Dependence orderings for some functionals of multivariate point processes

Abstract

We study dependence orderings for functionals of k-variate point processes F and C: We view the first process as a collection of counting measures, whereas the second as the sequences of interpoint distances. Subsequently, we establish regularity properties of stationary sequences which generalize known results for iid case. The theoretical results are illustrated by many special cases including comparison of multivariate sums and products, comparison of multivariate shock models and queueing systems. r

Dependence ordering for Markov processes on partially ordered spaces

“…There, stochastic monotonicity of the processes makes it possible to obtain necessary and sufficient conditions for supermodular ordering and concordance ordering for finite-dimensional distributions of stationary Markov chains in discrete and continuous time. Dependence orderings of some derived stationary sequences were studied by Kulik and Szekli (2004).…”

On negative association of some finite point processes on general state spaces

“…It follows that τ i=1 U i < cx P o(E(τ )) i=1 U i , where P o(E(τ )) denotes a Poisson random variable which is independent of (U i , i ≥ 1) (see e.g. [26], Corollary 4.5). From this we get η 1 (B) < cx Y, for Y described above since Eη 1 (B) = E(τ )E(U i ) and P o(E(τ )) i=1 U i has Poisson distribution.…”

“…An extensive study of dependence orderings for multivariate point processes on R is contained in [14]. Related results in the theory of point processes and stochastic geometry, where the directionally convex ordering is used to express more clustering in point patterns, are obtained by Błaszczyszyn and Yogeshwaran [5]; see also the references therein.…”

“…where κ is now assumed to be independent of (U i , i ≥ 1) (see, e.g. [14,Corollary 4.5]). From this we get η 1 (B) < cx η 2 (B), since Eη 1 (B) = E(τ )E(U i ) and κ i=1 U i has Poisson distribution.…”

Recent Advances and Applications of the Theory of Stochastic Convexity. Application to Complex Bio-Inspired and Evolution Models

“…At the best of our knowledge, recent results can be seen in (Miyoshi, 2004), and(Fernandez-Ponce, et al, 2008a,b). They are involved in the multivariate stochastic comparisons of some stochastic processes (see Kulik and Szekli, 2005). The following monographs include an exhaustive development of the theory, and/or some detailed applications for selected operational research problems.…”