TITLE: Intuitive explanation of Lyapunov condition for CLT
QUESTION [13 upvotes]: I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables:
Lyapunov CLT. Let $s_n^2 = \sum_{k=1}^n \text{Var}[Y_i]$ and let $Y=\sum_i Y_i$. If there exists $\ell>0$ s.t.: 
$\lim_{n\rightarrow\infty}\left( \frac{1}{s_n^{2+\ell}}\sum_{k=1}^n \text{E}\left[ |Y_k - \text{E}[Y_k]|^{2+\ell}\right] \right) = 0$, then $Z=(Y - E[Y])/\sqrt{Var[Y]}$ converges to the standard normal distribution. 
While I have no problem showing that this condition holds for certain exercises, I'm wondering what the intuition is behind this condition.

REPLY [14 votes]: Every CLT is basically a rendering of the fact that "many" "not-too-large" and "not-too-correlated" random increments average out in a bell shape. All three conditions ("many", "not-too-large", "not-too-correlated") are important. Lyapunov condition is one way of quantifying the "not-too-large increments" condition. 
Naturally each random variable $Y_k$ may be unbounded hence the condition bears on their moments. Lyapunov showed that some averaged control on some $(2+\delta)$-moment was enough to guarantee the conclusion. "Averaged" is good here since it allows for some exceptionally large individual $(2+\delta)$-moments. Uniform $(2+\delta)$-integrability is more restrictive, and some variants of the CLT weaken this condition.
The fact that a "not-too-large increments" condition of some sort is required should not come as a surprise, the idea being that if the size of some increment is not negligible then its realization might influence noticeably the result.