scholarly journals On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 477 ◽  
Author(s):  
Alexander Zeifman ◽  
Yacov Satin ◽  
Ksenia Kiseleva ◽  
Victor Korolev
Keyword(s):  
Markov Process ◽  
Rate Of Convergence ◽  
State Vector ◽  
The State ◽  
Death Process ◽  
General Situation ◽  
Two Dimensional ◽  
One Dimensional ◽  
Whole Process ◽  
Birth Death

We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered.

scholarly journals Approximating Quasistationary Distributions of Birth–Death Processes

2012 ◽  
Vol 49 (4) ◽  
pp. 1036-1051 ◽  
Author(s):  
Damian Clancy
Keyword(s):  
State Space ◽  
Population Biology ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
The State ◽  
Death Process ◽  
Sis Model ◽  
Finite State ◽  
Quasistationary Distributions ◽  
Birth Death

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (01) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

Evaluation of the decay parameter for some specialized birth-death processes

1992 ◽  
Vol 29 (4) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
The State ◽  
Death Process ◽  
Decay Parameter ◽  
Speed Of Convergence ◽  
Tridiagonal Matrices ◽  
Decay Parameters ◽  
Quasi Stationary Distribution ◽  
Birth Death

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn}, where µ0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

Approximating Quasistationary Distributions of Birth–Death Processes

2012 ◽  
Vol 49 (04) ◽  
pp. 1036-1051
Author(s):  
Damian Clancy
Keyword(s):  
State Space ◽  
Population Biology ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
The State ◽  
Death Process ◽  
Sis Model ◽  
Finite State ◽  
Quasistationary Distributions ◽  
Birth Death

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (1) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

scholarly journals Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

2021 ◽  
Vol 53 (1) ◽  
pp. 1-29
Author(s):  
Ari Arapostathis ◽  
Guodong Pang ◽  
Yi Zheng
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Regime Switching ◽  
Convergence Rates ◽  
Death Process ◽  
Ergodic Properties ◽  
State Diffusion ◽  
Markov Modulated ◽  
Steady State Diffusion ◽  
Birth Death

AbstractWe study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

Point-based polygonal models for random graphs

1993 ◽  
Vol 25 (2) ◽  
pp. 348-372 ◽  
Author(s):  
T. Arak ◽  
P. Clifford ◽  
D. Surgailis
Keyword(s):  
Computer Simulations ◽  
Random Graphs ◽  
Particle System ◽  
Two Dimensional ◽  
One Dimensional ◽  
Space And Time ◽  
Markov Random ◽  
Alternative Description ◽  
Polygonal Models ◽  
Birth Death

We define a class of two-dimensional Markov random graphs with I, V, T and Y-shaped nodes (vertices). These are termed polygonal models. The construction extends our earlier work [1]– [5]. Most of the paper is concerned with consistent polygonal models which are both stationary and isotropic and which admit an alternative description in terms of the trajectories in space and time of a one-dimensional particle system with motion, birth, death and branching. Examples of computer simulations based on this description are given.

Evaluation of the decay parameter for some specialized birth-death processes

1992 ◽  
Vol 29 (04) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
The State ◽  
Death Process ◽  
Decay Parameter ◽  
Speed Of Convergence ◽  
Tridiagonal Matrices ◽  
Decay Parameters ◽  
Quasi Stationary Distribution ◽  
Birth Death

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn }, where µ 0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ &lt; μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that lim n→∞ λn = λ and lim n→∞ µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

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