The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.
Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning.
Characterization
The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:
: <math>
\begin{align}
C_0 = {} & [0,1] \\[8pt]
C_1 = {} & [0,1/3]\cup[2/3,1] \\[8pt]
C_2 = {} & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\[8pt]
C_3 = {} & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\[4pt]
{} & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\[8pt]
C_4 = {} & \cdots
\end{align}
</math>
The Cantor distribution is the unique probability distribution for which for any C<sub>t</sub> (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in C<sub>t</sub> containing the Cantor-distributed random variable is identically 2<sup>−t</sup> on each one of the 2<sup>t</sup> intervals.
Moments
It is easy to see by symmetry and being bounded that for a random variable X having this distribution, its expected value E(X) = 1/2, and that all odd central moments of X are 0.
The law of total variance can be used to find the variance var(X), as follows. For the above set C<sub>1</sub>, let Y = 0 if X ∈ [0,1/3], and 1 if X ∈ [2/3,1]. Then:
: <math>
\begin{align}
\operatorname{var}(X) & = \operatorname{E}(\operatorname{var}(X\mid Y)) +
\operatorname{var}(\operatorname{E}(X\mid Y)) \\
& = \frac{1}{9}\operatorname{var}(X) +
\operatorname{var}
\left\{
\begin{matrix} 1/6 & \mbox{with probability}\ 1/2 \\
5/6 & \mbox{with probability}\ 1/2
\end{matrix}
\right\} \\
& = \frac{1}{9}\operatorname{var}(X) + \frac{1}{9}
\end{align}
</math>
From this we get:
:<math>\operatorname{var}(X)=\frac{1}{8}.</math>
A closed-form expression for any even central moment can be found by first obtaining the even cumulants
:<math>
\kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n
{n\, (3^{2n}-1)}, \,\!
</math>
where B<sub>2n</sub> is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.
References
Further reading
- This, as with other standard texts, has the Cantor function and its one sided derivates.
- This is more modern than the other texts in this reference list.
- This has more advanced material on fractals.