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## Poisson Process Approximation: From Palm Theory to Stein's Method

### Citations

367 |
Poisson processes
- Kingman
- 1993
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Citation Context ... central role in modeling the data on occurrence of rare events at random positions in time or space and is a building block for many other models such as Cox processes, marked Poisson processes (see =-=[24]-=-), compound Poisson processes and Lévy processes. To adapt the above idea of Poisson random variable approximation to Poisson process approximation, we need a probabilistic interpretation of Stein’s m... |

284 |
Poisson Approximation
- Barbour, Holst, et al.
- 1992
(Show Context)
Citation Context ...n’s method is. Furthermore, it is straightforward to use Stein’s method to study the quality of Poisson approximation to the sum of dependent random variables which has many applications (see [18] or =-=[8]-=- for more information). 2. Poisson process approximation Poisson process plays the central role in modeling the data on occurrence of rare events at random positions in time or space and is a building... |

233 |
Random Measures.
- Kallenberg
- 1976
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Citation Context ...o bounded by 1 but generates a weaker topology. We use δx to denote the point mass at x, let X : = { �k i=1 δαi : α1, . . .,αk ∈ Γ, k ≥ 1}, B(X) be the Borel σ–algebra generated by the weak topology (=-=[23]-=-, pp. 168–170): a sequence {ξn} ⊂ X converges weakly to ξ ∈ X if � Γ f(x)ξn(dx) → � Γ f(x)ξ(dx) as n → ∞ for all bounded continuous functions f on Γ. Such topology can also be generated by the metric ... |

133 |
Poisson approximation for dependent trials
- Chen
- 1975
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Citation Context ...process approximation for locally dependent point processes and for dependent superposition of point processes. 1. Poisson approximation Stein’s method for Poisson approximation was developed by Chen =-=[13]-=- which is based on the following observation: a nonnegative integer valued random variable W follows Poisson distribution with mean λ, denoted as Po(λ), if and only if IE{λf(W + 1) − Wf(W)} = 0 for al... |

125 |
Point Processes.
- Cox, Isham
- 1980
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Citation Context ... The configurations of Ξ are then obtained by deleting any point which is within distance r of another point, irrespective of whether the latter point has itself already been deleted [see Cox & Isham =-=[17]-=-, p. 170]. The point process is locally dependent with neighborhoods {B(α, 2r): α ∈ Γ}, where B(α, s) is the ball centered at α with radius s. Let λ be the intensity measure of Ξ, d0(α, β) = min{|α − ... |

43 |
On the rate of Poisson convergence.
- Barbour, Hall
- 1984
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Citation Context ... − fA(i). Further analysis shows that |∆fA(w)| ≤ 1−e−λ λ (see [6] for an analytical proof and [26] for a probabilistic proof). Therefore dTV (L(W), Po(λ)) ≤ � 1 ∧ 1 � �n p λ i=1 2 i. Barbour and Hall =-=[7]-=- proved that the lower bound of dTV (L(W), Po(λ)) above is of the same order as the upper bound. Thus this simple example of Poisson approximation demonstrates how powerful and effective Stein’s metho... |

42 |
Stein’s method and Poisson process convergence.
- Barbour
- 1988
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Citation Context ... processes. To adapt the above idea of Poisson random variable approximation to Poisson process approximation, we need a probabilistic interpretation of Stein’s method which was introduced by Barbour =-=[4]-=-. The idea is to split f by defining f(w) = g(w) −g(w −1) and rewrite the Stein equation (1) as (2) Ag(w): = λ[g(w + 1) − g(w)] + w[g(w − 1) − g(w)] = 1A(w) − Po(λ)(A), where A is the generator of an ... |

33 |
Stein’s method and point process approximation.
- Barbour, Brown
- 1992
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Citation Context ...imensions and process settings is, instead of considering Z+-valued immigrationdeath process, we now need an immigration-death process defined on X. More precisely, by adapting (2), Barbour and Brown =-=[5]-=- define the Stein equation as (4) � � Ag(ξ) : = [g(ξ + δx) − g(ξ)]λ(dx) + [g(ξ − δx) − g(ξ)]ξ(dx) Γ = h(ξ) − Po(λ)(h), Γ where Po(λ)(h) = IEh(ζ) with ζ ∼ Po(λ). The operator A is the generator of an X... |

21 |
Stein’s method, Palm theory and Poisson process approximation.
- Chen, Xia
- 2004
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Citation Context ...point of Palm theory. The connection between Stein’s method and Palm theory has been known to many others (e.g., T. C. Brown (personnel communication), [9]) and the exposition here is mainly based on =-=[14]-=- and [27]. There are two properties which distinguish a Poisson process from other processes: independent increments and the number of points on any bounded set follows Poisson distribution. Hence, a ... |

