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Which is the relation (can the 1st be derived knowing the 2nd) between the cumulative density function of positive rv $X>0$$CDF_X(x)=Prob(C<=x)$ and the characteristic function of $Y=X^2$: $CF_{Y}(z)=E[e^{izY}]$?

CONSTRAINT: The probability density function of Y cannot be Fourier-invertire from its CF, due to lack of sufficiently fast decay of the last.

Gabriele Pompa
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    Recall that the charachteristic function and the CDF are equivalent descriptions (knowing one gives you the other), and so we may assume we have the CDF of $Y$. Now, since $x^2$ is not an injective function, you can't invert it without further constraints on the domain, and so you can't get the CDF of $X$ given only the CDF of $Y$. In slightly different terms, for $c>0$, $F_Y(c) = F_X(c^{1/2}) - F_X(-c^{1/2})$, and you have two variables and only one equation. – stochasticboy321 Jul 18 '16 at 07:24
  • Thanks @stochasticboy321, I added the constraint X>0. Are you able to state explicitly the relation? And without using the probability density function of Y (that cannot be derived from its CF, since the anti-transform Fourier integral does not converge)? What I mean is: try to figure out your solution, without using anything else but the CF of Y – Gabriele Pompa Jul 18 '16 at 07:40

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