Cantor distribution

Cantor

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

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 it is a continuous function it is not absolutely continuous with respect to Lebesgue measure, nor has it 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

$\begin{align} C_{0} = & [0,1] \\ C_{1} = & [0,1/3]\cup[2/3,1] \\ C_{2} = & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\ C_{3} = & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\ & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\ C_{4} = & \cdots . \end{align}$

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 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:

$\begin{align} \operatorname{var}(X) & = \operatorname{E}(\operatorname{var}(X\mid Y)) + \operatorname{var}(\operatorname{E}(X\mid Y)) \\ & = \frac{1}{9}\operatorname{var}(X) + \operatorname{var} \left\{ \begin{matrix} 1/6 & \mbox{with probability}\ 1/2 \\ 5/6 & \mbox{with probability}\ 1/2 \end{matrix} \right\} \\ & = \frac{1}{9}\operatorname{var}(X) + \frac{1}{9} \end{align}$

From this we get:

$\operatorname{var}(X)=\frac{1}{8}.$

A closed-form expression for any even central moment can be found by first obtaining the even cumulants[1]

$\kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}} {n\, (3^{2n}-1)}, \,\!$

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

References

[1]KJ Falconer, GEometry of Fractal Sets, Cambridge Univ Press, Cambridge & New York, 1985 and later eds. (This has a chapter on Self-Similar Fractals; does not have the Cantor Function , graphed in this article))

[2]E.Hewitt & K Stromberg, Real and abstract analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1965 and later eds( This as withother standard texts has the Cantor Function and its one sided derivates)

[3])Tian-You Hu and Ka Sing Lau, Fourier Asymptotics of Cantor Type Measures atInfinity, Proc. A.M.S.,vol 130,Number9(2002) pp2711-2717( is related to the this article and is more modern than the Texts in this Reference list)

[4]O.Knill;Probability Theory&Stochastic Processes...Overseas Press (India) 2006 ( also directly related to this article)

[5]B.Mandelbrojt, The Fractal Geometry of Nature, WH Freeman &CO;San FRancisco CA (1982) ( This is where thepopular name"Fractals" was coined; According to a Reviewer for the AMS," written with daring originality".contains much valuable material. Most of it is by now rewritten in a style suitable for/ acceptable to pure mathematicians.; some of it by Falconer and in Mattillas books listed)

[6]P.Mattilla; Geometry of sets in Euclidean Spaces, CambridgeUniv Press , San FRancisco , 1995 and later edns..( This has more advanced material on Fractals)

[7]Stanislaw Saks, Theory of the Integral; PAN, Warsaw 1933 and later eds; Reprinted by Dover Publications, Mineola, NY(Classic Gem; contains a small section on Hausdorff Measures and Hausdorff dimension= Fractal Dimension)