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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 $\\sigma^2 = E[X_1^2] \\in (0, \\infty)$.\nLet $S_0 = 0$ and $S_k = X_1 + \\cdots + X_k$.\nFor $n \\ge 1$ and $t \\in [0, 1]$, let\n$$W_n(t) = \\frac{1}{\\sigma \\sqrt{n}} \\left( S_{\\floor{nt}} + (nt - \\floor{nt}) \\, X_{\\floor{nt} + 1} \\right),$$\nthe linear interpolation of the rescaled [[random walk|04VZE3FZXF]].\nLet $C[0, 1]$ be the space of [[continuous|0X3WGX95FE]] functions on $[0, 1]$ with the supremum norm.\nThen $W_n$ [[converges in distribution|0SRYJGXKR4]], as random elements of $C[0, 1]$, to a standard [[Wiener process|22735FGJV6]] $(W(t))_{t \\in [0, 1]}$.\n\n## Remarks\n\nThe proof has two steps.\nThe [[central limit theorem|0EPZC4Z1FC]], applied to the independent increments, gives the convergence of the finite-dimensional distributions of $W_n$ to those of $W$.\nA maximal inequality then shows that the laws of the $W_n$ are tight in $C[0, 1]$, and tightness together with convergence of finite-dimensional distributions gives [[weak convergence|MBW9JCGZC1]].\n\nExample: the supremum map $f \\mapsto \\max_{t \\in [0, 1]} f(t)$ is continuous on $C[0, 1]$.\nBy the [[continuous mapping theorem|CFRN0G7GDB]], $\\max_{k \\le n} S_k / (\\sigma \\sqrt{n})$ converges in distribution to $\\max_{t \\le 1} W(t)$.\nBy the [[reflection principle|K6R7GHB3VS]], this limit has the law of $\\abs{W(1)}$.\nSo the limit does not depend on the law of $X_1$, which is why the result is called an invariance principle.\n\nA second form of the theorem concerns the [[empirical process|XKFG1676CG]].\nIf $U_1, U_2, \\dots$ are i.i.d. uniform on $[0, 1]$ with empirical distribution function $F_n$, then $\\sqrt{n}(F_n(t) - t)$ converges in distribution to a [[Brownian bridge|3MCVMBAGD1]], in a suitable space of functions with jumps.\n\n## Explanations\n\n### L1 human\n\nDraw the path of a random walk with many small steps and shrink it so that it fits in a unit box.\nAs the steps get smaller, the picture looks more and more like the path of Brownian motion.\nThis holds for any step law with mean zero and finite variance, so Brownian motion is a universal limit.\n\n### L2 human\n\nDonsker's theorem is the functional form of the central limit theorem.\nWith the continuous mapping theorem, it gives the limit laws of functionals of random walks, computed once for Brownian motion.\nExamples are the maximum, the time spent above zero and the time of the last zero.\nIn statistics, the empirical-process form gives the limit law of the Kolmogorov–Smirnov statistic $\\sqrt{n} \\sup_x \\abs{F_n(x) - F(x)}$ for a continuous $F$.\n",
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