$, it looks like a [[Brownian bridge|3MCVMBAGD1]] for large $n$.\n",
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