18 |
Poisson approximation for some statistics based on exchangeable trials.
- Barbour, Eagleson
- 1983
(Show Context)
Citation Context ...IE{λfA(W + 1) − WfA(W)} = = n� piIEf(Wi + 1), i=1 n� piIE[fA(W + 1) − fA(Wi + 1)] i=1 n� i=1 p 2 i IE∆fA(Wi + 1), where ∆fA(i) = fA(i + 1) − fA(i). Further analysis shows that |∆fA(w)| ≤ 1−e−λ λ (see =-=[6]-=- for an analytical proof and [26] for a probabilistic proof). Therefore dTV (L(W), Po(λ)) ≤ � 1 ∧ 1 � �n p λ i=1 2 i. Barbour and Hall [7] proved that the lower bound of dTV (L(W), Po(λ)) above is of ... |

14 |
Stochastic point processes: limit theorems.
- Goldman
- 1967
(Show Context)
Citation Context ...e point processes on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes to a Poisson process under various conditions (see, e.g., =-=[16, 19, 21]-=- and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher [25] considered the Wasserstein distance between the weakly depend... |

13 |
On the convergence of sums of random step processes to a Poisson process.
- Grigelionis
- 1963
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Citation Context ...Poisson process approximation of Ξ = � n i=1 IiδUi is a special case of the following section. 5.3. Locally dependent superposition of point processes Since the publication of the Grigelionis Theorem =-=[20]-=- which states that the superposition of independent sparse point processes on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes t... |

11 | Limits of compound and thinned point processes.
- Kallenberg
- 1975
(Show Context)
Citation Context ...ergence of point processes to a Poisson process under various conditions (see, e.g., [16, 19, 21] and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in =-=[1, 2, 3, 11, 22]-=-. Schuhmacher [25] considered the Wasserstein distance between the weakly dependent superposition of sparse point processes and a Poisson process. Let Γ be a compact metric space, {Ξi: i ∈ I} be a col... |

10 | Compound Poisson process approximation
- Barbour, Mansson
- 2002
(Show Context)
Citation Context ...ake another approach to Stein’s method from the point of Palm theory. The connection between Stein’s method and Palm theory has been known to many others (e.g., T. C. Brown (personnel communication), =-=[9]-=-) and the exposition here is mainly based on [14] and [27]. There are two properties which distinguish a Poisson process from other processes: independent increments and the number of points on any bo... |

10 |
A martingale approach to the Poisson convergence of simple point processes.
- Brown
- 1978
(Show Context)
Citation Context ... on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes to a Poisson process under various conditions (see, e.g., [16, 19, 21] and =-=[10]-=-). Extensions to dependent superposition 1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher [25] considered the Wasserstein distance between the weakly dependent super... |

10 |
Stein’s method and Poisson process approximation. In: An Introduction to Stein’s Method, Eds.
- Xia
- 2005
(Show Context)
Citation Context ...ence {ξn} ⊂ X converges weakly to ξ ∈ X if � Γ f(x)ξn(dx) → � Γ f(x)ξ(dx) as n → ∞ for all bounded continuous functions f on Γ. Such topology can also be generated by the metric d1 defined below (see =-=[27]-=-, Proposition 4.2). A point process on Γ is defined as a measurable mapping from a probability space (Ω, F, IP) to (X, B(X)) (see [23], p. 13). We use Ξ to stand for a point process on Γ with finite i... |

9 |
Position dependent and stochastic thinning of point processes.
- Brown
- 1979
(Show Context)
Citation Context ...ergence of point processes to a Poisson process under various conditions (see, e.g., [16, 19, 21] and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in =-=[1, 2, 3, 11, 22]-=-. Schuhmacher [25] considered the Wasserstein distance between the weakly dependent superposition of sparse point processes and a Poisson process. Let Γ be a compact metric space, {Ξi: i ∈ I} be a col... |

9 |
On metrics in point process approximation.
- Brown, Xia
- 1995
(Show Context)
Citation Context ...,α ∈ M) = IE[Ξi(dα)1Vi∈M] for all M ∈ B(X) IEΞi(dα) and d ′ n� 1(ξ1, ξ2) = min d0(yi, zπ(i)) + (m − n) π: permutations of {1,...,m} i=1 for ξ1 = �n i=1 δyi and ξ2 = �m i=1 δzi with m ≥ n [Brown & Xia =-=[12]-=-]. Corollary 5.2 ([14]). For Ξ = � i∈I IiδUi and λ = � where Vi = � d2(L(Ξ), Po(λ)) ≤ IE � j�∈Ai Ij. � i∈I j∈Ai\{i} + � � i∈I j∈Ai � 3.5 λ i∈I pi defined in section 5.2, � 2.5 + IiIj Vi + 1 � � 2.5 � ... |

7 |
Superposition of point processes. Stochastic point processes: statistical analysis, theory, and applications (Conf., IBM Res.
- Cinlar
- 1972
(Show Context)
Citation Context ...e point processes on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes to a Poisson process under various conditions (see, e.g., =-=[16, 19, 21]-=- and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher [25] considered the Wasserstein distance between the weakly depend... |

7 |
On the weak convergence of superpositions of point processes.
- Jagers
- 1972
(Show Context)
Citation Context ...e point processes on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes to a Poisson process under various conditions (see, e.g., =-=[16, 19, 21]-=- and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher [25] considered the Wasserstein distance between the weakly depend... |

