Existence of non-trivial quasi-stationary distributions in the birth-death chain

1992 ◽  
Vol 24 (4) ◽  
pp. 795-813 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Servet Martínez ◽  
Pierre Picco
Keyword(s):  
Continued Fractions ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Birth And Death Chain ◽  
Birth Death

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

Existence of non-trivial quasi-stationary distributions in the birth-death chain

1992 ◽  
Vol 24 (04) ◽  
pp. 795-813 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Servet Martínez ◽  
Pierre Picco
Keyword(s):  
Continued Fractions ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Birth And Death Chain ◽  
Birth Death

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

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 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.

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

2003 ◽  
Vol 40 (03) ◽  
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 distributionν. 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 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.

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.

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.

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