In an answer to the question here What is the purpose of characteristic functions?

people answered the general question about characteristic functions. One answer mentioned that one can use it to short-cut proofs about distributions in cases where CDFs are unknown or don't even exist.

I have the following problem: I would like to prove that a particular central chi-squared variable first-order stochastically dominates the smallest eigenvalue of a certain non-central Wishart matrix.

Using the CDFs is impossible to do because the CDF of the smallest eigenvalue of a non-central Wishart matrix is impossibly hard to compute. I haven't been able to perform the integration, at least, since it's a Laplace transform over positive def. matrices. So I would like to use characteristic functions to do that. Can someone help me understand when or how that is possible?

Thank you so much!

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    $\begingroup$ User @charles.y.zheng commented on clever use of characteristic functions for proofs. $\endgroup$ – Hirek Feb 7 '15 at 10:28
  • $\begingroup$ Also, @mpiktas said that the primary use of characteristic functions is that they're useful to prove things. $\endgroup$ – Hirek Feb 7 '15 at 10:29
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    $\begingroup$ Thanks so much @Glen_b I took out that bit since it does exist for sure, I just am not able to do the integral that would lead to the CDF. $\endgroup$ – Hirek Feb 8 '15 at 10:56

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