Cantor distribution

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:
:
\begin
C_0 = & ,1\\ pt C_1 = & ,1/3cup /3,1\\ pt C_2 = & ,1/9cup /9,1/3cup /3,7/9cup /9,1\\ pt C_3 = & ,1/27cup /27,1/9cup /9,7/27cup /27,1/3cup \\ pt & /3,19/27cup 0/27,7/9cup /9,25/27cup 6/27,1\\ pt C_4 = & ,1/81cup /81,1/27cup /27,7/81cup /81,1/9cup /9,19/81cup 0/81,7/27cup \\ pt & /27,25/81cup 6/81,1/3cup /3,55/81cup 6/81,19/27cup 0/27,61/81cup \\ pt & 2/81,21/27cup /9,73/81cup 4/81,25/27cup 6/27,79/81cup 0/81,1\\ pt C_5 = & \cdots
\end

The Cantor distribution is the unique probability distribution for which for any ''C''_''t'' (''t'' ∈ ), 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 ''C''₁, let ''Y'' = 0 if ''X'' ∈  ,1/3 and 1 if ''X'' ∈  /3,1 Then:

: $\begin \operatorname(X) & = \operatorname(\operatorname(X\mid Y)) + \operatorname(\operatorname(X\mid Y)) \\ & = \frac\operatorname(X) + \operatorname \left\ \\ & = \frac\operatorname(X) + \frac \end$

From this we get:

:$\operatorname(X)=\frac.$

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

:$\kappa_ = \frac , \,\!$

where ''B''_2''n'' is the 2''n''th Bernoulli number, and then expressing the moments as functions of the cumulants.

References

Further reading

* ''This, as with other standard texts, has the Cantor function and its one sided derivates.''
* ''This is more modern than the other texts in this reference list.''
*
* ''This has more advanced material on fractals.''

{{Clear

Continuous distributions
Georg Cantor