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?

  • 3
    $\begingroup$ A random variable with a continuous distribution function that does not have a density with respect to the dominating measure in question (usually Lebesgue measure). $\endgroup$ – cardinal Nov 13 '11 at 22:44
  • $\begingroup$ en.wikipedia.org/wiki/Cantor_distribution $\endgroup$ – Nate Eldredge Jun 3 '12 at 1:45

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 $a\in \mathbb R$.


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