This is a not a very solid claim. The difference in calculations using a distribution with a finite variance versus one with an infinite variance could not be more different.
The approximation is not done for closeness, because they are not close at all, but for convenience and simplicity. Probability theory is amazingly difficult when you are dealing with infinite variance.
The approximation is not done for closeness, because they are not close at all, but for convenience and simplicity. Probability theory is amazingly difficult when you are dealing with infinite variance.