Show that ij,kl i,j,r k,l,sr,s, regarded as a p 2 p 2 symmetric

Show that κ

ij,kl − κ

i,j,rκ

k,l,sκr,s, regarded as a p 2 × p 2

symmetric matrix, is nonnegative definite. By examining the trace of this matrix in the case where κ

i = 0, κ

i,j = δ

ij

, show that

ρ4 ≥ ρ

2 23 − p − 1, with equality if and only if the joint distribution is concentrated on p+1 points not contained in any linear subspace of R p. Deduce that ρ4 = ρ

2 13 − p − 1 implies

ρ4 = ρ

2 13 − 2 and that ρ4 > ρ

2 13 − 2 implies ρ4 > ρ

2 13 − p − 1.