随机测度的方向凸序、散粒噪声场及其在无线通信中的应用
Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications
摘要：
方向凸（$dcx$）排序是一种用于比较随机向量的依赖结构的工具，其还考虑了边缘分布的可变性。当推广到随机场时，它涉及到所有有限维分布的比较。本文将局部有限测度看作测度值的非负域，以空间的有界Borel子集为指标，建立并研究了局部紧空间上随机测度的$dcx$序.我们表明，$dcx$秩序被保存下的一些自然的操作考虑随机措施和点过程，如确定性位移的点，独立的叠加和细化，以及独立，同分布的标记。进一步的操作，如位置相关的标记和位移的点，虽然不保持$dcx$为了所有的点过程，被证明是为了保持考克斯点过程。我们还研究了$dcx$顺序对二阶矩性质的影响，特别是对聚类和Palm分布的影响。Ripley函数、对相关函数以及实例的比较表明，$dcx$阶数越高的点过程越容易聚集。作为主要结果，我们表明，非负积分散粒噪声领域的$dcx$有序随机措施继承这种秩序的措施。这一结果的许多应用，特别是各种Cox过程和无线网络的一些性能指标的比较，在这两个散粒噪声场出现的关键成分。我们还提到几个相关的未决问题。
英文摘要：
Directionally convex ($dcx$) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. Viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the $dcx$ ordering of random measures on locally compact spaces. We show that the $dcx$ 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 such as position dependent marking and displacement of points though do not preserve the $dcx$ order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of $dcx$ order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that point processes higher in $dcx$ order cluster more. As the main result, we show that non-negative integral shot-noise fields with respect to $dcx$ 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.
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