Timeline for Interpreting definition of stable distributions
Current License: CC BY-SA 3.0
7 events
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| Jul 18, 2012 at 11:53 | comment | added | cardinal | Referring to your link, look under the "Classical CLT" heading or the Lindeberg-Levy CLT. This is the situation you describe. No additional moment finiteness is needed beyond the second moment. :) The Lyapunov variant drops the identical distribution assumption. Since we give something up, we have to add back some structure to retain the result. One way to do this is to require a slightly higher moment (though this is just a special case of full Lindeberg-Feller). :) | |
| Jul 18, 2012 at 11:01 | comment | added | Michael R. Chernick | @cardinal Sure that was the reason for my remark. Something more than second moments is needed and the Lyaponov condition is just one simple way to get it. This wikipedia article covers this well. en.wikipedia.org/wiki/Central_limit_theorem | |
| Jul 18, 2012 at 10:47 | comment | added | cardinal | Sorry I was not clear. I meant that requiring a finite moment strictly larger than the second to exist in order to apply the CLT is not necessary in the iid case. We just need the first condition you list (i.e., nontrivial finite variance). :) | |
| Jul 18, 2012 at 10:28 | comment | added | Michael R. Chernick | @cardinal Yes the CLT comes about in various forms with the Lindeberg-Feller conditions for example are more general than the version I mentioned. That is why I mentioned "one form" and my point was just that there are random variables that don't obey the CLT but do have a stable limit. But I was not aware or forgot about examples where finite variance was not needed in the IID case. Can you give me an example where the variance is infinite? | |
| Jul 18, 2012 at 10:16 | comment | added | cardinal | Just an aside: If our random variables are iid, we only need finite nonzero variance to get the CLT; the second moment condition listed is unnecessary. :) | |
| Jul 18, 2012 at 9:21 | vote | accept | upabove | ||
| Jul 17, 2012 at 23:14 | history | answered | Michael R. Chernick | CC BY-SA 3.0 |