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

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.

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.

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 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.

scholarly journals Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1123
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Transition Probabilities ◽  
Growth Models ◽  
Death Process ◽  
Bell Polynomials ◽  
Bell Polynomial ◽  
Conditional Mean ◽  
Finite State ◽  
Potential Applications ◽  
Intensity Functions ◽  
Birth Death

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

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 Weak convergence of conditioned birth-death processes in discrete time

1997 ◽  
Vol 34 (1) ◽  
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 as n → ∞ of the process conditioned on non-absorption until time n. 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.

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.

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