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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \\dots + X_n$.\nThen for each $t \\in \\mathbb{R}$, the [[characteristic functions|0MYDNRT7PP]] satisfy\n$$\\varphi_S(t) = \\varphi_{X_1}(t) \\, \\varphi_{X_2}(t) \\cdots \\varphi_{X_n}(t).$$\n\n## Remarks\n\nProof: $e^{itS} = e^{itX_1} \\cdots e^{itX_n}$. The factors are independent and bounded, so the [[product rule for expectations|05AW1G0WE1]] applies to their real and imaginary parts.\nFor i.i.d. $X_i$, $\\varphi_S(t) = \\varphi_{X_1}(t)^n$. The sample mean has $\\varphi_{S/n}(t) = \\varphi_{X_1}(t/n)^n$.\nThe converse is false: $\\varphi_{X+Y} = \\varphi_X \\varphi_Y$ does not imply that $X$ and $Y$ are independent.\n\n## Explanations\n\n### L1 human\n\nAdding independent variables corresponds to multiplying their characteristic functions.\nThis is the key step in the usual proof of the central limit theorem.\n",
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