Do singular distributions have any real-world applications?

Question

Do singular probability distributions have any real-world applications, or are they just a pure-mathematical curiosity? I can't imagine a real quantity that they would describe. But on the other hand, singular functions do have surprising applications, e.g. regarding the fractional quantum Hall effect in condensed-matter physics, so maybe singular probability distributions do as well.

By "real-world application" I don't mean a real phenomenon which could in principle be modeled by a singular distribution, but rather a situation in which engineers, scientists, financial analysts, or other non-mathematicians actually do use them in the context of a commercial application or other non-mathematical "product".

Answer 1

For example, consider an infinite sequence of fair coin-flips, corresponding to iid Bernoulli-$1/2$ random variables $X_n$. Then $2 \sum_{n=1}^\infty 3^{-n} X_n$ has a singular distribution whose cdf is the "devil's staircase".

EDIT: If you want to make it seem more "real-world", you might rephrase this in terms of return on investments or something similar. The point is that a rapidly converging sum of discrete random variables is quite likely to have a singular distribution.

Answer 2

Frequent examples involve probability measures over $\mathbb R^{n\ge2}$ with mass concentrated on a manifold $M$ of dimension less than $n$, such as any distribution over the unit sphere (a hypersurface of dimension $n-1$).

In such situations one can sometimes define a "(hyper)surface density" $\sigma$ which can be integrated over a measurable subset $A \subset M$ as $\mathbb P(A) = \int_A\sigma \ d\mu$, where $\mu$ measures the $(n - 1)$-dimensional volume on the manifold $M$.

Answer 3

Here is one application that might interest you. Consider an integrable function f on $\mathbb R$ such that $2f(x)=3f(3x)+3f(3x-1)$ a.e.. It turns out that $f=0$ a.e.. There seems to be no simple way of doing this even if you assume that f is smooth function. There is an elegant proof by relating it to cantor set and a singular distribution function. This is Problem 261 in 'Exercises in Analysis 201-300' at statmathbc.wordpress.com where complete solutions are provided on request.