central limit theoremnormality-assumptionresiduals
I have often read that thanks to the CLT, the residuals of a model are asymptotically normal. This argument always seemed odd to me since CLT states that
The sum of a number of independent and identically distributed random variables are asymptotically distributed.
Since a residual is generally a difference and not a sum of iid random variables, how can we can claim that the CLT applies?
The short answer as I understand your q is "yes, but ..." the rates of convergence on S, T, and any other moments are not necessarily the same -- check out determining bounds with the Berry-Esseen Theorem.
In case I misunderstand your q, Sn and Tn even hold to the CLT under conditions of weak dependence (mixing): check out Wikipedia's CLT for dependent processes.
CLT is such a general theorem -- the basic proof requires nothing more than the characteristic function of Sn and Tn converges to the characteristic function of the standard normal, then Levy Continuity Theorem says the convergence of the characteristic function implies convergence of the distribution.
John Cook provides a great explanation of CLT error here.
This figure shows the distributions of the means of $n=1$ (blue), $10$ (red), and $100$ (gold) independent and identically distributed (iid) normal distributions (of unit variance and mean $\mu$):
As $n$ increases, the distribution of the mean becomes more "focused" on $\mu$. (The sense of "focusing" is easily quantified: given any fixed open interval $(a,b)$ surrounding $\mu$, the amount of the distribution within $[a,b]$ increases with $n$ and has a limiting value of $1$.)
However, when we standardize these distributions, we rescale each of them to have a mean of $0$ and a unit variance: they are all the same then. This is how we see that although the PDFs of the means themselves are spiking upwards and focusing around $\mu$, nevertheless every one of these distributions is still has a Normal shape, even though they differ individually.
The Central Limit Theorem says that when you start with any distribution--not just a normal distribution--that has a finite variance, and play the same game with means of $n$ iid values as $n$ increases, you see the same thing: the mean distributions focus around the original mean (the Weak Law of Large Numbers), but the standardized mean distributions converge to a standard Normal distribution (the Central Limit Theorem).
Best Answer
What i think you are referring to is that estimators are often asymptotically normal with increase of the sample size. It's a mathematically correct statement that follows from more general formulation of CLT, Lyapunov's CLT. It doesn't require having identical distribution of variables, just independence. For example, OLS estimator is $\hat{\beta}=\beta+P_{x}\epsilon$, where projection operator $P_{x}=(X^{T}X)^{-1}X^{T}$, vector of errors is $\epsilon$. In other words the difference from the actual $\beta$ is linear combination of errors, which are independently distributed. Hence by Lyapunov CLT this difference is asympyotically normally distributed.
In terms of what you literally wrote, i.e. "residuals of a model are asymptotically normal" it is not clear what asymptotic limit we are talking about here. There is a sense in which errors themselves could be approximately normal, i.e. when they are modeled as a sum of very large number of unknown factors. When this assumption is applicable, once again Lyapunov CLT suggests they are approximately normal. Again, i am not sure that is what you are asking, because strictly there is no asymptotical equality here. Also, i suspect there is another confused notion about squared residuals being distributed normally. It would be more optimal to ask the question more specifically.