Physical processes described by pathological distributions

I am reading a decade-old paper entitled Above, below, and beyond Brownian motion which explains rather nicely random walk processes that are governed by probability distributions with infinite moments, such as the increasingly famous Lévy flight.

What I like about this paper is that it provides many real examples of processes that are explained by probability distributions with infinite moments. This means that usual descriptors of the random variable, such as its mean or variance, do not exist. One surprising example is the distribution of light rays hitting a wall from a point source emitting uniformly in all directions.

A random walk described by a "pathological" distribution (as they are sometimes described by mathematicians) is conceptually understood as some process with a very large spread in characteristic times or distances. By spread I mean that the random variable may take any value with a probability that is not exponentially vanishing (this property is also known as scale-free). Probability distributions modeled as power law decays are popular for encompassing this behavior since they decay slower than an exponential distribution.

I admit that much of the theory in the paper is best understood by someone who already is familiar with many of the ideas from probability theory, such as moment generating functions.