I’ve often heard folks use the term “long-tailed distribution” informally to refer to a probability distribution in which a lot of the probability mass is concentrated far from the mean. That got me wondering if there is a single probability distribution with the “longest tail,” in the sense that the maximum amount of probability mass is far from the mean.
To make this rigorous, let’s pin down some requirements. We’ll have $p : \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ be a function over nonnegative real numbers. We’ll require the following:
- $\int_0^\infty{p(x) dx} = 1$ ($p$ is a probability mass function).
- $p$ is nonincreasing. This ensures that the distribution shape is one where there really is a “tail,” rather than having several peaks far from the origin.
- $p$ is continuous. This ensures we’re looking at “nicely-shaped” functions rather than discrete stairsteps.
- For all $t$, the quantity $\int_0^t{p(x) dx}$ is minimal across all such functions. This ensures that the greatest amount of probability mass is far from the origin.
Is there a single pmf $p$ meeting these criteria? If so, how do we prove it? If not, how do we prove that for every such $p$ there’s a $p’$ that is “longer-tailed” according to the last metric?
Thanks!
