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Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Bartłomiej Błaszczyszyn  and
D. Yogeshwaran
Bartłomiej Błaszczyszyn*
Affiliation:
INRIA-ENS and University of Wrocław
D. Yogeshwaran*
Affiliation:
INRIA-ENS
*
∗ Postal address: Bureau 21 TREC, INRIA (5th floor), 23 Avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
∗ Postal address: Bureau 21 TREC, INRIA (5th floor), 23 Avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
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Abstract

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Directionally convex ordering is a useful tool for comparing the dependence structure of random vectors, which also takes into account the variability of the marginal distributions. It can be extended to random fields by comparing all finite-dimensional distributions. Viewing locally finite measures as nonnegative fields of measure values indexed by the bounded Borel subsets of the space, in this paper we formulate and study directionally convex ordering of random measures on locally compact spaces. We show that the directionally convex order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition, and thinning, as well as independent, identically distributed marking. Further operations on Cox point processes such as position-dependent marking and displacement of points are shown to preserve the order. We also examine the impact of the directionally convex order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions and pair correlation functions, as well as examples, seem to indicate that point processes higher in the directionally convex order cluster more. In our main result we show that nonnegative integral shot noise fields with respect to the directionally convex ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot noise fields appear as key ingredients. We also mention a few pertinent open questions.

Keywords

Stochastic orderingdirectional convexityrandom measurerandom fieldpoint processwireless network

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G60: Random fields 60G57: Random measures 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 623 - 646
Copyright
Copyright © Applied Probability Trust 2009 

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