scholarly journals Limiting Conditional Distributions for Birthdeath Processes

1997 ◽  
Vol 29 (01) ◽  
pp. 185-204
Author(s):  
M. Kijima ◽  
M. G. Nair ◽  
P. K. Pollett ◽  
E. A. Van Doorn
Keyword(s):  
Differential Equations ◽  
Transition Probabilities ◽  
Death Process ◽  
Transition Rates ◽  
Conditional Distributions ◽  
Stationary Distributions ◽  
Birth Death

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

scholarly journals Limiting Conditional Distributions for Birthdeath Processes

1997 ◽  
Vol 29 (1) ◽  
pp. 185-204 ◽  
Author(s):  
M. Kijima ◽  
M. G. Nair ◽  
P. K. Pollett ◽  
E. A. Van Doorn
Keyword(s):  
Differential Equations ◽  
Transition Probabilities ◽  
Death Process ◽  
Transition Rates ◽  
Conditional Distributions ◽  
Stationary Distributions ◽  
Birth Death

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

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.

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.

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.

Extinction times for a general birth, death and catastrophe process

2004 ◽  
Vol 41 (4) ◽  
pp. 1211-1218 ◽  
Author(s):  
Ben Cairns ◽  
P. K. Pollett
Keyword(s):  
Population Size ◽  
Rate Coefficients ◽  
Death Process ◽  
Transition Rates ◽  
Linear Rate ◽  
Expected Time ◽  
Current Population ◽  
Time To Extinction ◽  
Limited Class ◽  
Birth Death

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

scholarly journals On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Symmetry ◽  
2009 ◽  
Vol 1 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Transition Probabilities ◽  
Death Process ◽  
Birth Death
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