QuoteOriginally posted by: vixenQuoteOriginally posted by: NQuoteOriginally posted by: vixenQuoteOriginally posted by: NQuoteOriginally posted by: vixenQuoteOriginally posted by: N4. I claim that the limit of the sum of random variables is always a normal distribution. I know this concept nowdays is a bit controversial. It wasn't when I studied OR 30 years ago.Not true. Even for iid, this only applies to distributions with finite variance. Other distributions like alpha-stable ones ( Levy and Cauchy ) for the generalised CLT.Distributions like alpha-stable ones ( Levy and Cauchy ) aren't distributions at all. They're not stationary. They gotta be stationary to be a distribution.WTF!?? Please define 'distribution' and 'stationary'. I don't want to get sidetracked but are we using these words to mean the same thing?Distribution = PDF(x).Stationary = PDF is not a function of t.If we have PDF(x,t) that's not a distribution.Crow, A distribution can have just about any shape, but it can't be a function of time.Good. No one is talking about time dependence of PDF(x).Check the Wikipedia page Cauchy distribution. The relevant piece is here:Quote If X1,
, Xn are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean (X1 +
+ Xn)/n has the same standard Cauchy distribution ( the sample median, which is not affected by extreme values, can be used as a measure of central tendency). To see that this is true, compute the characteristic function of the sample mean:where is the sample mean. This example serves to show that the hypothesis of finite variance in the central limit theorem cannot be dropped. It is also an example of a more generalized version of the central limit theorem that is characteristic of all Lévy skew alpha-stable distributions, of which the Cauchy distribution is a special case. ( sorry, the equations did not copy/paste )Let me know if you disagree with this.Edit: Here is a link to the generalized central limit theoremIf you average over enough time to remove the time dependency, I agree with most. However, if you don't use the QM average the world approach, Cauchy dynamics is 100% deterministic scattering.