Heavy tailed rather than normal distribution for a stock simulator Have a program that simulates picking stocks 10000 times according to different parameters. I then look at the ending portfolio amounts, and the distribution is pretty heavy tailed. This doesn't violate 'normal distribution rules', correct? I remember in undergrad that the normal distribution would appear if I took a mean of all 10000 ending portfolio amounts, and then did that like, many times. Is my understanding correct? Is there a ELI5 version of this phenomenon?
Secondly, I aim to calculate the confidence intervals for the ending portfolio amounts. I generate 10000 samples for ending portfolio amount and then sort these results. For a 70 percent CI, I pick the 3000th and 7000th sample from the sorted results. Is this correct? I think that's not correct; I remember in undergrad when calculating CI, we had to collect a random sample from the 10000, and then resample with replacement, and then take percentiles of the resampled results? Vaguely remember something like that.
 A: *

*

You’re alluding to the central limit theorem (CLT), which says:
$$
\sqrt{n}\dfrac{
\bar X-\mu
}{
\sigma
}
\overset{d}{\rightarrow}
N(0,1)$$
(The $d$ above the arrow means a particular type of convergence in probability theory called convergence in distribution.)
Loosely speaking, the sample mean converges to a normal distribution as the sample size increases.
However, there are conditions on the observations used to calculate that $\bar X$. There are multiple variants of the CLT, but a commonality is a requirement that the observations used to calculate $\bar X$ come from distributions with finite variance.
Financial processes often are thought to be too heavy-tailed to have finite variance, so the central limit theorem would not apply.
Even if the variance is finite, the tails might be so heavy that a finite sample size does not get you particularly close to the limit. Vanderbilt’s Frank Harrell has tweeted about this, for instance.
2)
You’re alluding to bootstrap confidence intervals, particularly using a percentile bootstrap, which is easy to calculate but not considered a great method. Software like the boot package in R makes it easy to calculate preferred bootstrap confidence interval, such as bias-corrected and accelerated (BCa).
