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|  et/2∏k=1∞cosh⁡(t3k)
CF|  eit/2∏k=1∞cos⁡(t3k)

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.

## Contents

* 1 Characterization
* 2 Moments
* 3 References
* 4 Further reading

## Characterization

The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:

C0=[0,1]C1=[0,1/3]∪[2/3,1]C2=[0,1/9]∪[2/9,1/3]∪[2/3,7/9]∪[8/9,1]C3=[0,1/27]∪[2/27,1/9]∪[2/9,7/27]∪[8/27,1/3]∪[2/3,19/27]∪[20/27,7/9]∪[8/9,25/27]∪[26/27,1]C4=[0,1/81]∪[2/81,1/27]∪[2/27,7/81]∪[8/81,1/9]∪[2/9,19/81]∪[20/81,7/27]∪[8/27,25/81]∪[26/81,1/3]∪[2/3,55/81]∪[56/81,19/27]∪[20/27,61/81]∪[62/81,21/27]∪[8/9,73/81]∪[74/81,25/27]∪[26/27,79/81]∪[80/81,1]C5=⋯

The Cantor distribution is the unique probability distribution for which for any Ct (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in Ct containing the Cantor-distributed random variable is identically 2−t on each one of the 2t 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:

var⁡(X)=E⁡(var⁡(X∣Y))+var⁡(E⁡(X∣Y))=19var⁡(X)+var⁡{1/6with probability 1/25/6with probability 1/2}=19var⁡(X)+19

From this we get:

var⁡(X)=18.

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

κ2n=22n−1(22n−1)B2nn(32n−1),

where B2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

## References

1. ↑ Morrison, Kent (1998-07-23). "Random Walks with Decreasing Steps". Department of Mathematics, California Polytechnic State University. http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf.

## 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.

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