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 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.

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.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

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.

On a class of random motions of point processes involving interaction

1978 ◽  
Vol 10 (3) ◽  
pp. 613-632 ◽  
Author(s):  
Harry M. Pierson
Keyword(s):  
Large Class ◽  
Uniform Distribution ◽  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
General Setting ◽  
Invariant Distributions ◽  
Stationary Point Process ◽  
Random Motions ◽  
Interval Lengths

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.

Infinitely Divisible Point Processes

1979 ◽  
Vol 30 (5) ◽  
pp. 502
Author(s):  
W. D. Ray ◽  
K. Matthes ◽  
J. Kerstan ◽  
J. Mecke
Keyword(s):  
Point Processes ◽  
Infinitely Divisible

scholarly journals Simulation of cluster point processes without edge effects

2002 ◽  
Vol 34 (02) ◽  
pp. 267-280 ◽  
Author(s):  
Anders Brix ◽  
Wilfrid S. Kendall
Keyword(s):  
Edge Effects ◽  
Point Processes ◽  
Direct Method ◽  
A Priori ◽  
Simulation Method ◽  
Observation Window ◽  
Cox Processes ◽  
Cluster Distribution ◽  
Actual Observation ◽  
Parent Point

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

scholarly journals Empirical cumulant function based parameter estimation in stable laws

2019 ◽  
Vol 22 (2) ◽  
pp. 311-338 ◽  
Author(s):  
Annika Krutto
Keyword(s):  
Closed Form ◽  
Selection Rule ◽  
Empirical Characteristic Function ◽  
Challenging Problem ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Asymptotically Normal ◽  
Cumulant Function ◽  
Independent Identically Distributed ◽  
Infinite Moments

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

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