Weak convergence of conditioned birth-death processes in discrete time

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.

Download Full-text

Stochastic monotonicity of birth–death processes

Advances in Applied Probability ◽

10.1017/s0001867800033383 ◽

1980 ◽

Vol 12 (01) ◽

pp. 59-80 ◽

Cited By ~ 3

Author(s):

Erik A. Van Doorn

Keyword(s):

Initial Distribution ◽

Unit Vector ◽

Death Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Stochastic Monotonicity ◽

Initial State ◽

Distribution Vector ◽

Necessary And Sufficient ◽

Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t 1, t 2) if Pr {x(t) > i} is increasing (decreasing) with t on (t 1, t 2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

Download Full-text

Stochastic monotonicity of birth–death processes

Advances in Applied Probability ◽

10.2307/1426494 ◽

1980 ◽

Vol 12 (1) ◽

pp. 59-80 ◽

Cited By ~ 12

Author(s):

Erik A. Van Doorn

Keyword(s):

Initial Distribution ◽

Unit Vector ◽

Death Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Stochastic Monotonicity ◽

Initial State ◽

Distribution Vector ◽

Necessary And Sufficient ◽

Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

Download Full-text

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

Advances in Applied Probability ◽

10.1017/s0001867800025180 ◽

1993 ◽

Vol 25 (01) ◽

pp. 82-102

Author(s):

M. G. Nair ◽

P. K. Pollett

Keyword(s):

Invariant Measure ◽

Stationary Distribution ◽

Continuous Time ◽

Sufficient Condition ◽

Transition Rates ◽

Necessary And Sufficient Condition ◽

Absorbing State ◽

Necessary And Sufficient ◽

Forward Equations ◽

Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

Download Full-text

On weak convergence within the hnbue family of life distributions

Journal of Applied Probability ◽

10.1017/s0021900200028850 ◽

1984 ◽

Vol 21 (03) ◽

pp. 654-660 ◽

Cited By ~ 4

Author(s):

Sujit K. Basu ◽

Manish C. Bhattacharjee

Keyword(s):

Weak Convergence ◽

Exponential Distribution ◽

Limiting Distribution ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Moment Sequence ◽

Positive Order ◽

Life Distributions ◽

Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

Download Full-text

Extinction Probability in A Birth-Death Process with Killing

Journal of Applied Probability ◽

10.1017/s0021900200000152 ◽

2005 ◽

Vol 42 (01) ◽

pp. 185-198 ◽

Cited By ~ 4

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.

Download Full-text

Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

Journal of Function Spaces ◽

10.1155/2018/8430975 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Jinshu Chen

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Discrete Time ◽

Existence And Uniqueness ◽

Continuous Dependence ◽

Functional Space ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Convergence Of Operators

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

Download Full-text

Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

Abstract and Applied Analysis ◽

10.1155/2013/975461 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 2

Author(s):

Hong Shi ◽

Guangming Xie ◽

Wenguang Luo

Keyword(s):

Linear Systems ◽

Discrete Time ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Control Delay ◽

Discrete Time Systems ◽

Necessary And Sufficient ◽

And Control ◽

Time Systems ◽

Time Linear

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

Download Full-text

Extinction Probability in A Birth-Death Process with Killing

Journal of Applied Probability ◽

10.1239/jap/1110381380 ◽

2005 ◽

Vol 42 (1) ◽

pp. 185-198 ◽

Cited By ~ 19

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.

Download Full-text

Finite-Time Stability Analysis of Discrete-Time Linear Singular Systems

Mathematical Problems in Engineering ◽

10.1155/2014/579863 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Songlin Wo ◽

Xiaoxin Han

Keyword(s):

Discrete Time ◽

Finite Time ◽

Singular Systems ◽

Linear Matrix ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Time Stability ◽

Finite Time Stability ◽

Necessary And Sufficient ◽

Time Linear

The finite-time stability (FTS) problem of discrete-time linear singular systems (DTLSS) is considered in this paper. A necessary and sufficient condition for FTS is obtained, which can be expressed in terms of matrix inequalities. Then, another form of the necessary and sufficient condition for FTS is also given by using matrix-null space technology. In order to solve the stability problem expediently, a sufficient condition for FTS is given via linear matrix inequality (LMI) approach; this condition can be expressed in terms of LMIs. Finally, an illustrating example is also given to show the effectiveness of the proposed method.

Download Full-text