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Actually, the reason that real world isn't always Gaussian isn't only that sums take a long time to become Gaussian. Another factor is that the Gaussian isn't t

Actually, the reason that real world isn't always Gaussian isn't only that sums take a long time to become Gaussian. Another factor is that the Gaussian isn't the only distribution that's "stable" in summation. Yes, a sum of random variables that satisfies the central limit theorem will be Gaussian but this requires that the random variable have finite mean and variance and there are plenty of random variables/distributions not satisfying this. In general, the L-stable or stable family of describes the general case here. Notably, Benoit Mandlebrot's study of fractals began with the study of non-Gaussian distributions in financial markets and other situations. For example, Financial markets are a sum of price jumps but those jumps aren't of similar size - rather a few large jumps can have bigger impact than all the small together. See: https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution

fritzo 3y ago
I'd go farther and say the real central limit theorem is the generalized central limit theorem, and the finite variance central limit theorem is just a quirky special case.
rich_sasha 3y ago
While all true, I think variables with infinite variance are hard to come by. For a start, almost everything in the real world is kind of bounded (markets can't go below zero and also can't jump by more than a certain amount per day either, laws of physics determine min and max weight of people etc). But it's true that CLT requires some form of IID distribution, or something slightly weaker, and that often doesn't apply.
- fjkdlsjflkds 3y ago
  > markets can't go below zero and also can't jump by more than a certain amount per day either It depends on what you are looking at, and how you are modelling it, isn't it? For example: the value of log(asset_value_today/asset_value_yesterday) can become unbounded (and, thus, so can all its moments). Furthermore, the problem is not just the "finite variance assumption", but the fact that the CLT only talks about what happens asymptotically: for some reason people use t-scores in practice, rather than z-scores. CLT does not imply normality of (e.g.) sum-based estimators if you have finite number of samples (which is pretty much always the case).

fritzo 3y ago

I'd go farther and say the real central limit theorem is the generalized central limit theorem, and the finite variance central limit theorem is just a quirky special case.

rich_sasha 3y ago

While all true, I think variables with infinite variance are hard to come by. For a start, almost everything in the real world is kind of bounded (markets can't go below zero and also can't jump by more than a certain amount per day either, laws of physics determine min and max weight of people etc). But it's true that CLT requires some form of IID distribution, or something slightly weaker, and that often doesn't apply.

fjkdlsjflkds 3y ago
> markets can't go below zero and also can't jump by more than a certain amount per day either It depends on what you are looking at, and how you are modelling it, isn't it? For example: the value of log(asset_value_today/asset_value_yesterday) can become unbounded (and, thus, so can all its moments). Furthermore, the problem is not just the "finite variance assumption", but the fact that the CLT only talks about what happens asymptotically: for some reason people use t-scores in practice, rather than z-scores. CLT does not imply normality of (e.g.) sum-based estimators if you have finite number of samples (which is pretty much always the case).