Cantor 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
Excess kurtosis	−8/5
MGF	$e^{t/2}\prod _{k=1}^{\infty }\cosh \left({\frac {t}{3^{k}}}\right)$
CF	$e^{it/2}\prod _{k=1}^{\infty }\cos \left({\frac {t}{3^{k}}}\right)$

TheCantor distributionis theprobability distributionwhosecumulative distribution functionis theCantor function.

This distribution has neither aprobability density functionnor aprobability mass function,since although its cumulative distribution function is acontinuous function,the distribution is notabsolutely continuouswith respect toLebesgue 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 asingular distribution.

Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as theDevil's staircase,although that term has a more general meaning.

Characterization

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

${\begin{aligned}C_{0}={}&[0,1]\\[8pt]C_{1}={}&[0,1/3]\cup [2/3,1]\\[8pt]C_{2}={}&[0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]\\[8pt]C_{3}={}&[0,1/27]\cup [2/27,1/9]\cup [2/9,7/27]\cup [8/27,1/3]\cup \\[4pt]{}&[2/3,19/27]\cup [20/27,7/9]\cup [8/9,25/27]\cup [26/27,1]\\[8pt]C_{4}={}&[0,1/81]\cup [2/81,1/27]\cup [2/27,7/81]\cup [8/81,1/9]\cup [2/9,19/81]\cup [20/81,7/27]\cup \\[4pt]&[8/27,25/81]\cup [26/81,1/3]\cup [2/3,55/81]\cup [56/81,19/27]\cup [20/27,61/81]\cup \\[4pt]&[62/81,21/27]\cup [8/9,73/81]\cup [74/81,25/27]\cup [26/27,79/81]\cup [80/81,1]\\[8pt]C_{5}={}&\cdots \end{aligned}}$

The Cantor distribution is the unique probability distribution for which for anyC_t(t∈ { 0, 1, 2, 3,... }), the probability of a particular interval inC_tcontaining the Cantor-distributed random variable is identically 2^−ton each one of the 2^tintervals.

Moments

It is easy to see by symmetry and being bounded that for arandom variableXhaving this distribution, itsexpected valueE(X) = 1/2, and that all odd central moments ofXare 0.

Thelaw of total variancecan be used to find thevariancevar(X), as follows. For the above setC₁,letY= 0 ifX∈ [0,1/3], and 1 ifX∈ [2/3,1]. Then:

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

From this we get:

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

A closed-form expression for any evencentral momentcan be found by first obtaining the evencumulants

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

whereB_2nis the 2nthBernoulli number,and thenexpressing the moments as functions of the cumulants.

Further reading

• Hewitt, E.; Stromberg, K. (1965).Real and Abstract Analysis.Berlin-Heidelberg-New York: Springer-Verlag.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.Vol. 130, no. 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.