Many real-life data sets have power laws or Zipfian distributions. An integer-valued random variable *X* follows a power law with parameter *a* if *P*(*X* = *k*) is proportional to *k*^{–a}. Panos asked what the sum of two power laws was. He cites Wilke at al. who claim that the sum of two power laws *X* and *Y* with parameters *a* and *b* is a power law with parameter min(*a*, *b*).

I relate this problem to the sum of exponentials. Any engineer knows that if *a*>*b*, then *e*^{at} + *e*^{bt} will be approximately *e*^{at} for *t* sufficiently large. Hence, the sum of power law distributions *X* and *Y* is a power law distribution with parameter min(*a*, *b*) if you are only interested in large values of k in *P*(*X* + *Y* = *k*).

However, the sum of two power laws is not a power law. Egghe showed in The distribution of N-grams that even if the words follow a power law, the n-grams won’t!

Daniel Lemire, "On the sum of power laws," in *Daniel Lemire's blog*, January 25, 2008.

(1) Mandelbrot has proposed a generalization of Zipf’s Law. (2) Randomly generated strings follow Zipf’s Law, so some people argue that in some cases it is a statistical artifact.

http://en.wikipedia.org/wiki/Zipf%27s_law

It seems that power law is really the same as Pareto distribution . This paper gives some closed formulas for distributions of sums of Pareto, which are themselves not Pareto

If two power laws have different parameters, as you go to infinity, odds of encountering the one with higher a becomes vs. one with lower a goes to 0, so I also expect that for large values, heavier tail distribution will dominate

BTW, I also wondered about distribution of bigrams when unigrams are power-law distributed, David Cantrell in sci.math gave an approximate formula for the cdf involving Lambert’s W function

http://groups.google.com/group/sci.math/browse_thread/thread/8de7cee65f65ff70/810470b85f36523b?lnk=st&q=group%3Asci.math#810470b85f36523b

Yaroslav,

Thanks, very useful!

– Panos