Probability Inequalities for Multivariate Distributions with Dependence Structures

Abstract

Let X ₁, …, X_n be a sequence of random variables with a given positive or negative dependence structure. In this article we exploit the assumed dependence structure to construct a sequence of bounds for the P(X_i ε C_i ; i = 1, …, n), where C_i are infinite intervals of the same type. These bounds are superior to the well-known product bounds that are based solely on the marginal probabilities. Moreover, the new bounds can serve as respectable approximations for the P(X_i ε C_i ; i = 1, …, n).

Key Words:

• Multivariate probabilities
• Positive dependence
• Negative dependence
• Lower bounds
• Approximations