Dynamical systems defined on point processes

1998 ◽  
Vol 35 (3) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

Dynamical systems defined on point processes

1998 ◽  
Vol 35 (03) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

A note on the subcritical generalized age-dependent branching process

1976 ◽  
Vol 13 (4) ◽  
pp. 798-803 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching 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.

scholarly journals Characterization of rifampicin-resistant Mycobacterium tuberculosis in Taiwan

2003 ◽  
Vol 52 (3) ◽  
pp. 239-245 ◽  
Author(s):  
Hsing-Yu Hwang ◽  
Chung-Yu Chang ◽  
Lin-Li Chang ◽  
Shui-Feng Chang ◽  
Ya-Hui Chang ◽  
...  
Keyword(s):  
Mycobacterium Tuberculosis ◽  
Core Region ◽  
Banding Patterns ◽  
The Core ◽  
Resistant Tuberculosis ◽  
Proportion Method ◽  
The Mean ◽  
Novel Alleles ◽  
The Relationship

Sixty-three rifampicin-resistant (Rifr) isolates of Mycobacterium tuberculosis from Kaohsiung, Taiwan, were analysed for mutations in the core region (69 bp, codons 511–533) of the rpoB gene. Some 84.1 % (53/63) of the resistant isolates showed mutations in this region, especially in codons 531 (41.5 %), 526 (18.9 %), 516 (15.1 %) and 533 (7.5 %). Five novel alleles of a total of 16 different types of mutations were identified in Rifr isolates. Ten Rifr isolates (15.9 %) exhibited no mutations in the core region of rpoB. Also, they did not show mutations in another 365 bp fragment (codons 99–220) of rpoB. The agar proportion method was used to determine the relationship between the degree of rifampicin resistance and alterations in the core region of rpoB. The results revealed that the mean MIC was 92.38 μg ml−1 for the 53 isolates with a mutation in the core region, whereas the mean MIC of the other 10 isolates without mutations was only 24.8 μg ml−1. This indicates that the isolates with mutations in the core region had higher levels of resistance than those without mutations in this region. IS6110 restriction fragment length polymorphism (RFLP) was used for typing of 55 Rifr M. tuberculosis isolates. Isolates contained two to 19 copies of IS6110, with sizes ranging from 600 to 16 000 bp. The majority (85 %) contained six to 16 copies. No strains lacking IS6110 were found. A total of 54 of 55 RFLP types were defined at the 90 % similarity level. The observation of varied IS6110-associated banding patterns indicates that an outbreak of drug-resistant tuberculosis did not occur in this area.

Characterization of stochastic equilibrium controls by the Malliavin calculus

2021 ◽  
pp. 2150054
Author(s):  
Jiang Yu Nguwi ◽  
Nicolas Privault
Keyword(s):  
Malliavin Calculus ◽  
Linear Quadratic ◽  
Merton Problem ◽  
Classical Duality ◽  
The Mean ◽  
Time Inconsistent ◽  
Necessary And Sufficient ◽  
Mean Variance ◽  
Quadratic Case

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem.

Poisson limits and nonparametric estimation for pairwise interaction point processes

2000 ◽  
Vol 37 (1) ◽  
pp. 252-260 ◽  
Author(s):  
Wei-Bin Chang ◽  
John A. Gubner
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interpoint Distance ◽  
Interaction Point ◽  
Interaction Function ◽  
The Mean ◽  
Piecewise Continuous ◽  
Independent Identically Distributed

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Asymptotic independence of counts in isotropic planar point processes of phase-type

2000 ◽  
Vol 32 (2) ◽  
pp. 363-375
Author(s):  
Marie-Ange Remiche
Keyword(s):  
Poisson Process ◽  
Arbitrary Shape ◽  
Point Processes ◽  
The Other ◽  
Asymptotic Independence ◽  
Planar Point ◽  
Phase Type ◽  
Disjoint Sets ◽  
The Mean ◽  
The One

The isotropic planar point processes of phase-type are natural generalizations of the Poisson process on the plane. On the one hand, those processes are isotropic and stationary for the mean count, as in the case of the Poisson process. On the other hand, they exhibit dependence of counts in disjoint sets. In a recent paper, we have proved that the number of points in a square window has a Poisson distribution asymptotically as the window is located far away from the origin of the process. We extend our work to the case of a window of arbitrary shape.

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