4 ms·

A result of broader applicability is that of convergence to infinitely divisible distributions, more generally the stable distributions https://en.wikipedia.or

by srean 8mo ago

A result of broader applicability is that of convergence to infinitely divisible distributions, more generally the stable distributions https://en.wikipedia.org/wiki/Infinite_divisibility_(probability) https://en.wikipedia.org/wiki/Infinite_divisibility_(probabi... https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution This applies even when the variance is not finite. Note independence and identical nature of distribution is not necessary for Central Limit Theorem to hold. It is a sufficient condition, not a necessary one, however, it does speed up the convergence a lot. Gaussian distribution is a special case of the infinitely divisible distribution and is the most analytically tractable one in that family. Whereas, averaging gives you Gaussian as long as the original distribution is somewhat benign, the MAX operator also has nice limiting properties. They converge to one of three forms of limiting distributions, Gumbel being one of them. The general form of the limiting distributions when you take MAX of a sufficiently large sample are the extreme value distributions https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution https://en.wikipedia.org/wiki/Generalized_extreme_value_dist... Very useful for studying record values -- severest floods, world records of 100m sprints, world records of maximum rainfall in a day etc

topaz0 8mo ago
I think part of why we're much more likely to learn about the iid, finite-variance CLT is that it's a lot easier to prove than the more general ones.
- srean 8mo ago
  Yes that's a big part. The proofs get hairier otherwise. But I think there is more to it, the convergence to Gaussian also gets slower. In practice, we deal with finite averaging, so speed of convergence matters. For some non-iid case, the convergence may be so slow that the distribution cannot be approximated well by a Gaussian.

by srean 8mo ago

topaz0 8mo ago

I think part of why we're much more likely to learn about the iid, finite-variance CLT is that it's a lot easier to prove than the more general ones.

srean 8mo ago
Yes that's a big part. The proofs get hairier otherwise. But I think there is more to it, the convergence to Gaussian also gets slower. In practice, we deal with finite averaging, so speed of convergence matters. For some non-iid case, the convergence may be so slow that the distribution cannot be approximated well by a Gaussian.