Hard-core statistics question: fractal vs. normal distribution

[Randy Burgess]

Reading an interesting book on risk in the financial markets by Mandelbrot, the guy who brought us fractals. He asserts that for markets, the Gaussian bell-curve is not an accurate description of how prices actually behave. He proposes "fat tails," etc. Apparently others are coming to agree, at least in part.

Anyone ever look at whether poker results truly tend to fall into a normal distribution or whether there are often excesssive outliers that in theory should not be occuring? I expect the answer is that no, no one has really looked at this with any degree of rigor; after all the subject is rather esoteric and not a lot is riding on the answer - vs. the financial markets where global billions of bucks are at stake.

One can argue that the curve *should* be normal since we assume results with a properly shuffled deck are independent of each other - but therein lies the question: are they actually independent of each other? For a long time this was the assumption in the markets, and it has proven not to be the case. Granted the inputs are far simpler when playing poker, but that by itself doesn't argue that all results are independent. For example if you habitually play worse during a bad streak, right away we have a problem with that assumption.

[Glenn]

Fat tails are no secret. For instance the 87 crash is something like a 1/10^120 event if stock prices were actually was normally distributed. The reason the normal distribution is used is that is it a good starting point and a reasonable approximation in many cases. You can use it IN CONCERT with other information you have. The same goes for poker. Your mean and standard deviation are not constant, and your results may not be independent, so technically the central limit theorem does not apply. However it is a very simple and reasonably accurate starting point. This, combined with common sense gives results accurate enough to be helpful.

[pzhon]

[ QUOTE ]
Reading an interesting book on risk in the financial markets by Mandelbrot, the guy who brought us fractals. He asserts that for markets, the Gaussian bell-curve is not an accurate description of how prices actually behave. He proposes "fat tails," etc. Apparently others are coming to agree, at least in part.

[/ QUOTE ]

Be wary that anything with "fractal" in its title is likely to be 99% hype or technobabble.

The Mandelbrot set is a neat idea, but Mandelbrot is not considered a particularly good mathematician by mathematicians. The contributions of Julia and Siegel are more fundamental to the foundations of complex dynamics, and are more impressive given that they predated the pretty pictures from computers.

The idea of "fat tails" is not new. The distribution of light on a wall from a point source is a Cauchy (http://encyclopedia.thefreedictionary.com/Cauchy) distribution, which has tails so fat that there is no expectation. Otherwise, Huygens' principle (http://www.mathpages.com/home/kmath242/kmath242.htm) in optics would fail. The idea of fat tails predates modern probability theory.

[Randy Burgess]

[ QUOTE ]
Be wary that anything with "fractal" in its title is likely to be 99% hype or technobabble.

The Mandelbrot set is a neat idea, but Mandelbrot is not considered a particularly good mathematician by mathematicians.

[/ QUOTE ]

Even so, Mandelbrot has apparently made some bold assertions in the financial arena in particular that others have had to take seriously, judging by a quick survey of the academic financial literature. He seems to be one of these agent provacateurs who give "serious" academics of all stripes the willies - half the time he's full of it, the other half of the time they have to slow down and actually think about what he said. Very disturbing to the orthodoxy, as he intends.

I should mention that he does reference Cauchy and doesn't pretend to have invented anything new there. His primary contention is that the normal curve isn't adequate for describing markets, in contrast to what Markowitz, Sharp, and all the folks pushing VAR have to say. And bear in mind that CAPM and the "efficient market" hypothesis are still dogma for anyone going for a financial degree, even though they clearly should be thrown in the trash can.

[Randy Burgess]

That's more or less the answer I expected. But I still think it's interesting that most of the time this sort of question isn't even asked. Not very important ... but interesting!

[playerfl]

i'm aware of the higher kurtosis in the financial markets but I was not aware that had anything to do with a Fractal Distribution( i'm not a mathematician ).

Does anybody have an easy explanation or maybe a web link that explains fractal distribution ?

[Randy Burgess]

Try these links:

http://www.aci.net/kalliste/chaos_index.htm

http://www.economymodels.com/factalmarkets.asp

www.mises.org/journals/qjae/pdf/qjae7_1_8.pdf

[playerfl]

cool, thx

[GStrot]

[ QUOTE ]
The reason the normal distribution is used is that is it a good starting point and a reasonable approximation in many cases. You can use it IN CONCERT with other information you have. The same goes for poker. Your mean and standard deviation are not constant, and your results may not be independent, so technically the central limit theorem does not apply. However it is a very simple and reasonably accurate starting point. This, combined with common sense gives results accurate enough to be helpful.

[/ QUOTE ]

I guess it depends on how you define "poker results". But starting hands are independent, and have the same mean and "standard deviation". Ending hands will have very fat tails to the right (assuming right is better) as you will see more people stay in the game with a flush than 9 high.

But, the reason starting poker hands are normally distributed is because it is a closed sample. There are 52 cards. Period. The stock market is different because there are infinite returns to the right and a finite set of returns to the left (you can't lose more than 100%). So, fat tails can exist (and they do) and the distribution is not normally distributed but skewed to the right.

I think we are saying the same thing, I just wanted to clarify.

If your starting hands are not normally distributed, you need to find a new game cuz someone is cheating!