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!