Palm measure

2001 ◽  
pp. 601-601
Author(s):  
Saul I. Gass ◽  
Carl M. Harris
Keyword(s):  
Palm Measure

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (2) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

Perturbation analysis of functionals of random measures

1995 ◽  
Vol 27 (2) ◽  
pp. 306-325 ◽  
Author(s):  
François Baccelli ◽  
Maurice Klein ◽  
Sergei Zuyev
Keyword(s):  
Phase Space ◽  
Random Measure ◽  
Random Processes ◽  
Stationary Processes ◽  
Light Traffic ◽  
Random Measures ◽  
First Order ◽  
Order Expansion ◽  
Palm Measure ◽  
New Algorithms

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (02) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

Perturbation analysis of functionals of random measures

1995 ◽  
Vol 27 (02) ◽  
pp. 306-325 ◽  
Author(s):  
François Baccelli ◽  
Maurice Klein ◽  
Sergei Zuyev
Keyword(s):  
Phase Space ◽  
Random Measure ◽  
Random Processes ◽  
Stationary Processes ◽  
Light Traffic ◽  
Random Measures ◽  
First Order ◽  
Order Expansion ◽  
Palm Measure ◽  
New Algorithms

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

scholarly journals Large Deviations for the Graph Distance in Supercritical Continuum Percolation

2011 ◽  
Vol 48 (1) ◽  
pp. 154-172 ◽  
Author(s):  
Chang-Long Yao ◽  
Ge Chen ◽  
Tian-De Guo
Keyword(s):  
Large Deviation ◽  
Geometric Graph ◽  
Graph Distance ◽  
Homogeneous Poisson Process ◽  
Random Geometric Graph ◽  
Asymptotic Shape ◽  
Deviation Inequality ◽  
Large Deviation Inequality ◽  
Palm Measure ◽  
Infinite Component

Denote the Palm measure of a homogeneous Poisson process Hλ with two points 0 and x by P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C∞ of the random geometric graph G(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

scholarly journals Stationary partitions and Palm probabilities

2006 ◽  
Vol 38 (3) ◽  
pp. 602-620 ◽  
Author(s):  
Günter Last
Keyword(s):  
Probability Measure ◽  
Voronoi Tessellation ◽  
Random Partition ◽  
Weighted Version ◽  
Stationary Point Process ◽  
Typical Cell ◽  
Palm Measure ◽  
Special Case ◽  
Palm Probabilities ◽  
Shift Coupling

A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

scholarly journals Large Deviations for the Graph Distance in Supercritical Continuum Percolation

2011 ◽  
Vol 48 (01) ◽  
pp. 154-172 ◽  
Author(s):  
Chang-Long Yao ◽  
Ge Chen ◽  
Tian-De Guo
Keyword(s):  
Large Deviation ◽  
Geometric Graph ◽  
Graph Distance ◽  
Homogeneous Poisson Process ◽  
Random Geometric Graph ◽  
Asymptotic Shape ◽  
Deviation Inequality ◽  
Large Deviation Inequality ◽  
Palm Measure ◽  
Infinite Component

Denote the Palm measure of a homogeneous Poisson process H λ with two points 0 and x by P0,x . We prove that there exists a constant μ ≥ 1 such that P0,x (D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C ∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C ∞ of the random geometric graph G(H λ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

scholarly journals A more comprehensive complementary theorem for the analysis of Poisson point processes

2006 ◽  
Vol 38 (03) ◽  
pp. 581-601 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Poisson Point Processes ◽  
Palm Measure ◽  
Open Issues

In this paper we discuss the complementary theorem applied to the typical n-tuple of a Poisson point process. The theorem was first presented by Miles in 1970 and discussed by Santaló in 1976 and, within a Palm measure framework, by Møller and Zuyev in 1996. The theorems put forward by these authors are not correct for all the examples that they present, suggesting that further consideration of their work is needed if one wishes to bring all those examples within the ambit of the complementary theorem. We give alternative analyses of the errant examples and, with a modification of the technicalities in the work of the above authors, move toward a more comprehensive complementary theorem. Some open issues still remain.

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