scholarly journals Quasi-stationary distribution for the birth–death process with exit boundary

2015 ◽  
Vol 427 (1) ◽  
pp. 114-125 ◽  
Author(s):  
Wu-Jun Gao ◽  
Yong-Hua Mao
Keyword(s):  
Stationary Distribution ◽  
Death Process ◽  
Exit Boundary ◽  
Quasi Stationary Distribution ◽  
Birth Death

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.

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.

scholarly journals Quasi-stationary distributions for birth-death processes with killing

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Pauline Coolen-Schrijner ◽  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Stationary Distribution ◽  
Transition Probabilities ◽  
The State ◽  
Death Process ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Quasi Stationary Distribution ◽  
Birth Death

The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state (killing) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains largely intact as long as killing is possible from only finitely many states. In particular, the existence of a quasi-stationary distribution is ensured in this case if absorption is certain and the state probabilities tend to zero exponentially fast.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

scholarly journals Stochastic models for interference between searching insect parasites

1989 ◽  
Vol 30 (3) ◽  
pp. 268-277 ◽  
Author(s):  
Phil Diamond
Keyword(s):  
Stationary Distribution ◽  
Intraspecific Competition ◽  
Death Process ◽  
Confluent Hypergeometric Functions ◽  
Large Populations ◽  
The Mean ◽  
The One ◽  
Globally Stable ◽  
Insect Parasites ◽  
Birth Death

AbstractCompetition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

1995 ◽  
Vol 32 (01) ◽  
pp. 25-38
Author(s):  
Servet Martínez ◽  
Maria Eulália Vares
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Conditional Distribution ◽  
Transition Probabilities ◽  
The Matrix ◽  
Quasi Stationary Distribution ◽  
Birth Death

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018 ◽  
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic

2003 ◽  
Vol 40 (3) ◽  
pp. 821-825 ◽  
Author(s):  
Damian Clancy ◽  
Philip K. Pollett
Keyword(s):  
Probability Distributions ◽  
Explicit Representation ◽  
Death Process ◽  
Conditional Distributions ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Likelihood Ratio Ordering ◽  
Positive Integers ◽  
Quasi Stationary Distribution ◽  
Birth Death

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distribution ν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

Quasi-stationary distributions and one-dimensional circuit-switched networks

1987 ◽  
Vol 24 (04) ◽  
pp. 965-977 ◽  
Author(s):  
Ilze Ziedins
Keyword(s):  
Communication Networks ◽  
Stationary Distribution ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Switched Networks ◽  
One Dimensional ◽  
Special Cases ◽  
Quasi Stationary Distribution

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

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