One power law to rule them all?

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

Preferential attachment [1] can generate power law.

Barabási–Albert model [2] is one model of that process.

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

[2] https://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_m...

Preferential attachment:

The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.

Source: https://en.m.wikipedia.org/wiki/Preferential_attachment

"A Brief History of Generative Models for Power Law and Lognormal Distributions": https://projecteuclid.org/euclid.im/1089229510

If you have 2 memory less processes with positive outcomes which depend on each other. For example: An exponential decay where decay rates are exponentially distributed. In this process decay times will be power law distributed.

Lognormal distributions can be generated via processes that follow geometric Brownian motion.

Power laws can be produced using stochastic processes called sample space reducing (SSR) processes. Stefan Thurner from Vienna has done a lot of work on this.

Hope this helps.

Log normal is something like Y = X1 * X2 * X3 * .. * XN, where Xi are iid positive random variables?

Through central limit theorem one could argue Y ~ exp(N * normal) for large N.

Strictly speaking, 1/N log Y converges to a normal rv in distribution.

Return times of random walks are power law distributed in some graphs. For example return to an interval on the line.