Lyapunov CLT
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@ -35,7 +35,8 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$,
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and $S_n \coloneqq \sqrt{\sum_{i=1}^n \sigma_i^2}$.
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Then, assume that, for some $\delta > 0$,
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\[
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\lim_{n \to \infty} \sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0
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\lim_{n \to \infty} \frac{1}{S_n^{2+\delta}}
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\sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0
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\]
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(\vocab{Lyapunov condition}).
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Then the CLT holds.
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