In previous problem, suppose that Σ is positive semidefinite of rank q ⊤ 1 , X⊤ 2 , where X1 is a q-vector for which Σ11 is of full rank q. Then show that X2−Σ22Σ −1 11 X has a degenerated...

In previous problem, suppose that Σ is positive semidefinite of rank q <>⊤ 1 , X⊤ 2 , where X1 is a q-vector for which Σ11 is of full rank q. Then show that X2−Σ22Σ −1 11 X has a degenerated distribution at the point θ2−Σ21Σ −1 11 θ1. Hence, or otherwise, show that (8.4) holds with the norm kx−θkΣ replaced by kx−θkΣ11 . Work out a characterization of elliptically symmetric distributions bypassing the existence of the density function.