Let (X1j ,...,Xpj ), j = 1,...,n, be a sample from a p-variate normal distribution with mean (ξ1,...,ξp) and covariance matrix Σ = (σij ), where σ2 ij = σ2 when j = i, and σ2 ij = ρσ2 when j = i. Show that the covariance matrix is positive definite if and only if ρ > −1/(p − 1).

[For fixed σ and ρ < 0, the quadratic form (1/σ2)

σij yiyj = y2 i +

ρ

yiyj takes on its minimum value over y2 i = 1 when all the y’s are equal.]