Characteristic functions (cf) are closely related to cdfs and pdfs of random variables, for example
Question: Is there any link (integral representation) between a cf and the inverse cdf (or quantile function)?
The inverse of a cdf $F : \mathbf{R} \mapsto [0,1]$ is usually $$ F^{-1}(p) = \inf\{ x : F(x) \ge p \}. $$
The way you invert a c.f. $\phi$ to get measures of intervals is $$ \mu([a,b]) = \lim_{T\rightarrow \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-ita}-e^{-itb}}{it} \phi(t) dt. $$
So you can put these together: $$ F^{-1}(p)= \inf\{ x : \lim_{a \to -\infty}\lim_{T\rightarrow \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-ita}-e^{-itx}}{it} \phi(t) dt \ge p \}. $$ Does this satisfy your needs?
