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.

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.

Examples of Stationary Sequences with Approximate Negative Dependence

2019 ◽  
pp. 305-318
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Spectral Density ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Stationary Processes ◽  
Negative Dependence ◽  
Stationary Sequences ◽  
Determinantal Point Process ◽  
Dependent Random Variables ◽  
Associated Sequences

In this chapter, we treat several examples of stationary processes which are asymptotically negatively dependent and for which the results of Chapter 9 apply. Many systems in nature are complex, consisting of the contributions of several independent components. Our first examples are functions of two independent sequences, one negatively dependent and one interlaced mixing. For instance, the class of asymptotic negatively dependent random variables is used to treat functions of a determinantal point process and a Gaussian process with a positive continuous spectral density. Another example is point processes based on asymptotically negatively or positively associated sequences and displaced according to a Gaussian sequence with a positive continuous spectral density. Other examples include exchangeable processes, the weighted empirical process, and the exchangeable determinantal point process.

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 point process with independent locations

1975 ◽  
Vol 12 (3) ◽  
pp. 435-446 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Independent Random Variables ◽  
General Conditions

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (01) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X 1, X 2, … are non-constant mutually independent random variables with X 1,X 2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X 1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (1) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X1,X2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X1,n,…,Xn,n be the sequence of order statistics of X1,…,Xn. For a sequence (cn)n≥1 of positive constants, the smallest fit off-line counting random variable is defined by Ne(cn) := max {j ≤ n : X1,n + … + Xj,n ≤ cn}. The asymptotic joint distributional comparison is given between the off-line count Ne(cn) and on-line counts Nnτ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑j≥1XτjI(τj≤n) ≤ cn. Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (cn)n≥1, we find sequences of positive constants (Bn)n≥1, (Δn)n≥1 and (Δ'n)n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

A bivariate point process connected with electronic counters

1977 ◽  
Vol 356 (1685) ◽  
pp. 149-160 ◽  
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Dead Time ◽  
Poisson Processes ◽  
Constant Time ◽  
Coincidence Rate ◽  
Time Period

Consider three independent Poisson processes of point events of rates λ 1 , λ 2 and λ 12 . There are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters have constant dead-time, i.e. following the recording of an event on a counter no further event can be recorded on that counter until the appropriate constant time has elapsed. Two ways of estimating λ 12 are via a coincidence rate, i.e. the rate of occurrence of pairs of events separated by less than a suitable small tolerance, and via the covariance of the numbers of events recorded on the two counters in a suitable time period. The theoretical values of these quantities are calculated allowing for dead-time. The techniques used illustrate the study of bivariate point processes.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (1) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

On the variance to mean ratio for random variables from Markov chains and point processes

1998 ◽  
Vol 35 (2) ◽  
pp. 303-312 ◽  
Author(s):  
Timothy C. Brown ◽  
Kais Hamza ◽  
Aihua Xia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Point Process ◽  
Continuous Time ◽  
Infinitesimal Generator ◽  
Point Processes ◽  
Random Variables ◽  
Simple Point ◽  
Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

Thinning of Point Processes—Martingale Method

1990 ◽  
Vol 4 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Shengwu He ◽  
Jiagang Wang
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Classical Problem ◽  
Arbitrary Point ◽  
Affirmative Answer ◽  
Poisson Processes ◽  
Jump Processes ◽  
Martingale Method ◽  
Distribution Law ◽  
Nonhomogeneous Poisson Processes

By using the martingale method, we show that thinning of an arbitrary point process produces independent thinned processes if and only if the original point process is (nonhomogeneous) Poisson. Thinning is a classical problem for point processes. It is well-known that independent homogeneous Poisson processes result from constant Bernoulli thinnings of homogeneous Poisson processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. It is easy to see that the converse is also true, i.e., if the thinned processes are independent nonhomogeneous Poisson processes, so are the original processes. But if we only suppose the thinned processes are independent, nothing is concerned with their distribution law, the problem of whether or not the original processes are (nonhomogeneous) Poisson becomes interesting and challenging, which is the objective of this paper. It is considerably surprising for us to arrive at the affirmative answer. So far as this problem is concerned, the most work was done under the renewal assumption. For example, Bremaud [1] showed that for arbitrary delayed renewal processes, the existence of a pair of uncorrelated thinned processes is sufficient to guarantee that the original process is Poisson. It is natural that the mathematical tools to solve the problem in this case be typical ones for renewal theory, such as renewal equations and Laplace-Stieltjes transformations. Obviously, they are not available for nonrenewal processes. We find out that the martingale method is the most efficient one to solve this problem in the general case. More precisely, we use mainly the dual predictable projections of point processes. In fact, the distribution law of a point process is determined uniquely by its dual predictable projection (see [5]), and Ma [6] offered us a very useful criterion of independence of jump processes having no common jump time through their dual predictable projections. Based on these results, it is not a long way to arrive at the destination.

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