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Is it possible to construct a characteristic function (for a distribution) $\phi(t)$ such that $\phi(t) = t^{-1/4}$ for $16\leq t \leq 20$?

I tried the inversion formula, but didn't know how to prove that the resulting set function is a probability measure.

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    If you've answered the question using an [inversion formula](http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29#Inversion_formulas), I think you should post it.2011-12-07

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Is it possible to construct a characteristic function (for a distribution) $\phi(t)$ such that $\phi(t) = t^{-1/4}$ for $16\leq t \leq 20$?

Pólya’s theorem can be used to construct such a $\phi$. Here's a simple example:

graph of <span class=\phi">

$ \phi(t) = \begin{cases} 1 - \frac{t}{32} & \text{if } 0 \leq t < 16 \\ \\ t^{-\frac{1}{4}} & \text{if } t \geq 16 \\ \\ \phi(-t) & \text{if t} \lt 0 \end{cases} $

It's easily verified that the conditions of the theorem are met, so this $\phi$ is the characteristic function of an absolutely continuous symmetrical distribution.

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    @epsilon: Perhaps you should edit your question to clarify which approaches you consider acceptable/unacceptable. (Presently, there's nothing to suggest avoiding Pólya’s theorem. Also, contrary to your comment, it seems quite easy to obtain the required convexity.)2011-12-06