Does a vast amount of probability and statistical literature make a mistake when they make use of CLT/asymptotic normality?

Question

Suppose we toss a fair coin $N$ times and we are interested in the probability that we get at least $cN$ heads for $c\in [0,1]$. We can model this situation by letting $S_N = \sum_{i=1}^N X_i$ where $X_i$ are i.i.d. Bernoulli random variables with parameter $p=1/2$.

On page 13-14 of these notes, for the case of $c=3/4$, the author uses the fact that by the central limit theorem $S_N$ is asymptotically normally distributed to find a bound on the tails of the distribution. That is, $$ Z_N = \frac{S_N - N/2}{\sqrt{N/4}} \to \mathcal{N}(0,1). $$ Thus $P(S_N \ge \frac{3}{4}N) = P(Z_N \ge \sqrt{N/4})$.

Let $g \sim \mathcal{N}(0,1)$. We have the following approximation $$ P\bigg(Z_N \ge \sqrt{\frac{N}{4}}\bigg) = P\bigg(g \ge \sqrt{\frac{N}{4}}\bigg) + \varepsilon(N), $$ where the approximation error is $$ \varepsilon(N) = P\bigg(Z_N \ge \sqrt{\frac{N}{4}}\bigg) - P\bigg(g \ge \sqrt{\frac{N}{4}}\bigg). $$

The author then shows that $P(g \ge t) \le (1/\sqrt{2\pi})e^{-N/8}$, that is, the probability decays exponentially fast with increasing $N$. So it looks like the probability of getting at least $(3/4)N$ heads decays very fast.

However, on page 14, he shows that the approximation error $\varepsilon(N) = O(1/\sqrt{N})$ so the approximation is no good since the error is the dominating term.

On pages 15-19, the author then goes on to

'resolve this issue, by developing alternative, direct approaches to concentration, which bypasses the central limit theorem.'

Using these alternative methods shows that $P(S_N \ge \frac{3}{4}N)$ does indeed decay exponentially fast.

Now, from what I understand though, the central limit theorem and asymptotic normality is used all over the place (here's one example) to find an estimate for the probability that some quantity is less than or greater than some value, or between two values. But in every one of these situations, the result will be invalid for the reason given above (the approximation error is too big and dominates the approximation).

1. So am I missing something or is a vast amount of the probability/statistical literature simply incorrect when they use the CLT and asymptotic normality to find an estimate for the probability of some event? Since they ignore the approximation error which is generally bigger than the result itself.
2. In Example 9.1.2 here, the CLT/asymptotic normality is invoked to estimate the probability that the number of heads lies behind $40$ and $60$ out of $100$ coin tosses and the result given is $0.9642$ which matches the true answer up to three decimal places. Here we see the CLT/asymptotic normality approximation has worked out very well. But why did it work so well when the approximation error is $O(1/\sqrt{N})$ so we should generally expect the results to be very inaccurate instead of accurate to three decimal places?

Answer 1

The author of the link given by the OP states near the end of p. 14 (emphasis mine)

This big error is a roadblock toward proving concentration properties for $S_N$ with light, exponentially decaying tails.

The author does not say that the approximation error is so big that it destroys the validity and usefulness of the approximation itself.

The author says that the approximation error in the CLT is big enough to not allow us to obtain useful bounds and rate of decays.

And the ubiquitous presence and use of the CLT is related to it functioning as an approximation, not as a tool for the latter purposes.

Answer 2

In statistics the central limit theorem is almost always used to characterize a statistic that is a sum whose aim is to estimate its expected value (e.g., $\bar x$ estimating $\mu$). Here the approximation applies to the neighborhood of the mode of the distribution, where the error term is relatively small.

In your example the statistic is a sum but we're not asking about its behavior near the distribution mode. We're asking about tail behavior and that's where the CLT approximation will eventually be dominated by the error term.

To take this point to an extreme, consider the probability of obtaining $S_N<0$, or $S_N>N$. The CLT approximation will give positive probability where the binomial distribution obviously gives zero.

On the other hand, your second example is a question about behavior near the mode and the CLT gives a good approximation.