Cantor Distribution - Characterization

Characterization

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

$\begin{align} C_{0} = & \\ C_{1} = & \cup \\ C_{2} = & \cup\cup\cup \\ C_{3} = & \cup\cup\cup\cup \\ & \cup\cup\cup \\ C_{4} = & \cdots . \end{align}$

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 2t intervals.

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