For power law distributions, if you have sums of identical independent random variables that are power law with exponent 1 < a < 3, then the limiting distribution of their sum is power law in it's tails. The limiting distribution is called Levy stable [1] [2], where the 'stability' means it converges to this distribution. Levy stable distributions are power laws in their tails. When the exponent is 3 or more, the resulting stable distribution is Gaussian (aka a Normal distribution).

Another is to have a process where the value at each time grows exponentially with a certain rate, r, but the probability of continuing decreases exponentially with a different rate, s. The resulting random variable is power law [3].

[1] https://en.wikipedia.org/wiki/Stable_distribution

[2] "Stable Distributions - Models for Heavy Tailed Data" by J. Nolan (chapter 1) - http://fs2.american.edu/jpnolan/www/stable/chap1.pdf

[3] "Random multiplicative processes: An elementary tutorial" S. Redner - http://physics.bu.edu/~redner/pubs/pdf/ajp58p267.pdf