# Characteristic functions can establish stochastic dominance?

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!

• User @charles.y.zheng commented on clever use of characteristic functions for proofs. – Hirek Feb 7 '15 at 10:28
• Also, @mpiktas said that the primary use of characteristic functions is that they're useful to prove things. – Hirek Feb 7 '15 at 10:29
• 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. – Hirek Feb 8 '15 at 10:56