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We study the first and second orders of the asymptotic expansion, as thedimension goes to infinity, of the moments of the Hilbert-Schmidt norm of auniformly distributed matrix in the p-Schatten unit ball. We consider the caseof matrices with real, complex or quaternionic entries, self-adjoint or not.When p > 3, this asymptotic expansion allows us to establish a generalizedversion of the variance conjecture for the family of p-Schatten unit balls ofself-adjoint matrices.

Asymptotics of the Inertia Moments and the Variance Conjecture in Schatten Balls