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Lyapunov CLT

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Josia Pietsch 2023-07-13 00:17:26 +02:00
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@ -35,7 +35,8 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$,
and $S_n \coloneqq \sqrt{\sum_{i=1}^n \sigma_i^2}$. and $S_n \coloneqq \sqrt{\sum_{i=1}^n \sigma_i^2}$.
Then, assume that, for some $\delta > 0$, Then, assume that, for some $\delta > 0$,
\[ \[
\lim_{n \to \infty} \sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0 \lim_{n \to \infty} \frac{1}{S_n^{2+\delta}}
\sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0
\] \]
(\vocab{Lyapunov condition}). (\vocab{Lyapunov condition}).
Then the CLT holds. Then the CLT holds.