Positive dependence orderings and stopping times (Q1336528)

The authors establish a connection between the concept of positive dependence [cf. \textit{E. L. Lehmann}, Ann. Math. Stat. 37, 1137-1153 (1966; Zbl 0146.406); \textit{G. Kimeldorf} and \textit{A. R. Sampson}, Ann. Inst. Stat. Math. 39, 113-128 (1987; Zbl 0617.62006)] and the properties of certain stopping or hitting times. Given two bivariate processes such that the marginal processes have the same law, if the components of one are more positively dependent, then its difference process will tend to be smaller, so the first time its absolute value crosses some threshold will tend to be larger. In several special cases such as families of Brownian motion, necessary and sufficient conditions are found for the stochastic orderings of these stopping times which are meaningful in terms of dependence. These ideas are also applied to dependent random walks and exchangeable sequences of elliptically contoured random variables, and in most cases only sufficient conditions are found. Methods employed include use of Laplace order, Slepian's inequality, the Sudakov-Fernique inequality, and covariance of the increments.