Almost sure convergence in Markov branching processes with infinite mean

If {Zn } is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn } and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt } with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

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On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.2307/3213010 ◽

1977 ◽

Vol 14 (2) ◽

pp. 387-390 ◽

Cited By ~ 3

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

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On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.1017/s0021900200105078 ◽

1977 ◽

Vol 14 (02) ◽

pp. 387-390 ◽

Cited By ~ 1

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn } exists such that {Xn/cn } converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn } such that {Xn /c n} converges in law to a proper limit distribution function F, with F(0 +) < 1.

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On the identification of continuous-time Markov chains with a given invariant measure

Journal of Applied Probability ◽

10.1017/s0021900200099435 ◽

1994 ◽

Vol 31 (04) ◽

pp. 897-910

Author(s):

P. K. Pollett

Keyword(s):

Invariant Measure ◽

Continuous Time ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Birth Process ◽

Continuous Time Markov Chains ◽

Branching Process With Immigration ◽

Birth Processes ◽

Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

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On branching processes allowing immigration

Journal of Applied Probability ◽

10.1017/s0021900200094651 ◽

1972 ◽

Vol 9 (01) ◽

pp. 24-31 ◽

Cited By ~ 8

Author(s):

Y. S. Yang

Keyword(s):

Invariant Measure ◽

Discrete Time ◽

Continuous Time ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Positive Recurrence ◽

Discrete Time Case ◽

Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

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On branching processes allowing immigration

Journal of Applied Probability ◽

10.2307/3212633 ◽

1972 ◽

Vol 9 (1) ◽

pp. 24-31 ◽

Cited By ~ 16

Author(s):

Y. S. Yang

Keyword(s):

Invariant Measure ◽

Discrete Time ◽

Continuous Time ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Positive Recurrence ◽

Discrete Time Case ◽

Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

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Almost sure convergence of measure-valued branching processes: a critical exponent

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500028675 ◽

1994 ◽

Vol 124 (4) ◽

pp. 811-823

Author(s):

Alison M. Etheridge

Keyword(s):

Markov Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Upper And Lower Bounds ◽

Spatial Process ◽

Sufficient Condition ◽

Time Process ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Occupation Time Process

A large class of measure-valued critical branching processes can be classified in terms of a parameter ρ which arises as a measure of the recurrence of the underlying spatial Markov process. By establishing upper and lower bounds for the total weighted occupation time process, it is shown that if a measure-valued process is started from an invariant measure of its underlying spatial process, then a necessary and sufficient condition for (a.s.) local extinction is that ρ > 0.

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On the identification of continuous-time Markov chains with a given invariant measure

Journal of Applied Probability ◽

10.2307/3215315 ◽

1994 ◽

Vol 31 (4) ◽

pp. 897-910 ◽

Cited By ~ 3

Author(s):

P. K. Pollett

Keyword(s):

Invariant Measure ◽

Continuous Time ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Birth Process ◽

Continuous Time Markov Chains ◽

Branching Process With Immigration ◽

Birth Processes ◽

Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

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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.

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Binomial subsampling

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1622 ◽

2006 ◽

Vol 462 (2068) ◽

pp. 1181-1195 ◽

Cited By ~ 12

Author(s):

Carsten Wiuf ◽

Michael P.H Stumpf

Keyword(s):

Mathematical Biology ◽

Power Series ◽

Generating Functions ◽

Binomial Distribution ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

The Family ◽

Necessary And Sufficient ◽

Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X , p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X , p ) to mean that given X = x , Z is a draw from the binomial distribution Bi( x , p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

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