On Probabilities of Rectangles in Multivariate Student Distributions: Their Dependence on Correlations

Recently several authors (cf. 5, 6, 8, 9) have established for arbitrary positive numbers c₁, , cₖ the inequality equation*1P\|X₁| c₁, , |Xₖ| cₖ\ ᵏ₈=₁ P\|Xᵢ| cᵢ\equation* valid for a random vector X = (X₁, , Xₖ) having a multivariate normal distribution with mean values 0 and with an arbitrary covariance matrix. A question then arises whether also an analogue to (1) for multivariate Student distributions holds true, i. e. the inequality equation*2P\|X₁|/S₁ c₁, , |Xₖ|/Sₖ cₖ\ ᵏ₈=₁ P\|Xᵢ|/Sᵢ cᵢ\equation* where X = (X₁, , Xₖ) is as before, while Sᵢ = (ᵖ=₁ Z²₈) ^1{2}, i = 1, , k, where Z_ = (Z₈, , Z₊), = 1, , p, is a random sample of p vectors, which are mutually independent and independent of X, and each of which has, in the simplest case, the same normal distribution as X. More generally, the Z_'s have some normal distributions with mean values 0 and with some covariance matrices which need not coincide with that of X and even need not be identical. A certain proof of (2) was presented by A. Scott 6 but we shall give here a counterexample showing that, unfortunately, this proof is incorrect. However, if the correlations between Xᵢ and Xⱼ have the form ᵢⱼ₈₉ (i, j = 1, , k; i j) where |ᵢ| 1 (i = 1, , k) and where \₈₉\ is any fixed correlation matrix, and if the correlations between Z₈ and Z₉ have the form ₈₉ (i, j = 1, , k; i j; = 1, , p) where |₈| < 1 (i = 1, , k; = 1, , p), we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each |ᵢ| and each |₈|; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.