How can I determine if experimental data comes from a distribution where the variance is undefined (e.g. the Cauchy distribution)?
I honestly have no idea how to attack this problem in a sensible way, but I imagine one could use random matrix theory to help you. For example, the eigenvalues of a (N,N) non-symmetric matrix form a pattern depending on the distribution. For a standard normal distribution, the eigenvalues on the complex plane form a beautiful unit circle:
For the Cauchy distribution the results are quite different:
So random matrix theory predicts a signal, though it may be an over-engineered solution.
Question: What statistical test can determine if a distribution has undefined variance?