A simple process that produces power law distributions is exponential growth over a period of time t where t is sampled from an exponential distribution. A contrived example might be growing bacteria in a dish until an atomic nucleus decays. A more realistic example would be the total profits a company makes over its lifetime, where a very simple model would be to say that the company grows exponentially until it is acquired or is destroyed by some disaster. (Assuming the the chance of getting acquired/​destroyed in a given month stays constant.)
Some power law distributions are weirder than others. If the doubling period of the growth is at least as large as the half life of the process, then the expected value is infinite, even though the distribution itself is still perfectly well defined. Other, slightly less extreme, power law distributions have a finite mean, but infinite variance.
A simple process that produces power law distributions is exponential growth over a period of time t where t is sampled from an exponential distribution. A contrived example might be growing bacteria in a dish until an atomic nucleus decays. A more realistic example would be the total profits a company makes over its lifetime, where a very simple model would be to say that the company grows exponentially until it is acquired or is destroyed by some disaster. (Assuming the the chance of getting acquired/​destroyed in a given month stays constant.)
Some power law distributions are weirder than others. If the doubling period of the growth is at least as large as the half life of the process, then the expected value is infinite, even though the distribution itself is still perfectly well defined. Other, slightly less extreme, power law distributions have a finite mean, but infinite variance.