Cantor distribution

Short description: Probability distribution

Cantor

Cumulative distribution function
Parameters	none
Support	Cantor set, a subset of [0,1]
pmf	none
CDF	Cantor function
Mean	1/2
Median	anywhere in [1/3, 2/3]
Mode	n/a
Variance	1/8
Skewness	0
Kurtosis	−8/5
MGF	[math]\displaystyle{ e^{t/2} \prod_{k=1}^\infty \cosh\left(\frac{t}{3^k}\right) }[/math]
CF	[math]\displaystyle{ e^{it/2} \prod_{k=1}^\infty \cos\left(\frac{t}{3^k}\right) }[/math]

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]\displaystyle{ \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 = {} & [0,1/81]\cup[2/81,1/27]\cup[2/27,7/81]\cup[8/81,1/9]\cup[2/9,19/81]\cup[20/81,7/27]\cup \\[4pt] & [8/27,25/81]\cup[26/81,1/3]\cup[2/3,55/81]\cup[56/81,19/27]\cup[20/27,61/81]\cup \\[4pt] & [62/81,21/27]\cup[8/9,73/81]\cup[74/81,25/27]\cup[26/27,79/81]\cup[80/81,1] \\[8pt] C_5 = {} & \cdots \end{align} }[/math]

The Cantor distribution is the unique probability distribution for which for any C_t (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in C_t containing the Cantor-distributed random variable is identically 2^−t on each one of the 2^t 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 C₁, let Y = 0 if X ∈ [0,1/3], and 1 if X ∈ [2/3,1]. Then:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}} {n\, (3^{2n}-1)}, \,\! }[/math]

where B_2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

Further reading

• Hewitt, E.; Stromberg, K. (1965). Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer-Verlag. https://archive.org/details/realabstractanal00hewi_0.  This, as with other standard texts, has the Cantor function and its one sided derivates.
• Hu, Tian-You; Lau, Ka Sing (2002). "Fourier Asymptotics of Cantor Type Measures at Infinity". Proc. AMS 130 (9): pp. 2711–2717.  This is more modern than the other texts in this reference list.
• Knill, O. (2006). Probability Theory & Stochastic Processes. India: Overseas Press. 
• Mattilla, P. (1995). Geometry of Sets in Euclidean Spaces. San Francisco: Cambridge University Press.  This has more advanced material on fractals.