An order statistic characterization of the poisson renewal process

1985 ◽  
Vol 22 (3) ◽  
pp. 717-722 ◽  
Author(s):  
Uri Liberman
Keyword(s):  
Poisson Process ◽  
Order Statistic ◽  
Renewal Process ◽  
Point Processes

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

An order statistic characterization of the poisson renewal process

1985 ◽  
Vol 22 (03) ◽  
pp. 717-722
Author(s):  
Uri Liberman
Keyword(s):  
Poisson Process ◽  
Order Statistic ◽  
Renewal Process ◽  
Point Processes

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

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.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

A characterization of the Poisson process

1986 ◽  
Vol 23 (01) ◽  
pp. 233-235 ◽  
Author(s):  
Pushpa Lata Gupta ◽  
Ramesh C. Gupta
Keyword(s):  
Poisson Process ◽  
Real Number ◽  
Renewal Process ◽  
Residual Life ◽  
Mild Conditions

Denoting by v(t) the residual life of a component in a renewal process, Çinlar and Jagers (1973) and Holmes (1974) have shown that if E(v(t)) is independent of t for all t, then the process is Poisson. In this note we prove, under mild conditions, that if E(G(v(t))) is constant, then the process is Poisson. In particular if E((v(t))r) for some specific real number r ≧ 1 is independent of t, then the process is Poisson.

A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously

1985 ◽  
Vol 22 (02) ◽  
pp. 314-323
Author(s):  
A. Deffner ◽  
E. Haeusler
Keyword(s):  
Order Statistic ◽  
Real Line ◽  
Time Scale ◽  
Point Processes ◽  
Present Note ◽  
Poisson Processes ◽  
Scale Transformation ◽  
The Real ◽  
Mixed Sample

The results of Nawrotzki (1962), Feigin (1979) and Puri (1982) show that the class of all point processes (on the real line) with the order statistic property consists of all mixed Poisson processes up to a time-scale transformation, and of all mixed sample processes. The present note characterizes those order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously.

scholarly journals Renewal characterization of Markov modulated Poisson processes

1989 ◽  
Vol 2 (1) ◽  
pp. 53-70 ◽  
Author(s):  
Marcel F. Neuts ◽  
Ushio Sumita ◽  
Yoshitaka Takahashi
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Renewal Process ◽  
Jump Process ◽  
Probability Vector ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated ◽  
Bivariate Markov Chain ◽  
Characterization Theorems

A Markov Modulated Poisson Process (MMPP) M(t) defined on a Markov chain J(t) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov chain J(t) is in state i. M(t) is called strongly renewal (SR) if M(t) is a renewal process for an arbitrary initial probability vector of J(t) with full support on P={i:λi>0}. M(t) is called weakly renewal (WR) if there exists an initial probability vector of J(t) such that the resulting MMPP is a renewal process. The purpose of this paper is to develop general characterization theorems for the class SR and some sufficiency theorems for the class WR in terms of the first passage times of the bivariate Markov chain [J(t),M(t)]. Relevance to the lumpability of J(t) is also studied.

A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously

1985 ◽  
Vol 22 (2) ◽  
pp. 314-323 ◽  
Author(s):  
A. Deffner ◽  
E. Haeusler
Keyword(s):  
Order Statistic ◽  
Real Line ◽  
Time Scale ◽  
Point Processes ◽  
Present Note ◽  
Poisson Processes ◽  
Scale Transformation ◽  
The Real ◽  
Mixed Sample

The results of Nawrotzki (1962), Feigin (1979) and Puri (1982) show that the class of all point processes (on the real line) with the order statistic property consists of all mixed Poisson processes up to a time-scale transformation, and of all mixed sample processes. The present note characterizes those order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously.

On the characterization of point processes with the order statistic property

1979 ◽  
Vol 16 (2) ◽  
pp. 297-304 ◽  
Author(s):  
Paul D. Feigin
Keyword(s):  
Order Statistic ◽  
Real Line ◽  
Point Processes ◽  
Martingale Theory ◽  
The Real ◽  
Probabilistic Proof

We provide a probabilistic proof of the characterization of point processes (on the real line) with the order statistic property. The characterization is used to investigate the homogeneity of such processes and is also related to the martingale theory associated with point processes.

A characterization of the Poisson process

1974 ◽  
Vol 11 (01) ◽  
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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