1. Under what assumptions is the CLT valid? There seem to be many real world situations where it is not valid. E.g. finite variance is one such assumption.
2. How easily can one check that the assumptions are valid in a given case? The very existence f fat tails can make it hard to notice that there are fat tails and easy to assume that tails are not fat.
2. Even when valid, how fast does the ensemble distribution converge, in particular how fast does the “zone of normality” spread? Even when CLT applies to the mean of the distribution it may not apply to the far tails for a long time.
Key questions for me.
1. Under what assumptions is the CLT valid? There seem to be many real world situations where it is not valid. E.g. finite variance is one such assumption.
2. How easily can one check that the assumptions are valid in a given case? The very existence f fat tails can make it hard to notice that there are fat tails and easy to assume that tails are not fat.
2. Even when valid, how fast does the ensemble distribution converge, in particular how fast does the “zone of normality” spread? Even when CLT applies to the mean of the distribution it may not apply to the far tails for a long time.