Natural statistics for spectral samples

Spectral sampling is associated with the group of unitary transformations acting on matrices in the same way that simple random sampling is associated with the symmetric group acting on vectors. This parallel extends to symmetric functions of spectral samples, k-statistics and polykays. According to the moment method em- ployed within random matrix theory, we introduce cumulants of a random matrix as cumulants of its trace powers and construct spectral polykays as unbiased estimators of their product. When referred to spectral samples, we prove that spectral polykays are natural statistics and return products of free cumulants when the sampling is from an innite population. Moreover we dene generalized cumulants of spectral polykays and study their asymptotic behaviour.