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Directionally convex ($dcx$) ordering is a tool for comparison of dependencestructure of random vectors that also takes into account the variability of themarginal distributions. When extended to random fields it concerns comparisonof all finite dimensional distributions. Viewing locally finite measures asnon-negative fields of measure-values indexed by the bounded Borel subsets ofthe space, in this paper we formulate and study the $dcx$ ordering of randommeasures on locally compact spaces. We show that the $dcx$ order is preservedunder some of the natural operations considered on random measures and pointprocesses, such as deterministic displacement of points, independentsuperposition and thinning as well as independent, identically distributedmarking. Further operations such as position dependent marking and displacementof points though do not preserve the $dcx$ order on all point processes, areshown to preserve the order on Cox point processes. We also examine the impactof $dcx$ order on the second moment properties, in particular on clustering andon Palm distributions. Comparisons of Ripley's functions, pair correlationfunctions as well as examples seem to indicate that point processes higher in$dcx$ order cluster more. As the main result, we show that non-negativeintegral shot-noise fields with respect to $dcx$ ordered random measuresinherit this ordering from the measures. Numerous applications of this resultare shown, in particular to comparison of various Cox processes and someperformance measures of wireless networks, in both of which shot-noise fieldsappear as key ingredients. We also mention a few pertinent open questions.

Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications