On the asymptotic behaviour of the extinction time of the simple branching process

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

Advances in Applied Probability ◽

10.1017/s000186780001853x ◽

1989 ◽

Vol 21 (02) ◽

pp. 243-269 ◽

Cited By ~ 6

Author(s):

Anthony G. Pakes

Keyword(s):

Population Size ◽

Large Scale ◽

Branching Process ◽

Branching Processes ◽

Initial Population ◽

Extinction Time ◽

Asymptotic Results ◽

Time To Extinction ◽

Initial Population Size ◽

The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

Advances in Applied Probability ◽

10.2307/1427159 ◽

1989 ◽

Vol 21 (2) ◽

pp. 243-269 ◽

Cited By ~ 13

Author(s):

Anthony G. Pakes

Keyword(s):

Population Size ◽

Large Scale ◽

Branching Process ◽

Branching Processes ◽

Initial Population ◽

Extinction Time ◽

Asymptotic Results ◽

Time To Extinction ◽

Initial Population Size ◽

The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Asymptotic analysis of the general stochastic epidemic with variable infectious periods

Journal of Applied Probability ◽

10.1017/s0021900200018477 ◽

2001 ◽

Vol 38 (01) ◽

pp. 18-35 ◽

Cited By ~ 4

Author(s):

A. N. Startsev

Keyword(s):

Asymptotic Analysis ◽

Asymptotic Behaviour ◽

Size Distribution ◽

Random Variables ◽

Independent Random Variables ◽

Boundary Crossing ◽

Total Size ◽

Stochastic Epidemic ◽

General Stochastic

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

Continuity for multi-type branching processes with varying environments

Journal of Applied Probability ◽

10.1017/s0021900200016910 ◽

1999 ◽

Vol 36 (01) ◽

pp. 139-145 ◽

Cited By ~ 1

Author(s):

Owen Dafydd Jones

Keyword(s):

Linear Combination ◽

Branching Process ◽

Branching Processes ◽

Random Variables ◽

Main Tool ◽

Concentration Function ◽

Independent Random Variables ◽

Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

On the Sequential Detection Problem

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319890301 ◽

1989 ◽

Vol 38 (3-4) ◽

pp. 129-146

Author(s):

Uttam Bandyopadhyay

Keyword(s):

Infinite Sequence ◽

Random Variables ◽

Stopping Rule ◽

Independent Random Variables ◽

Sequential Detection ◽

Detection Problem ◽

Asymptotic Results ◽

Rule Based ◽

Unknown Point ◽

Cumulative Sums

In this paper, for an infinite sequence of independent random variables, we have considered the problem of estimation of an unknown point ( q) where a change in the distribution of the random variables occurs. Attaching suitable scores for the observed values. of the random variables, a stopping rule based on the cumulative sums of these scores has been proposed. Some asymptotic results useful for studying the performance of the proposed procedure havo beon obtained.

Tree-dependent extreme values: the exponential case

Journal of Applied Probability ◽

10.1017/s002190020003847x ◽

1990 ◽

Vol 27 (01) ◽

pp. 124-133 ◽

Cited By ~ 1

Author(s):

Vijay K. Gupta ◽

Oscar J. Mesa ◽

E. Waymire

Keyword(s):

Branching Process ◽

Extreme Values ◽

Rooted Tree ◽

Domain Of Attraction ◽

Extreme Value Distribution ◽

Value Theory ◽

Extreme Value ◽

Extinction Time ◽

Brownian Excursion ◽

Weighted Trees

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

Extreme values of phase-type and mixed random variables with parallel-processing examples

Journal of Applied Probability ◽

10.1017/s002190020001696x ◽

1999 ◽

Vol 36 (01) ◽

pp. 194-210 ◽

Cited By ~ 1

Author(s):

Sungyeol Kang ◽

Richard F. Serfozo

Keyword(s):

Parallel Processing ◽

Asymptotic Distribution ◽

Extreme Values ◽

Random Variables ◽

Extreme Value Distribution ◽

Rates Of Convergence ◽

Value Theory ◽

Extreme Value ◽

Independent Random Variables ◽

Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

Bounds on the extinction time distribution of a branching process

Advances in Applied Probability ◽

10.2307/1426296 ◽

1974 ◽

Vol 6 (2) ◽

pp. 322-335 ◽

Cited By ~ 25

Author(s):

Alan Agresti

Keyword(s):

Generating Function ◽

Generating Functions ◽

Time Distribution ◽

Branching Process ◽

Probability Generating Function ◽

Extinction Time ◽

Expected Time ◽

Time To Extinction ◽

Special Case ◽

Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

Maximal branching processes and ‘long-range percolation’

Journal of Applied Probability ◽

10.1017/s0021900200026966 ◽

1970 ◽

Vol 7 (01) ◽

pp. 89-98

Author(s):

John Lamperti

Keyword(s):

Probability Distribution Function ◽

Branching Process ◽

Branching Processes ◽

Random Variables ◽

Random Variable ◽

Independent Random Variables ◽

Transition Matrices ◽

A Chain ◽

Image Position ◽

The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.