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  "body": "\n## Definition\n\nLet $X_1, X_2, \\dots$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]] with finite [[variances|0477R9PYAE]].\nLet $\\mu_k = E[X_k]$, let $\\sigma_k^2 = \\operatorname{Var}(X_k)$, and let $s_n^2 = \\sigma_1^2 + \\cdots + \\sigma_n^2$, with $s_n \u003e 0$ for all $n$.\nThe sequence satisfies *Lindeberg's condition* if, for every $\\varepsilon \u003e 0$,\n$$\\lim_{n \\to \\infty} \\frac{1}{s_n^2} \\sum_{k=1}^{n} E\\left[(X_k - \\mu_k)^2 \\, \\mathbf{1}_{\\set{\\abs{X_k - \\mu_k} \u003e \\varepsilon s_n}}\\right] = 0.$$\n\n## Remarks\n\nThe Lindeberg central limit theorem states that under this condition $(S_n - \\sum_{k \\le n} \\mu_k) / s_n$ converges in distribution to a standard normal variable, where $S_n = X_1 + \\cdots + X_n$.\n\nExample: an i.i.d. sequence with variance $\\sigma^2 \\in (0, \\infty)$ satisfies the condition.\nHere $s_n^2 = n \\sigma^2$, and the sum equals $\\sigma^{-2} E[(X_1 - \\mu_1)^2 \\mathbf{1}_{\\set{\\abs{X_1 - \\mu_1} \u003e \\varepsilon \\sigma \\sqrt{n}}}]$.\nIt tends to $0$ by dominated convergence, since $E[(X_1 - \\mu_1)^2] \u003c \\infty$.\n\nNon-example: let $X_1$ be standard normal and $X_k = 0$ for $k \\ge 2$.\nThen $s_n = 1$ for all $n$, and the sum equals $E[X_1^2 \\mathbf{1}_{\\set{\\abs{X_1} \u003e \\varepsilon}}] \u003e 0$, which does not tend to $0$.\nIn this case one variable carries all the variance.\n\nThe condition implies Feller's condition $\\max_{k \\le n} \\sigma_k^2 / s_n^2 \\to 0$.\nThe Lyapunov condition, $s_n^{-2-\\delta} \\sum_{k \\le n} E[\\abs{X_k - \\mu_k}^{2+\\delta}] \\to 0$ for some $\\delta \u003e 0$, implies Lindeberg's condition and is often easier to check.\n",
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