Convergence of sums of Markov renewal processes to a branching poisson process

Cybernetics ◽  
1974 ◽  
Vol 8 (1) ◽  
pp. 102-106
Author(s):  
M. A. Yastrebenetskii
Keyword(s):  
Poisson Process ◽  
Renewal Processes ◽  
Markov Renewal Processes

On the superposition of m-dimensional point processes

1968 ◽  
Vol 5 (1) ◽  
pp. 169-176 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Natural Extension ◽  
Renewal Processes ◽  
Independent Vector ◽  
One Dimensional ◽  
Independent Components ◽  
Markov Renewal Processes ◽  
The One ◽  
Vector Valued

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

scholarly journals On convergence of the sums of the Markov renewal processes to a Poisson process

1966 ◽  
Vol 6 (2) ◽  
pp. 271-277
Author(s):  
J. Sapagovas
Keyword(s):  
Poisson Process ◽  
Renewal Processes ◽  
Markov Renewal Processes

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: И. Сапаговас. О сходимости сумм марковских процессов восстановления к процессу Пуассона J. Sapagovas, Apie Markovo atstatymo procesų sumų konvergavimą į Puasono procesą

scholarly journals On the convergence of the sums of the Markov renewal processes to a many-dimensional Poisson process

1969 ◽  
Vol 9 (4) ◽  
pp. 817-826
Author(s):  
J. Sapagovas
Keyword(s):  
Poisson Process ◽  
Renewal Processes ◽  
Markov Renewal Processes

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: И. Сапаговас. О сходимости сумм марковских процессов восстановления к многомерному процессу Пуассона J. Sapagovas. Apie Markovo atstatymo procesų sumų konvergenciją į daugiamatį Puasono procesą

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.

On the superposition of m-dimensional point processes

1968 ◽  
Vol 5 (01) ◽  
pp. 169-176 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Natural Extension ◽  
Renewal Processes ◽  
Independent Vector ◽  
One Dimensional ◽  
Independent Components ◽  
Markov Renewal Processes ◽  
The One ◽  
Vector Valued

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov 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 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.

scholarly journals Limit Theorems for Markov Renewal Processes

1964 ◽  
Vol 35 (4) ◽  
pp. 1746-1764 ◽  
Author(s):  
Ronald Pyke ◽  
Ronald Schaufele
Keyword(s):  
Limit Theorems ◽  
Renewal Processes ◽  
Markov Renewal Processes
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