6 |
On superpositions of random measures and point processes. Mathematical statistics and probability theory
- Banys
- 1980
(Show Context)
Citation Context ...ergence of point processes to a Poisson process under various conditions (see, e.g., [16, 19, 21] and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in =-=[1, 2, 3, 11, 22]-=-. Schuhmacher [25] considered the Wasserstein distance between the weakly dependent superposition of sparse point processes and a Poisson process. Let Γ be a compact metric space, {Ξi: i ∈ I} be a col... |

6 | Distance estimates for dependent superpositions of point processes.
- Schuhmacher
- 2005
(Show Context)
Citation Context ... Poisson process under various conditions (see, e.g., [16, 19, 21] and [10]). Extensions to dependent superposition 1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher =-=[25]-=- considered the Wasserstein distance between the weakly dependent superposition of sparse point processes and a Poisson process. Let Γ be a compact metric space, {Ξi: i ∈ I} be a collection of point p... |

5 | A probabilistic proof of Stein’s factors.
- Xia
- 1999
(Show Context)
Citation Context ...iIEf(Wi + 1), i=1 n� piIE[fA(W + 1) − fA(Wi + 1)] i=1 n� i=1 p 2 i IE∆fA(Wi + 1), where ∆fA(i) = fA(i + 1) − fA(i). Further analysis shows that |∆fA(w)| ≤ 1−e−λ λ (see [6] for an analytical proof and =-=[26]-=- for a probabilistic proof). Therefore dTV (L(W), Po(λ)) ≤ � 1 ∧ 1 � �n p λ i=1 2 i. Barbour and Hall [7] proved that the lower bound of dTV (L(W), Po(λ)) above is of the same order as the upper bound... |

3 |
The convergence of sums of dependent point processes to Poisson processes.
- Banis
- 1975
(Show Context)
Citation Context ... a Wasserstein pseudometric on the distributions of point processes on Γ through a pseudo-metric on Γ as shown in the following chart: imsart-lnms ver. 2006/03/07 file: Chen36.tex date: July 31, 2006 4 L. H. Y. Chen & A. Xia Carrier space Γ Configuration space X Space of the distributions of point processes ρ0 −→ ρ1 −→ ρ2 (≤ 1) (≤ 1) (≤ 1) As a simple example, we consider a Bernoulli process defined as Ξ = n∑ i=1 Xiδ i n , where, as before, X1, · · · , Xn are independent Bernoulli random variables with IP(Xi = 1) = 1 − IP(Xi = 0) = pi, 1 ≤ i ≤ n. Then Ξ is a point process on carrier space Γ = [0, 1] with intensity measure λ = ∑n i=1 piδ in . With the metric ρ0(x, y) = |x − y |: = d0(x, y), we denote the induced metric ρ2 by d2. Using the Stein equation (4), we have IEh(Ξ)− Po(λ)(h) = IE {∫ Γ [gh(Ξ + δx)− gh(Ξ)]λ(dx) + ∫ Γ [gh(Ξ− δx)− gh(Ξ)]Ξ(dx) } = n∑ i=1 piIE { [gh(Ξ + δ i n )− gh(Ξ)]− [gh(Ξi + δ i n )− gh(Ξi)] } = n∑ i=1 p2i IE { [gh(Ξi + 2δ i n )− gh(Ξi + δ i n )]− [gh(Ξi + δ i n )− gh(Ξi)] } , where Ξi = Ξ−Xiδ i n . It was shown in [Xia (2005), Proposition 5.21] that sup h∈H,α,β∈Γ |gh(ξ + δα + δβ)− gh(ξ + δα)− gh(ξ + δβ) + gh(ξ) |≤ 3.5 λ + 2.5 |ξ|+ 1 , (6) where, and in the sequel, ... |

3 | A Poisson limit theorem for rare events of a discrete random field. - Banis - 1985 |

2 |
The convergence of sums of dependent point processes to Poisson processes.
- Banys
- 1975
(Show Context)
Citation Context ...ined as n� Ξ = i=1 Xiδ i n , where, as before, X1, . . .,Xn are independent Bernoulli random variables with IP(Xi = 1) = 1 − IP(Xi = 0) = pi, 1 ≤ i ≤ n. Then Ξ is a point process on carrier space Γ = =-=[0, 1]-=- with intensity measure λ = �n i=1 piδ i . With the metric n ρ0(x, y) = |x − y|: = d0(x, y), we denote the induced metric ρ2 by d2. Using the Stein equation (4), we have IEh(Ξ) − Po(λ)(h) �� � = IE [g... |

2 |
A Poisson limit theorem for rare events of a discrete random field.
- Banys
- 1985
(Show Context)
Citation Context |

1 | Poisson process approximation for dependent superposition of point processes - Chen, Xia - 2006 |

1 |
Poisson and compound Poisson approximation. In: An Introduction to Stein’s Method
- Erhardsson
- 2005
(Show Context)
Citation Context ...ive Stein’s method is. Furthermore, it is straightforward to use Stein’s method to study the quality of Poisson approximation to the sum of dependent random variables which has many applications (see =-=[18]-=- or [8] for more information). 2. Poisson process approximation Poisson process plays the central role in modeling the data on occurrence of rare events at random positions in time or space and is a b... |