I've been reading through Grimmett's and Stirzaker's Probability and Random Processes and on page 33 they state:
For the moment we are concerned only with discrete variables and continuous variables. There is another sort of random variable, called 'singular', for a discussion of which the reader should look elsewhere. A common example of this phenomenon is based upon the Cantor ternary set (see Grimmett and Welsh 1986, or Billingsley 1995.) Other variables are 'mixtures' of discrete, continuous, and singular variables.
I looked on Google Scholar for the two references mentioned in the excerpt, but haven't had any luck locating them.
Can someone explain what a singular random variable is and how it differs from the continuous and discrete variants?
Best Answer
This segregation for random variables is based on the Lebesgue decomposition Theorem for measures (you apply here to the law of your random variable).
More precisely, a random variable $X$ taking values in $\mathbb{R}^d$ is singular if its law is purely singular continuous, that is it satisfies $P(X=a)=0$ for any $a\in\mathbb{R}^d$, but is orthogonal to the Lebesgue measure $\lambda$ of $\mathbb{R}^d$, namely there exists $A\subset\mathbb{R}^d$ such that $P(X\in A)=1$ and $\lambda(A)=0$.
A classical example for singular random variable on $\mathbb{R}$ is indeed the one which has for distribution function the Cantor ladder, but on $\mathbb{R}^2$ just take $(X,0)$, where $X$ is a random variable on $\mathbb{R}$ satisfying $P(X=a)=0$ for any $a\in \mathbb R$.