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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 

References

Baccelli, F. and Błaszczyszyn, B. (2001). On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications. Adv. App. Prob. 33, 293–323.Google Scholar
Baccelli, F., Błaszczyszyn, B. and Mühlethaler, P. (2006). An aloha protocol for multihop mobile wireless networks. IEEE Trans. Inf. Theory 52, 421–436.Google Scholar
Bassan, B. and Scarsini, M. (1991). Convex orderings for stochastic processes. Comment. Math. Univ. Carolin. 32, 115–118.Google Scholar
Chang, C.-S., Chao, X. and Pinedo, M. (1991). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Adv. App. Prob. 23, 210–228.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Dousse, O. et al. (2006). Percolation in the signal to interference ratio graph. J. Appl. Prob. 43, 552–562.CrossRefGoogle Scholar
Ganti, R. and Haenggi, M. (2008). Interference and outage in clustered wireless ad hoc networks. Preprint. Available at http://arxiv.org/abs/0706.2434 Google Scholar
Gilbert, E. N. (1961). Random plane networks. J. SIAM 9, 533–543.Google Scholar
Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inf. Theory 42, 388–404.Google Scholar
Hall, P. G. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Heinrich, L. and Molchanov, I. S. (1994). Some limit theorems for extremal and union shot-noise processes. Math. Nachr. 168, 139–159.CrossRefGoogle Scholar
Hellmund, G., Prokešová, M. and Vedel Jensen, E. B. (2008). Lévy-based Cox point processes. Adv. App. Prob. 40, 603–629.Google Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Virag, B. (2006). Determinantal processes and independence. Prob. Surveys 3, 206–229.Google Scholar
Huffer, F. (1984). Inequalities for M/G/∞ queues and related shot noise processes. Tech. Rep. 351, Department of Statistics, Stanford University.Google Scholar
Huffer, F. (1987). Inequalities for the M/G/∞ queue and related shot noise processes. J. Appl. Prob. 24, 978–989.Google Scholar
Kallenberg, O. (1983). Random Measures. Academic Press, London.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Kostlan, E. (1992). On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164, 385–388.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, London.Google Scholar
Meester, L. E. and Shanthikumar, J. G. (1993). Regularity of stochastic processes: a theory based on directional convexity. Prob. Eng. Inf. Sci. 7, 343–360.Google Scholar
Meester, L. E. and Shanthikumar, J. G. (1999). Stochastic convexity on general space. Math. Operat. Res. 24, 472–494.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.CrossRefGoogle Scholar
Miyoshi, N. (2004). A note on bounds and monotonicity of spatial stationary Cox shot noise. Prob. Eng Inf. Sci. 18, 561–571.Google Scholar
Miyoshi, N. and Rolski, T. (2004). Ross-type conjectures on monotonicity of queues. Austral. N. Z. J. Statist. 46, 121–131.Google Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. App. Prob. 35, 614–640.Google Scholar
Møller, J. and Torrisi, G. L. (2005). Generalized shot noise Cox processes. Adv. App. Prob. 37, 48–74.CrossRefGoogle Scholar
Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451–482.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Rolski, T. (1986). Upper bounds for single server queues with doubly stochastic Poisson arrivals. Math. Operat. Res. 11, 442–450.CrossRefGoogle Scholar
Rolski, T. and Szekli, R. (1991). Stochastic ordering and thinning of point processes. Stoch. Process. Appl. 37, 299–312.Google Scholar
Ross, S. M. (1978). Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 15, 602–609.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Ann. Inst. Statist. Math. 42, 509–531.Google Scholar
Stoyan, D. (1983). Inequalities and bounds for variances of point processes and fibre processes. Math. Operationsforsch. Statist. Ser. Statist. 14, 409–419.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar