A limit theorem for point processes by applications

1978 ◽  
Vol 15 (04) ◽  
pp. 726-747
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorem ◽  
Point Process ◽  
Random Vector ◽  
Arbitrary Function ◽  
Point Processes ◽  
Probability Generating Function ◽  
Mild Conditions ◽  
Dimensional Distribution ◽  
Image Position

Let 0 ≦ T 1 ≦ T 2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN (t), t – TN (t)–1, · ··, t – TN (t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ 1, · ··, ξ k ) necessarily satisfying P(0 ≦ ξ 1 ≦ ξ 2 ≦ ·· ·≦ ξ k) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

A limit theorem for point processes by applications

1978 ◽  
Vol 15 (4) ◽  
pp. 726-747 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorem ◽  
Point Process ◽  
Random Vector ◽  
Arbitrary Function ◽  
Point Processes ◽  
Probability Generating Function ◽  
Mild Conditions ◽  
Dimensional Distribution ◽  
Image Position

Let 0 ≦ T1 ≦ T2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN(t), t – TN(t)–1, · ··, t – TN(t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ1, · ··, ξ k) necessarily satisfying P(0 ≦ ξ1 ≦ ξ2 ≦ ·· ·≦ ξk) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

A limit theorem for spatial point processes

1986 ◽  
Vol 18 (03) ◽  
pp. 646-659 ◽  
Author(s):  
Steven P. Ellis
Keyword(s):  
Limit Theorem ◽  
Poisson Process ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Point Processes ◽  
Affine Transformations ◽  
Density Estimates ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
As If

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

A limit theorem for spatial point processes

1986 ◽  
Vol 18 (3) ◽  
pp. 646-659 ◽  
Author(s):  
Steven P. Ellis
Keyword(s):  
Limit Theorem ◽  
Poisson Process ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Point Processes ◽  
Affine Transformations ◽  
Density Estimates ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
As If

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

scholarly journals Point process stabilization methods and dimension estimation

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Point Processes ◽  
Dimension Estimation ◽  
Stabilization Methods ◽  
International Audience

International audience We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.

scholarly journals A GENERALIZED CONTAGION PROCESS WITH AN APPLICATION TO CREDIT RISK

2017 ◽  
Vol 20 (01) ◽  
pp. 1750003 ◽  
Author(s):  
ANGELOS DASSIOS ◽  
HONGBIAO ZHAO
Keyword(s):  
Credit Risk ◽  
Point Process ◽  
Point Processes ◽  
Probability Generating Function ◽  
Jump Processes ◽  
Hawkes Process ◽  
Cox Process ◽  
Model Framework ◽  
Special Cases ◽  
The Laplace Transform

We introduce a class of analytically tractable jump processes with contagion effects by generalizing the classical Hawkes process. This model framework combines the characteristics of three popular point processes in the literature: (1) Cox process with CIR intensity; (2) Cox process with Poisson shot-noise intensity; (3) Hawkes process with exponentially decaying intensity. Hence, it can be considered as a self-exciting and externally-exciting point process with mean-reverting stochastic intensity. Essential probabilistic properties such as moments, the Laplace transform of intensity process, and the probability generating function of point process as well as some important asymptotics have been derived. Some special cases and a method for change of measure are discussed. This point process may be applicable to modeling contagious arrivals of events for various circumstances (such as jumps, transactions, losses, defaults, catastrophes) in finance, insurance and economics with both endogenous and exogenous risk factors within one framework. More specifically, these exogenous factors could contain relatively short-lived shocks and long-lasting risk drivers. We make a simple application to calculate the default probability for credit risk and to price defaultable zero-coupon bonds.

A new graph related to the directions of nearest neighbours in a point process

2003 ◽  
Vol 35 (1) ◽  
pp. 47-55 ◽  
Author(s):  
S. N. Chiu ◽  
I. S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Point Process ◽  
Process Simulation ◽  
Point Processes ◽  
Nearest Neighbour ◽  
Simulation Studies ◽  
Nearest Neighbours ◽  
Typical Point

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

scholarly journals On Risk Processes with Stochastic Intensity function

1971 ◽  
Vol 6 (2) ◽  
pp. 116-128
Author(s):  
Jan Grandell
Keyword(s):  
Stochastic Process ◽  
Point Process ◽  
Intensity Function ◽  
Insurance Company ◽  
Stochastic Intensity ◽  
Image Position ◽  
Risk Processes

In this paper we are going to study some properties of a stochastic process, which has been proposed by Cramér (1968) as a model of the claims arising in an insurance company. This process has been studied by Cox in a different context. A few elementary results, concerning moments, are given by Cox and Lewis (1966). The present paper will be a survey of some results derived by the author (1970:1) and (1970:2). For detailed proofs we refer to these papers.Let λ(t) be a real-valued stochastic process, such that P{λ(t) < o} = o. We further assume that Eλ(t) = 1 and that Eλ2(t) < ∞ for every fixed value of t. We denote the covarianceThe process λ(t) will play the role of an intensity function. That means, that for every fixed realization of the process, the probability ofand that the number of events in disjoint intervals are independent.We now define a point process N(t), where N(t) is the number of events which have occurred in (o, t]). With this definition we getwhereThe integral is assumed to exist almost surely. This process will be called the N-process.

Infinite intensity mixtures of point processes

1982 ◽  
Vol 92 (1) ◽  
pp. 109-114
Author(s):  
D. J. Daley
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Mixing Distribution ◽  
Palm Distribution ◽  
Stationary Point Process ◽  
Image Position

AbstractLet the stationary point process N(·) be the mixture of scaled versions of a stationary orderly point process N1(·) of unit intensity with mixing distribution G(·), so thatWithN(·) has finite or infinite intensity as is finite or infinite, and it is Khinchin orderly when the function γ(·) is slowly varying at infinity. Conditions for N(·) to be orderly involve both G(·) and the Palm distribution of N1(·).

Two inverse problems concerning the superposition of recurrent point processes

1965 ◽  
Vol 2 (02) ◽  
pp. 449-454 ◽  
Author(s):  
R. V. Ambartzumian
Keyword(s):  
Inverse Problems ◽  
Point Process ◽  
Point Processes ◽  
Recurrent Point ◽  
Image Position ◽  
Mutually Independent

Suppose that we observe an infinite realisation of a point process Π, which is a superposition of a number of mutually independent recurrent point processes Π i , such that

Two inverse problems concerning the superposition of recurrent point processes

1965 ◽  
Vol 2 (2) ◽  
pp. 449-454 ◽  
Author(s):  
R. V. Ambartzumian
Keyword(s):  
Inverse Problems ◽  
Point Process ◽  
Point Processes ◽  
Recurrent Point ◽  
Image Position ◽  
Mutually Independent

Suppose that we observe an infinite realisation of a point process Π, which is a superposition of a number of mutually independent recurrent point processes Πi, such that

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