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# On the sum of power laws

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!