The Law of Large Numbers Under Fat Tails（肥尾下的大数定律）.pdf

REAL WORLD RISK INSTITUTE, LLC The Law of Large Numbers Under Fat Tails Nassim Nicholas Taleb Tandon School of Engineering, New York University and Real World Risk Institute, LLC. I. INTRODUCTION You observe data and get some confidence that the average is represented by the sample thanks to a standard metrified n. Now what if the data were fat tailed? How much more do you need? What if the model were uncertain –we had uncertainty about the parameters or the probability distribution itself? Let us call sample equivalence the sample size that is needed to correspond to a Gaussian sample size of n. It appears that 1) the statistical literature has been silent on the subject of sample equivalence –since the sample mean is not a good estimator under fat tailed distributions, 2) errors in the estimation of the mean can be several order of magnitudes higher than under corresponding thin tails, 3) many operators writing scientific papers aren’t aware of it (which includes T many statisticians), 4) model error compounds the issue. We show that fitting tail exponents via ML methods have a small error in delivering the mean. Main Technical Results In addition to the qualitative F discussions about commonly made errors in violating the sample equivalence, the technical contribution is as follows: • explicit extractions of partial expectations for alpha stable distributions • the expression of how uncertainty about parameters A (quantified in terms of parameter volatility) trans- lates into a larger (or smaller) required n. In other words, the effect of m