Lyapunov CLT

inputs/lecture_13.tex

@@ -35,7 +35,8 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$,
     and $S_n \coloneqq  \sqrt{\sum_{i=1}^n \sigma_i^2}$.
     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}).
     Then the CLT holds.