Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (04) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (4) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for 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.

scholarly journals Geomatric ergodicity and quasi-stationarity in discrete-time birth-death processes

1995 ◽  
Vol 37 (2) ◽  
pp. 121-144 ◽  
Author(s):  
Erik A. van Doorn ◽  
Pauline Schrijner
Keyword(s):  
Discrete Time ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Death Process ◽  
Geometric Ergodicity ◽  
Decay Parameter ◽  
Stationary Distributions ◽  
The Common ◽  
Step Transition ◽  
Birth Death

AbstractWe study two aspects of discrete-time birth-death processes, the common feature of which is the central role played by the decay parameter of the process. First, conditions for geometric ergodicity and bounds for the decay parameter are obtained. Then the existence and structure of quasi-stationary distributions are discussed. The analyses are based on the spectral representation for the n-step transition probabilities of a birth-death process developed by Karlin and McGregor.

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.

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.

Domain of Attraction of the Quasistationary Distribution for Birth-and-Death Processes

2013 ◽  
Vol 50 (01) ◽  
pp. 114-126 ◽  
Author(s):  
Hanjun Zhang ◽  
Yixia Zhu
Keyword(s):  
Probability Measure ◽  
Domain Of Attraction ◽  
Death Process ◽  
Extinction Time ◽  
Decay Parameter ◽  
Birth And Death Processes ◽  
Absorbing State ◽  
Positive Integers ◽  
Initial Distributions ◽  
Birth Death

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λ n ,n≥0, and death coefficients μ n ,n≥0. If we define A=∑ n=1 ∞ 1/λ n π n and S=∑ n=1 ∞ (1/λ n π n )∑ i=n+1 ∞ π i , where {π n ,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λ C >0 and there is precisely one quasistationary distribution, namely, {a j (λ C )}, where λ C is the decay parameter of {X(t),t≥0} in C={1,2,...} and a j (x)≡μ1 -1π j xQ j (x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={m i , i=1,2,...}, we have lim t→∞ℙ M (X(t)=j∣ T>t)= a j (λ C ), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

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 Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (1) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (01) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

scholarly journals Weak convergence of conditioned birth-death processes in discrete time

1997 ◽  
Vol 34 (01) ◽  
pp. 46-53
Author(s):  
Pauline Schrijner ◽  
Erik A. Van Doorn
Keyword(s):  
Weak Convergence ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Limiting Behaviour ◽  
Necessary And Sufficient ◽  
Birth Death

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

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