I know what kind of process generates a normal distribution, but what about power law and lognormal? I’ve googled for an answer and never found anything (just found the formula for distributions, etc).
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].
The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.
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.
