scholarly journals Lévy-based Cox point processes

2008 ◽  
Vol 40 (03) ◽  
pp. 603-629 ◽  
Author(s):  
Gunnar Hellmund ◽  
Michaela Prokešová ◽  
Eva B. Vedel Jensen
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Kernel Smoothing ◽  
Random Measure ◽  
Infinitely Divisible ◽  
Mixing Properties ◽  
Cox Processes ◽  
Special Cases ◽  
Spatio Temporal ◽  
Product Densities

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

scholarly journals Lévy-based Cox point processes

2008 ◽  
Vol 40 (3) ◽  
pp. 603-629 ◽  
Author(s):  
Gunnar Hellmund ◽  
Michaela Prokešová ◽  
Eva B. Vedel Jensen
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Kernel Smoothing ◽  
Random Measure ◽  
Infinitely Divisible ◽  
Mixing Properties ◽  
Cox Processes ◽  
Special Cases ◽  
Spatio Temporal ◽  
Product Densities

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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

2009 ◽  
Vol 41 (03) ◽  
pp. 623-646 ◽  
Author(s):  
Bartłomiej Błaszczyszyn ◽  
D. Yogeshwaran
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Pair Correlation ◽  
Dependence Structure ◽  
Convex Order ◽  
Random Measures ◽  
Convex Ordering ◽  
Cox Processes ◽  
The Impact ◽  
Directionally Convex Order

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.

Generalized Gamma measures and shot-noise Cox processes

1999 ◽  
Vol 31 (04) ◽  
pp. 929-953 ◽  
Author(s):  
Anders Brix
Keyword(s):  
Shot Noise ◽  
Random Measures ◽  
Intensity Measures ◽  
Mixing Properties ◽  
Point Patterns ◽  
Poisson Cluster Process ◽  
Cox Processes ◽  
The Family ◽  
Poisson Cluster ◽  
Markov Properties

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes. We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

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

2009 ◽  
Vol 41 (3) ◽  
pp. 623-646 ◽  
Author(s):  
Bartłomiej Błaszczyszyn ◽  
D. Yogeshwaran
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Pair Correlation ◽  
Dependence Structure ◽  
Convex Order ◽  
Random Measures ◽  
Convex Ordering ◽  
Cox Processes ◽  
The Impact ◽  
Directionally Convex Order

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.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (2) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

scholarly journals The permanental process

2006 ◽  
Vol 38 (4) ◽  
pp. 873-888 ◽  
Author(s):  
Peter McCullagh ◽  
Jesper Møller
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Point Processes ◽  
Infinitely Divisible ◽  
Determinantal Processes ◽  
Cox Processes

We extend the boson process first to a large class of Cox processes and second to an even larger class of infinitely divisible point processes. Density and moment results are studied in detail. These results are obtained in closed form as weighted permanents, so the extension is called a permanental process. Temporal extensions and a particularly tractable case of the permanental process are also studied. Extensions of the fermion process along similar lines, leading to so-called determinantal processes, are discussed.

scholarly journals Bartlett spectrum and mixing properties of infinitely divisible random measures

2007 ◽  
Vol 39 (04) ◽  
pp. 893-897
Author(s):  
Emmanuel Roy
Keyword(s):  
Gaussian Process ◽  
Spectral Measure ◽  
Random Measure ◽  
Stationary Gaussian Process ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Weak Mixing ◽  
Mixing Properties

We prove that the Bartlett spectrum of a stationary, infinitely divisible (ID) random measure determines ergodicity, weak mixing, and mixing. In this context, the Bartlett spectrum plays the same role as the spectral measure of a stationary Gaussian process.

Generalized Gamma measures and shot-noise Cox processes

1999 ◽  
Vol 31 (4) ◽  
pp. 929-953 ◽  
Author(s):  
Anders Brix
Keyword(s):  
Shot Noise ◽  
Random Measures ◽  
Intensity Measures ◽  
Mixing Properties ◽  
Point Patterns ◽  
Poisson Cluster Process ◽  
Cox Processes ◽  
The Family ◽  
Poisson Cluster ◽  
Markov Properties

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes.We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (02) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

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