Let the distribution function $CDF(X,t)$ of a random variable $X$ be defined as $0$ for $ t <0, \text{Cantor function}(t)$ for $t \ge 0$ and $ t \le 1, 1$ for $ t > 1$ (for example, see http://en.wikipedia.org/wiki/Cantor_function for the used definition). What is the characteristic function of $X$? It would be an example of the characteristic function of a singular distribution.
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This should probably be a comment but I'm 9 points short.
The answer's on wikepedia. It's $e^{\tfrac{it}2} \prod_{i=1}^\infty cos\left(\frac t{3i}\right)$.
http://en.wikipedia.org/wiki/Cantor_distribution
I'm pretty sure there's no closed form of the infinite product.
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$\begingroup$ Can you kindly give a reference for the proof? $\endgroup$– MarkApr 17, 2013 at 19:11
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$\begingroup$ There's also a reference on the wikipedia page. calpoly.edu/~kmorriso/Research/RandomWalks.pdf The proof is on page 4. $\endgroup$ Apr 17, 2013 at 19:40