scholarly journals Construction of algorithms for discrete-time quasi-birth-and-death processes through physical interpretation

2020 ◽  
Vol 36 (2) ◽  
pp. 193-222
Author(s):  
Aviva Samuelson ◽  
Małgorzata M. O’Reilly ◽  
Nigel G. Bean
Keyword(s):  
Discrete Time ◽  
Physical Interpretation ◽  
Birth And Death Processes

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

scholarly journals Poisson’s equation for discrete-time quasi-birth-and-death processes

2013 ◽  
Vol 70 (9) ◽  
pp. 564-577 ◽  
Author(s):  
Sarah Dendievel ◽  
Guy Latouche ◽  
Yuanyuan Liu
Keyword(s):  
Discrete Time ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Birth And Death Processes

Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes

1976 ◽  
Vol 8 (1) ◽  
pp. 58-87 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Markov Process ◽  
Markov Chains ◽  
Discrete Time ◽  
Compact Set ◽  
Two Dimensional ◽  
Birth And Death Processes ◽  
Deterministic Process ◽  
Displacement Vectors ◽  
The Mean

Criteria are established for a discrete-time Markov process {Xn}n≧0 in Rd to have strictly positive, respectively zero, probability of escaping to infinity. These criteria are mainly in terms of the mean displacement vectors μ(y) = E{Xn+1|Xn = y} – y, and are essentially such that they force a deterministic process w.p.1 to move off to infinity, respectively to return to a compact set infinitely often. As an application we determine of most two-dimensional birth and death processes with rates linearly dependent on the population, whether they can escape to infinity or not.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

On a relationship between the inverse of a stationary covariance matrix and the linear interpolator

1990 ◽  
Vol 27 (01) ◽  
pp. 156-170 ◽  
Author(s):  
R. J. Bhansali
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Physical Interpretation ◽  
Stationary Process ◽  
Covariance Matrices ◽  
Interpolation Error ◽  
Linear Least Squares ◽  
Linear Interpolator

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices ofxB(k) = [x′–1,x′–2, · ··,x′–k]′,xF(k) = [x′1,x′2· ··x′k] andx= [·· ·x′–2,x′–1,x′0,x′1,x′2· ··]′ respectively, wherek≧ 1, is finite or infinite. Also letφm,n(j) and δm,n(u) be the coefficients ofxt+jandxt–urespectively in the linear least-squares interpolator ofxtfromxt+ 1, · ··,xt+m;xt− 1, · ··,xt–n, wherem, n≧ 0, 0 ≦j≦m, 0 ≦u≦nare integers,zt(m, n) denote the interpolation error and τ2(m, n) =E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1and Γ–1is given by relating these components to theφm,n(j)δm,n(u) andτ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–land the interpolation parameters when these have been obtained from a realization of lengthTof {xt}. Some of the applications of the results are considered.

On a relationship between the inverse of a stationary covariance matrix and the linear interpolator

1990 ◽  
Vol 27 (1) ◽  
pp. 156-170 ◽  
Author(s):  
R. J. Bhansali
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Physical Interpretation ◽  
Stationary Process ◽  
Covariance Matrices ◽  
Interpolation Error ◽  
Linear Least Squares ◽  
Linear Interpolator

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices of xB(k) = [x′–1, x′–2, · ··, x′–k]′, xF(k) = [x′1, x′2 · ·· x′k] and x = [·· ·x′–2, x′–1, x′0, x′1, x′2 · ··]′ respectively, where k ≧ 1, is finite or infinite. Also let φ m,n(j) and δm,n(u) be the coefficients of xt+ j and xt– u respectively in the linear least-squares interpolator of xt from xt+ 1, · ··, xt+ m; xt− 1, · ··, xt– n, where m, n ≧ 0, 0 ≦ j ≦ m, 0 ≦ u ≦ n are integers, zt(m, n) denote the interpolation error and τ2(m, n) = E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1 and Γ–1 is given by relating these components to the φm,n(j) δm,n(u) and τ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–l and the interpolation parameters when these have been obtained from a realization of length T of {xt}. Some of the applications of the results are considered.

Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

2000 ◽  
Vol 32 (3) ◽  
pp. 844-865 ◽  
Author(s):  
Wilfrid S. Kendall ◽  
Jesper Møller
Keyword(s):  
Point Process ◽  
Discrete Time ◽  
Point Processes ◽  
Random Point ◽  
Perfect Simulation ◽  
Stable Point ◽  
Birth And Death Processes ◽  
The Past ◽  
Coupling From The Past ◽  
Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes

1976 ◽  
Vol 8 (01) ◽  
pp. 58-87 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Markov Process ◽  
Markov Chains ◽  
Discrete Time ◽  
Compact Set ◽  
Two Dimensional ◽  
Birth And Death Processes ◽  
Deterministic Process ◽  
Displacement Vectors ◽  
The Mean

Criteria are established for a discrete-time Markov process {Xn } n≧0 in R d to have strictly positive, respectively zero, probability of escaping to infinity. These criteria are mainly in terms of the mean displacement vectors μ(y) = E{X n+1|Xn = y} – y, and are essentially such that they force a deterministic process w.p.1 to move off to infinity, respectively to return to a compact set infinitely often. As an application we determine of most two-dimensional birth and death processes with rates linearly dependent on the population, whether they can escape to infinity or not.

Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

2000 ◽  
Vol 32 (03) ◽  
pp. 844-865 ◽  
Author(s):  
Wilfrid S. Kendall ◽  
Jesper Møller
Keyword(s):  
Point Process ◽  
Discrete Time ◽  
Point Processes ◽  
Random Point ◽  
Perfect Simulation ◽  
Stable Point ◽  
Birth And Death Processes ◽  
The Past ◽  
Coupling From The Past ◽  
Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

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