# Cantor distribution

The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.

Parameters Cumulative distribution function none Cantor set none Cantor function 1/2 anywhere in [1/3, 2/3] n/a 1/8 0 −8/5 ${\displaystyle e^{t/2}\prod _{k=1}^{\infty }\cosh \left({\frac {t}{3^{k}}}\right)}$ ${\displaystyle e^{it/2}\prod _{k=1}^{\infty }\cos \left({\frac {t}{3^{k}}}\right)}$

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:

{\displaystyle {\begin{aligned}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{aligned}}}

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 2t on each one of the 2t 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:

{\displaystyle {\begin{aligned}\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{aligned}}}

From this we get:

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

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

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

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

## References

• Falconer, K. J. (1985). Geometry of Fractal Sets. Cambridge & New York: Cambridge Univ Press.
• Hewitt, E.; Stromberg, K. (1965). Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer-Verlag.
• Hu, Tian-You; Lau, Ka Sing (2002). "Fourier Asymptotics of Cantor Type Measures at Infinity". Proc. A.M.S. 130 (9). pp. 2711–2717.
• Knill, O. (2006). Probability Theory & Stochastic Processes. India: Overseas Press.
• Mandelbrot, B. (1982). The Fractal Geometry of Nature. San Francisco, CA: WH Freeman & Co.
• Mattilla, P. (1995). Geometry of Sets in Euclidean Spaces. San Francisco: Cambridge University Press.
• Saks, Stanislaw (1933). Theory of the Integral. Warsaw: PAN. (Reprinted by Dover Publications, Mineola, NY.