scholarly journals Ratio Limits and Limiting Conditional Distributions for Discrete-Time Birth-Death Processes

1995 ◽  
Vol 190 (1) ◽  
pp. 263-284 ◽  
Author(s):  
E.A. Vandoorn ◽  
P. Schrijner
Keyword(s):  
Discrete Time ◽  
Conditional Distributions ◽  
Birth Death

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.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

A birth and death model of neuron firing

1974 ◽  
Vol 11 (02) ◽  
pp. 369-373
Author(s):  
B. F. Logan ◽  
L. A. Shepp
Keyword(s):  
Discrete Time ◽  
Random Variables ◽  
Nerve Cells ◽  
Neuron Firing ◽  
The Times ◽  
Number Of Particles ◽  
Birth Death

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 386-395
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Matrix Equation ◽  
Nonnegative Solution ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

scholarly journals Estimates of the convergence rate in the limit theorems on transition phenomena for branching random processes

2021 ◽  
pp. 15-28
Author(s):  
Ш.Ю. Жураев ◽  
А.Ф. Алиев
Keyword(s):  
Convergence Rate ◽  
Discrete Time ◽  
Limit Theorems ◽  
Random Processes ◽  
Initial Moment ◽  
Conditional Distributions ◽  
Number Of Particles

В данной работе рассматриваются ветвящиеся случайные процессы с дискретным временем в двух предположениях: в начальный момент времени имеется одна частица или в начальный момент времени существует большое число частиц. В переходных явлениях для таких ветвящихся случайных процессов получены оценки скорости сходимости условных законов распределений к предельному распределению. We consider branching random processes with discrete time in two assumptions: at the initial moment of time there is one particle and there are large number of particles. In transition phenomena for such branching random processes, estimates of the convergence rate of conditional distributions are obtained.

scholarly journals On the convergence to stationarity of birth-death processes

2001 ◽  
Vol 38 (03) ◽  
pp. 696-706 ◽  
Author(s):  
Pauline Coolen-Schrijner ◽  
Erik A. Van Doorn
Keyword(s):  
Discrete Time ◽  
Death Process ◽  
Speed Of Convergence ◽  
Recent Proposal ◽  
The Many ◽  
Birth Death

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dtas a measure of the speed of convergence towards stationarity of the process {X(t),t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t),t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

Sign in / Sign up

Export Citation Format

Share Document