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### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 3:47 pm**

by **N**

QuoteOriginally posted by: gardener3QuoteOriginally posted by: NQuoteOriginally posted by: outrunQuoteReturns are only random if the distribution is Gaussian (think of sums of returns - central limit theorem). Is there anyone around who believes returns have a Gaussian distribution? Therefore Avellaneda's book is worthless.Your probably talking about independence (instead of random) with your central limit theorem. Returns can be dependent & still be random. Mean reverting processes are random but have autocorrelation. You can also have non gaussian distribution (e.g. the stable distributions) that keep the same (scaled) non gaussian distribution under summation.what I;ve seen: returns are non-Gaussian, have dependency in time & are not deterministic.If there is autocorrelation, one can simply use a linear predicitive filter to get the next return. A slam dunk for making money. Unfortunately there is absolutely no autocorrelation in returns of any financial instruments.zero linear correlation does NOT equal independenceWhere'd you come up with that one?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:03 pm**

by **Rez**

QuoteOriginally posted by: N(...) Things not random are deterministic.This is not correct. Things that can be random for one observer can be deterministic for another.It is a matter of information.K

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:14 pm**

by **Rez**

QuoteOriginally 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.5. Suppose returns were actually a superpostion of sine waves. Would you still use Gaussian assumption for tractability?4. This is correct under some technical conditions, but what does that have to do with the discussion here? You need the things that you add up to be iid, and they are not. That's why you don't have an unconditional normal.In the Black-Scholes case you add up normals that are iid => you get a normalIn the Heston case you add up normals that have different variances => you get high kurtosiseven levy processes can be thought as subordinated brownian motions => you add up normals with different variances => unconditional kurtosisNo?5. Do you know the coefficients that multiply the waves? Are they constant through time?K edit: added levy

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:19 pm**

by **N**

QuoteOriginally posted by: RezQuoteOriginally posted by: N(...) Things not random are deterministic.This is not correct. Things that can be random for one observer can be deterministic for another.It is a matter of information.KGreat. I knew we'd eventually get to Bell inequalities. Want to be Alice or Bob?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:26 pm**

by **Rez**

what do bell's inequalities have to do with it?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:29 pm**

by **vixen**

QuoteOriginally posted by: NQuoteOriginally posted by: RezQuoteOriginally posted by: N(...) Things not random are deterministic.This is not correct. Things that can be random for one observer can be deterministic for another.It is a matter of information.KGreat. I knew we'd eventually get to Bell inequalities. Want to be Alice or Bob?At last! How did you guess my real name!? Now all we need is Bob. Is he around?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 4:33 pm**

by **vixen**

QuoteOriginally 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 ) follow the generalised CLT.Edit: tyop

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 5:16 pm**

by **N**

QuoteOriginally 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.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 5:22 pm**

by **crowlogic**

QuoteOriginally 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.I thought any function such that f(-inf)=0 and f(+inf)=1 and diff(f(x),x)>=0 could be considered a distribution no?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 5:23 pm**

by **vixen**

QuoteOriginally 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?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 5:44 pm**

by **N**

QuoteOriginally 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.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 5:57 pm**

by **crowlogic**

QuoteOriginally posted by: NCrow, A distribution can have just about any shape, but it can't be a function of time.Right. But what about sequences of different distributions packed arbitrarily close together in time? Like stacking pages in a book.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 6:11 pm**

by **vixen**

QuoteOriginally 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 theorem

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 6:34 pm**

by **N**

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.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 6:54 pm**

by **vixen**

QuoteIf 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.We are not talking about time dependence or dynamics in the context of the CLT!Mean of a large number of iid rv from a Cauchy distribution follows a Cauchy distribution. Do you agree or not?