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

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

Selective interaction between two independent stationary recurrent point processes

1970 ◽  
Vol 7 (02) ◽  
pp. 476-482 ◽  
Author(s):  
S. K. Srinivasan ◽  
G. Rajamannar
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Short Note ◽  
Secondary Process ◽  
Statistical Features ◽  
Inhibitory Process ◽  
Primary Event ◽  
Recurrent Point ◽  
Selective Interaction ◽  
Interval Distribution

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).

Selective interaction between two independent stationary recurrent point processes

1970 ◽  
Vol 7 (2) ◽  
pp. 476-482 ◽  
Author(s):  
S. K. Srinivasan ◽  
G. Rajamannar
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Short Note ◽  
Secondary Process ◽  
Statistical Features ◽  
Inhibitory Process ◽  
Primary Event ◽  
Recurrent Point ◽  
Selective Interaction ◽  
Interval Distribution

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).

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.

scholarly journals Limits of compound and thinned point processes

1975 ◽  
Vol 12 (2) ◽  
pp. 269-278 ◽  
Author(s):  
Olav Kallenberg
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Random Measure ◽  
Poisson Processes ◽  
Limiting Behaviour ◽  
Mutually Independent

Let η =Σjδτj be a point process on some space S and let β,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σjβjδτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

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(·).

scholarly journals Limits of compound and thinned point processes

1975 ◽  
Vol 12 (02) ◽  
pp. 269-278 ◽  
Author(s):  
Olav Kallenberg
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Random Measure ◽  
Poisson Processes ◽  
Limiting Behaviour ◽  
Mutually Independent

Let η =Σ j δτj be a point process on some space S and let β,β 1,β 2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σ j β j δτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

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.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

scholarly journals Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

2021 ◽  
Vol 58 (2) ◽  
pp. 469-483
Author(s):  
Jesper Møller ◽  
Eliza O’Reilly
Keyword(s):  
Finite Number ◽  
Point Process ◽  
Point Processes ◽  
Parametric Models ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
The Difference ◽  
Weaker Conditions ◽  
Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

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