Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

1985 ◽  
Vol 22 (04) ◽  
pp. 828-835 ◽  
Author(s):  
J. Chandramohan ◽  
Lung-Kuang Liang
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Renewal Process ◽  
Point Processes ◽  
Renewal Processes ◽  
Single Pair ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Non Homogeneous Poisson Process ◽  
Dependent Point

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

1985 ◽  
Vol 22 (4) ◽  
pp. 828-835 ◽  
Author(s):  
J. Chandramohan ◽  
Lung-Kuang Liang
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Renewal Process ◽  
Point Processes ◽  
Renewal Processes ◽  
Single Pair ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Non Homogeneous Poisson Process ◽  
Dependent Point

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (2) ◽  
pp. 455-470 ◽  
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (02) ◽  
pp. 455-470
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Thinning of point processes—covariance analyses

1985 ◽  
Vol 17 (01) ◽  
pp. 127-146
Author(s):  
Jagadeesh Chandramohan ◽  
Robert D. Foley ◽  
Ralph L. Disney
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Arbitrary Point ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Zero Correlation

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

scholarly journals Thinning of point processes—covariance analyses

1985 ◽  
Vol 17 (1) ◽  
pp. 127-146 ◽  
Author(s):  
Jagadeesh Chandramohan ◽  
Robert D. Foley ◽  
Ralph L. Disney
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Arbitrary Point ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Zero Correlation

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

scholarly journals Testing and preventive maintenance scheduling optimization for aging systems modeled by generalized renewal process

2009 ◽  
Vol 29 (3) ◽  
pp. 563-576 ◽  
Author(s):  
Vinícius Correa Damaso ◽  
Pauli Adriano de Almada Garcia
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Point Processes ◽  
Water System ◽  
Actual State ◽  
Feed Water ◽  
Repairable Systems ◽  
Homogeneous Poisson Process ◽  
Availability Analysis ◽  
Non Homogeneous Poisson Process

The use of stochastic point processes to model the reliability of repairable systems has been a regular approach to establish survival measures in failure versus repair scenarios. However, the traditional processes do not consider the actual state in which an item returns to operational condition. The traditional renewal process considers an "as-good-as-new" philosophy, while a non-homogeneous Poisson process is based on the minimal repair concept. In this work, an approach based on the concept of Generalized Renewal Process (GRP) is presented, which is a generalization of the renewal process and the non-homogeneous Poisson process. A stochastic modeling is presented for systems availability analysis, including testing and/or preventive maintenances scheduling. To validate the proposed approach, it was performed a case study of a hypothetical auxiliary feed-water system of a nuclear power plant, using genetic algorithm as optimization tool.

On doubly stochastic Poisson processes

1964 ◽  
Vol 60 (4) ◽  
pp. 923-930 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Poisson Process ◽  
Stationary Point ◽  
Point Processes ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Stationary Point Process ◽  
Special Case ◽  
Stationary Point Processes ◽  
Made In

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

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