A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.

Let P(c) = P(X1 ≦ c1, · ··, Xp ≦ cp) for a random vector (X1, · ··, Xp). Bounds are considered of the form where T is a spanning tree corresponding to the bivariate probability structure and di is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.

Sufficient conditions under which two random vectors are ordered in the sense of uniform conditional stochastic order (Whitt (1980), (1982)) with respect to planar regions are given. A natural classification of distributions based on this notion of stochastic order is defined and studied. A negative dependence property of Block et al. (1985) is shown to hold for this class of distributions. An application of these results in statistics is also presented.

A direct approach to derive dependence properties among the hitting times of bivariate processes has been initiated by Ebrahimi (1987) and explored further by Ebrahimi and Ramallingam (1988). In this paper, new results are obtained for multivariate processes, which help us to identify positive and negative dependence structures among the hitting times of the processes. Applications of our theorems to reliability of systems are given.