# Is power law distribution for extreme event special like normal distribution?

The power-law distribution is defined as below in Wikipedia article:

The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. $$\mathrm{Pr}[X>x] \sim x^{-\alpha} \ \ \mathrm{as} \ \ x \rightarrow \infty, \ \ \alpha > 0$$

then the distribution is said to have a fat tail if $$\alpha$$ is small.

I read numerous articles on power-law distribution to explain extreme events. Especially, the power-law distribution is ubiquitous in finance and economy.

I understand that normal distribution can be ubiquitous because of the central limit theorem(CLT). The combined effects consisting of many stochastic sub-effects might follow a normal distribution from CLT. However, I don't know if there is any universal theorem like CLT for the power-law distribution.

It seems nature favors power-law when it decides to make an extreme event. Is power-law distribution special like CLT? Or is the power-law just an artificial convenient concept to approximate the true distribution which is not power-law?

• Perhaps that's a question best directed at the economics or financial SE communities. – whuber Jun 30 '20 at 13:18
• This bactra.org/weblog/cat_power_laws.html has a set of posts on both reasons people are motivated to fit power laws and the unfortunate fact that they tend not to fit well. – Thomas Lumley Jul 1 '20 at 0:58
• A partial explanation comes from normalization and integrability. The probability density needs to be normalized. On an unbounded domain, say $[0, +\infty)$, this necessity excludes asymptotic behaviours such as $\exp(x)$, or $x^n$ with $n>0$, or even $1/x$. So if you want a density that allows for relatively large probabilities for large values of $x$ but that's normalizable, and mathematically not too complicated, you end up with $x^n$ with $n<-1$. Note that this density may fit extreme events, it doesn't explain them. – pglpm Jul 6 '20 at 7:16
• Regarding the question whether it's "common" or not, I don't even know whether we can give real meaning to such a question. As a matter of usage, it surely depends on the specific scientific field and on time-variable fashion in science. Is it "common in nature"? well, how can we quantify what's common and uncommon in nature? which space do we mean, and what kind of measure do we put there to make such a quantification? – pglpm Jul 6 '20 at 7:21

With respect to the first question, whether there is a convergence theorem leading to power-law distributions, the Extremal Types Theorem should be mentioned (Fisher and Tippett, 1928, Gnedenko 1934), leading to the three max-stable Extreme Value Distributions (Type I to III, also known as Gumbel, Fréchet and Weibull distributions). The Fréchet distribution or the Generalised Extreme Value (GEV) distribution with shape parameter $$\xi>0$$ is characterised by a fat tail. Maxima of long (asymptotically) sequences of e.g. the Cauchy, Student, or Pareto (power-law) distribution are Fréchet distributed, and it can also be shown that the Fréchet distribution is the so-called Penultimate distribution for maxima of the Gaussian distribution, even though the latter are asymptotically Gumbel (light tailed) distributed. So there is a general theorem related to the power-law distribution.

With respect to the second question, I can point you to the book of Prof. Didier Sornette, "Critical Phenomena in Natural Sciences", where he provides extensive motivation from statistical physics for fat-tailed natural phenomena.

PS: This is my first answer on a StackExchange site, in case you have tips or criticism, I'm happy for feedback!

$$\bullet$$ First, heavy-tailed/Pareto distributions (aka power laws) frequently occur outside of finance and economy. They are used to describe the size/height/magnitude/severity distribution of countless natural phenomena.
A few examples are ocean waves, volcanic eruptions, asteroid impacts (a well-known example is the size of the craters on the moon), tornadoes, forest fires, streamflow and floods, solar flares, landslides, rainfall... Other interesting human-related examples are the size of human settlements, size of files transferred on the web, Google's PageRank numbers of webpages...

This is closely related to the field of extreme value analysis (hundred-year floods, rogue waves...) - see the answer by Joel listing some famous extreme value distributions.

It is also to be noted that the empirical distributions (histograms, KDEs) of the phenomena above are heavy-tailed. In other words, we already observe the power-law naturally emerging in these phenomena. So yes, we do use parametric models to approximate the naturally occurring distributions, simulate values, etc., but I don't think we can say that it is artificial. (The paper Clauset et al. 2009 linked to by Sycorax in the comments above seems to be a good reference here.)

$$\bullet$$ Second, as far as your question, while power-law distributions are frequently observed, the underlying physical processes by which they arise are still mostly unknown. Indeed, natural phenomena are complex. E.g., floods are produced by the interplay between meteorological and hydrological processes, but are also influenced by infrastructure (e.g., dams) and human activities (e.g., land use). Nevertheless, there are many underlying processes that are believed to generate fat tails in the distributions of natural and other human-related phenomena, such as:

References

Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323-351.

El Adlouni, S., B. Bobée, and T. B. M. J. Ouarda. "On the tails of extreme event distributions in hydrology." Journal of Hydrology 355.1-4 (2008): 16-33.

Mitzenmacher, Michael. "A brief history of generative models for power law and lognormal distributions." Internet mathematics 1.2 (2004): 226-251.

Clauset, Aaron, Cosma Rohilla Shalizi, and Mark EJ Newman. "Power-law distributions in empirical data." SIAM review 51.4 (2009): 661-703.