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  "body": "\n## Statement\n\nLet $X, X_1, X_2, \\dots$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\\varphi, \\varphi_1, \\varphi_2, \\dots$.\n1. If $X_n \\to X$ [[in distribution|0JDAQHWA9N]], then $\\varphi_n(t) \\to \\varphi(t)$ for each $t \\in \\mathbb{R}$.\n2. Conversely, suppose that $\\varphi_n(t)$ converges to a limit $\\psi(t)$ for each $t \\in \\mathbb{R}$, and that $\\psi$ is continuous at $0$. Then $\\psi$ is the characteristic function of a random variable $Y$, and $X_n \\to Y$ in distribution.\n\n## Remarks\n\nPart 1 follows from the definition of convergence in distribution with the bounded continuous functions $\\cos(tx)$ and $\\sin(tx)$.\nThe continuity at $0$ in part 2 is necessary. For $X_n \\sim \\mathcal{N}(0, n)$, $\\varphi_n(t) = e^{-nt^2/2}$ tends to $1_{\\{t = 0\\}}$, which is not continuous at $0$, and $X_n$ does not converge in distribution.\nThe theorem holds in the same form for random vectors in $\\mathbb{R}^k$.\n\n## History\n\nThe theorem is named after Paul Lévy.\n\n## Explanations\n\n### L1 human\n\nTo show that random variables converge in distribution, it is enough to show that their characteristic functions converge.\nThis is often much easier, because characteristic functions of sums of independent variables are products.\n",
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