Probability Inequalities for n-Dimensional Rectangles via Multivariate Majorization

Abstract

Inequalities for the probability content $$ P\left[ { \cap _{j\, = \,1}^n\left\{ {\,{a_{1j\,}}\, \leqslant \,{X_j}\, \leqslant \,{a_{2j}}} \right\}} \right] $$are obtained, via concepts of multivariate majorization (which involves the diversity of elements of the 2xn matrix A = (aij)). A special case of the general result is that $$ P\left[ { \cap _{j = 1}^n\left\{ {{a_{1j}} \leqslant {X_j} \leqslant {a_{2j}}} \right\}} \right] \leqslant P\left[ { \cap _{j = 1}^n\left\{ {{{\bar a}_1} \leqslant {X_j} \leqslant {{\bar a}_2}} \right\}} \right] $$for $$ {\bar a_{i\,}} = \,\frac{1}{n}\,\sumolimits_{j = 1}^n {{a_{ij}}\,\left( {i\, = \,1,\,2} \right).} $$. The main theorems apply in most important cases, including the exchangeable normal, t, chi-square and gamma, F, beta, and Dirichlet distributions. The proofs of the inequalities involve a convex combination of an n-dimensional rectangle and its permutation sets.