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  "body": "\n## Definition\n\nLet $X, X_1, X_2, \\dots$ be real random variables, possibly on different probability spaces, with [[cumulative distribution functions|0MXMS3AE13]] $F, F_1, F_2, \\dots$.\nThe sequence $(X_n)$ converges in distribution to $X$ if\n$$\\lim_{n \\to \\infty} F_n(x) = F(x)$$\nfor each $x \\in \\mathbb{R}$ at which $F$ is continuous. One writes $X_n \\xrightarrow{d} X$.\n\n## Remarks\n\nBy the portmanteau theorem, an equivalent condition is $E[g(X_n)] \\to E[g(X)]$ for each bounded continuous $g \\colon \\mathbb{R} \\to \\mathbb{R}$. This form extends to random variables in metric spaces.\nThe condition depends only on the laws. So it is the weak convergence of the laws of the $X_n$ to the law of $X$.\nThe restriction to continuity points is necessary. For $X_n = 1/n$ and $X = 0$, $F_n(0) = 0$ for all $n$, but $F(0) = 1$.\nThe central limit theorem is a result about convergence in distribution.\n\n## Explanations\n\n### L1 human\n\nConvergence in distribution means that the probabilities for $X_n$ come close to those for $X$. The values of $X_n$ themselves need not come close to the values of $X$.\nFor example, the number of heads in $n$ tosses, after standardization, converges in distribution to a standard normal variable.\n",
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