Singular distributions: Applications and Instances

Question

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the typical example of such distribution.

I was wondering whether there are instances in nature that are explained using singular distributions or any applications of singular distribution in engineering milieu.

One interesting observation about Cantor set is that it is a self-similar set, like Sierpinski triangle and Koch snowflake. Therefore one candidate is obviously fractals.

My personal Google inquiry resulted in some papers and works in the fractal electro-dynamics theory and even economics. But I prefer to hear from experts on each domain.

I also wonder how a phenomenon related to singular distributions differs or is expected to differ from those with non-singular ones.

Answer 1

In the so-called red and black, a player starts with a given fortune and wants to reach a given target. The reader may want to have a look at the exposition How to Gamble If You Must by Kyle Siegrist for the further reference.

For concreteness, let us call the player Milan, and for simplicity let us say that the original fortune is $x$, $0<x<1$, and the target is $1$. In each round, Milan bets some part of his fortune and wins with some fixed probability $p$ and looses with probability $1-p$. In case of a win, Milan gets his bet back and additionally the same amount of money as the bet was. In case of a loss, Milan looses the amount that he bet. The game is over if Milan reaches the goal or if he has no money left.

One of the strategies is called bold play: if the fortune is less than one half, Milan bets everything. Otherwise the bet is exactly such that in case of a win, the target is reached. Thus the probability of winning is in case of $0\leq x<\frac{1}{2}$ \begin{equation} \varphi(x)=p\varphi(2x), \end{equation} and \begin{equation} \varphi(x)=p+(1-p)\varphi(2x-1) \end{equation} if $\frac{1}{2}\leq x\leq 1$ (here we also allow the cases $x=0$ and $x=1$). Equivalently stated \begin{align} \varphi\left(\frac{x}{2}\right)&=p\varphi(x),\\ \varphi\left(\frac{x+1}{2}\right)&=(1-p)\varphi(x)+p \end{align} for all $0\leq x\leq 1$.

If the probability $p$ is not one half, then $\varphi$ is singular.

In our case, the success function of a strategy is the probability that Milan reaches his target starting with $x$. A strategy $S$ is optimal if any other strategy's success function is bounded from above by $S$'s success function for every admissible bet. It turns out that if $p$ is less or equal than one half, bold play is an optimal strategy (but not the only optimal one, see the section about Unfair Trials in Siegrist's paper).

As far as I can tell, George de Rham was the first to study such kind of systems (in a different context allowing $p$ to be a complex number with absolute value less than 1) in the paper Sur quelques courbes definies par des equations fonctionnelles (Univ. e Politec. Torino. Rend. Sem. Mat. 16 1956/1957 101--113). He shows that the unique bounded solution is a continuous function and the derivative, if it exists, can only be $0$. Further, he points out that the continuous function that solves the system has been studied before. The reader may be interested in the paper Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions by E. de Amo, M. Díaz Carrillo, and J. Fernández-Sánchez, (Acta Appl. Math. 119 (2012), 129--148), especially Proposition 2, where the here relevant properties of $\varphi$ are listed.

Answer 2

I don't think that we should expect an economic phenomenon which is modelled with the Cantor distribution.

In applied sciences, in particular in economics (with which I’m most familiar), the Cantor distribution is mentioned to show that the results obtained are very general as they hold even assuming such a ‘pathological’ case. See e.g.

Machina, Mark, and John Pratt. "Increasing risk: some direct constructions." Journal of Risk and Uncertainty 14.2 (1997): 103-127. Link

When we model economic (or other real-world) situations we want the output of a model to be robust to changes in underlying assumptions. Thus, obtaining the result under less restrictive assumptions is seen as a progress in applied sciences.

Answer 3

My colleagues and collaborators, Lutwak and Zhang, have worked together with others to solve Minkowski type problems, where a Borel measure is given on the unit sphere and one solves for a convex body for which a certain geometric measure is equal to the given Borel measure. For continuous measures, these problems always have a solution, but when the measure is singular, there are obstructions.

Answer 4

The notion of Smale horseshoe in dynamical systems involves construction of cantor sets. Smale horseshoe is a central concept in modern dynamical systems theory, and as such, provides a model for various chaotic systems found in nature: from the three-body problem to turbulence to double pendulum and so on.. you can literally find horseshoes in data collected from fluid dynamics in nature, or simulating your favorite chaotic dynamical system on the computer.