# Do singular distributions have any real-world applications?

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".

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.

• That's interesting, although I wouldn't call it a "real-world application". Feb 21, 2018 at 3:50
• @tparker, that is a very weird comment. A sequence of Bernoulli variables models any sequence of binary experiments: one talks about a coin simply it is convenient to think of the abstract situation. Feb 21, 2018 at 4:21
• @MarianoSuárez-Álvarez I've edited my question to clarify the scope of what I consider a "real-world application". I'd welcome any examples, but I'm skeptical that non-mathematicians actually use these singular distributions in practice rather than just using absolutely continuous probability distributions. Feb 21, 2018 at 17:33
• @MarianoSuárez-Álvarez An $N \to \infty$ limit is almost always artificial in the real world, but applied types are happy to use that idealization if the result is easier to work with than the finite case. It often is, but in this case the limiting case is arguably more complicated than the finite one, so I'm not sure whether it's useful in (non-mathematical) practice. Feb 21, 2018 at 17:36
• On the other hand, the reason we usually consider the limit is precisely that for many (most?) situations it is easier to deal with that the finite case. This is appreciated very immediately when one tries to do things Feb 21, 2018 at 17:59

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 $$M$$.

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.