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

Błaszczyszyn, Bartłomiej; Yogeshwaran, D.

doi:10.1017/s0001867800003499

Cited by 15 publications

(54 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this regard, our initial choice was directionally convex (dcx) order 2 (to be formally defined later). It has its roots in [5], where one shows various results as well as examples indicating that the dcx order on point processes implies ordering of several well-known clustering characteristics in spatial statistics such as Ripley's K-function and second moment densities. Namely, a point process that is larger in the dcx order exhibits more clustering, while having equal mean number of points in any given set.…”

Section: 2mentioning

confidence: 99%

Clustering and percolation of point processes

Błaszczyszyn

Yogeshwaran

2013

Electron. J. Probab.

View full text Add to dashboard Cite

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are $k$-coverage in the Boolean model (coverage by at least $k$ grains) and SINR-coverage (coverage if the signal-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the \v{C}ech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.Comment: 25 pages, 1 figure. This paper complements arXiv:1111.6017. arXiv admin note: substantial text overlap with arXiv:1105.429

show abstract

Section: 2mentioning

confidence: 99%

Clustering and percolation of point processes

Błaszczyszyn

Yogeshwaran

2013

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…Another observation, proved in [4], says that the operations transforming some random measure L into a Cox point process Cox L (cf. Definition 4) preserves the dcx-order.…”

Section: Definitions and Basic Resultsmentioning

confidence: 99%

“…Using Proposition 8 one can prove (cf. [4]) that Poisson-Poisson cluster point processes and, more generally, Lévybased Cox point processes are super-Poisson.…”

Section: Definition 16 (Sub-and Super-poisson Point Process)mentioning

confidence: 99%

Clustering Comparison of Point Processes, with Applications to Random Geometric Models

Błaszczyszyn

Yogeshwaran

2014

Stochastic Geometry, Spatial Statistics and Random Fields

Self Cite

View full text Add to dashboard Cite

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the Boolean model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers forČech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some "sub-Poisson" processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.

show abstract

“…Then of course F dcx ⊆ F sm . Typical examples from F dcx class of functions are f (x) = ψ( n i=1 x i ), for ψ convex, or f (x) = max 1≤i≤n x i , but there are many other useful functions in this class, see for example [3].…”

Section: Dependence Orderings and Negative Correlations For Vectorsmentioning

confidence: 99%