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*}{1}P\{|X_1| c_1,⋯, |X_k| c_k\} Π^kᵢ₌₁ P\{|X_i| c_i\}{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*}{2}P\{|X_1|/S_1 c_1,⋯, |X_k|/S_k c_k\} Π^kᵢ₌₁ P\{|X_i|/S_i c_i\}{equation*} where X = (X₁,⋯, Xₖ) is as before, while Sᵢ = (∑ᵖν=1 Z²iν)1/2, i = 1,⋯, k, where Z_ν = (Ziν,⋯, Zkν), ν = 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_ν'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 λᵢλⱼρᵢⱼ (i,j = 1,⋯, k; i ≠ j) where |λᵢ| 1 (i = 1,⋯, k) and where \ρᵢⱼ\ is any fixed correlation matrix, and if the correlations between Ziν and Zjν have the form τiντjν (i,j = 1,⋯, k; i ≠ j; ν = 1,⋯, p) where |τiν| < 1(i = 1,⋯, k; ν = 1,⋯, p), we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each |λᵢ| and each |τiν|; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.