Extreme events in power law distribution After reading Difference between power law distribution and exponential decay, I wanted to know more about power law distributions. I stumbled across Nassim Nicholas Taleb's "Technical Incerto - The Statistical Consequences of Fat Tails", where (p.8) - in relation to subexponential distributions - he claims that
"[...] extreme events away from the centre of the distribution play a very large role. Black Swans are not "more frequent" (as it is commonly misinterpreted), they are more consequential."
This seems to conflict with other sources I found (e.g. https://science.sciencemag.org/content/sci/335/6069/665.full.pdf) and with my intuition. Why shouldn't an extreme event (say an event larger than 10SD) be more frequent in a power law distribution than in, for example, a normal distribution? 
 A: The first point is that all distributions with a finite standard deviation or variance satisfy Chebyshev's inequality $$\mathbb P(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2}$$  so, to take your example, no more than $0.01$ i.e. $1\%$ of the distribution can be $10$ standard deviations or more from the mean.  
But for most distributions this is a very loose bound:


*

*For a normal distribution $2\Phi(-10)\approx 1.5 \times 10^{-23}$ so that is the proportion of the distribution which is $10$ standard deviations or more from the mean, much much less than the Chebyshev bound.  

*For an example of a power law distribution with a finite standard deviation, say a Pareto distribution with minimum $1$ and shape $3$, so with mean $\frac32$ and standard deviation $\frac{\sqrt{3}}{2}$,  the proportion of the distribution $10$ standard deviations or more from the mean is about $0.00095$, around a tenth of the Chebyshev bound.
So that seems to confirm your intuition.  
But I think Taleb may be saying an extreme event should not be defined by how many standard deviations it is from the mean, but simply by how unlikely it (or something more extreme) is to occur.  In that sense there are as many extreme events with one continuous distribution as with another, and the difference between them is how far such extreme events are from the mean.  
For example setting "extreme" as a probability of $10^{-6}$ and with the same mean $\frac32$ and s.d. $\frac{\sqrt{3}}{2}$,


*

*a normal distribution would have "extreme" values being those below $-2.74$ and above $5.74$     

*a Pareto distribution with minimum $1$ and shape $3$ would have "extreme" values being those above $100$
and in this sense "extreme" values are more consequential in the Pareto case
