General Branching Processes Research Articles

General branching processes as Markov fields

The natural Markov structure for population growth is that of genetics: newborns inherit types from their mothers, and given those they are independent of the history of their earlier ancestry. This leads to Markov fields on the space of sets of individuals, partially ordered by descent. The structure of such fields is investigated. It is proved that this Markov property implies branching, i.e. the conditional independence of disjoint daughter populations. The process also has the strong Markov property at certain optional sets of individuals. An intrinsic martingale (indexed by sets of individuals) is exhibited, that catches the stochastic element of population development. The deterministic part is analyzed by Markov renewal methods. Finally the strong Markov property found is used to divide the population into conditionally independent subpopulations. On those classical limit theory for sums of independent random variables can be used to catch the asymptotic population development, as real time passes.

Some limit theorems for continuous-state branching processes

AbstractThe most general continuous time and state branching (C.B.) process (Xt) can be constructed as a certain random time transformation of a spectrally positive Levy process. When the generating process is compound Poisson with a superimposed negative linear drift and the C.B. process is not supercritical, then there is a random timeTsuch thatXt+T= e-ctXTwherec> 0 is the drift parameter. ThusTis the last epoch of random variation.

Properties of the nonextinction probability of general critical branching processes under weak constrainis

Properties of the nonextinction probability of general critical branching processes under weak constrainis

Average efficiency of data structures for binary image processing

This letter adresses the problem of computing the average values associated to geometric decompositions of binary quadtrees, under a faisly general branching process probabilistic model adapted to these types of trees.

Expected population size in the generation-dependent branching process

"A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered."

Expected population size in the generation-dependent branching process

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.

A stochastic population projection system based on general age-dependent branching processes

"Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections...are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations."

A stochastic population projection system based on general age-dependent branching processes

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

Asymptotic Analysis of a Cell Cycle Model Based on Unequal Division

We provide an analysis of the asymptotic behavior of a novel cell cycle model offering a uniform description of the processes of RNA production and division and explaining the cell generation time variability, at least in certain cell lines. We prove that the distribution $m( t,x )$ of the RNA level $( x )$ in the dividing cells at a given time $( t )$ tends to $\exp ( \lambda^* t )\mu^* ( x )$, as t tends to infinity, i.e., that there exists the so-called exponential steady state for our model, and that perturbed cell populations reach this state in the limit.The result is obtained by applying the semigroup theory to a functional integral equation describing the evolution of $m( t,x )$ in time. For a variety of reasons, the analysis requires explicit characterization of the semigroup spectrum. Our asymptotic result may be viewed as a generalization of similar results for the generalized branching processes with continuous time, under less restrictive assumptions. Also, we discuss related models and asymp...

Fokker-Planck stochastic Koba-Nielsen-Olesen solutions, branching processes, and path to hadronization

We have analyzed a large class of Markov processes for the properties of multiplicity distribution at finite n\ifmmode\bar\else\textasciimacron\fi{} and for their Koba-Nielsen-Olesen (KNO) scaling and scaling violation by using a Fokker-Planck stochastic differential equation of the KNO-scaling functions, QCD branching processes, and/or simple supercluster models. Through this analysis, the distinction between the simple Fokker-Planck formulation of the KNO function and the general branching processes is seen. Our solutions allow in general an arbitrary initial condition and demonstrate the richness and flexibility of many possible paths of hadronization. They can be used for the study of distribution of the preexisting gluons and quarks.

Projection of mean and variance functions for population processes with time-homogeneous laws of evolution.

This paper provides algorithms for projection of mean and covariance functions for stochastic population processes governed by time-homogeneous laws of fertility and mortality. The theoretical foundation of the algorithms is general age-dependent branching processes in discrete time. The algorithms are employed in several illustrative projections, based on 1982 Chinese data, of a population experiencing an abrupt transition to below replacement fertility. Methods of constructing confidence limits for total population size are illustrated. Also developed are procedures for projecting mean and variance functions for populations which may be heterogeneous with respect to mortality or fertility. The projections performed yield two observations. First, the coefficient of variation in population size appears to be inversely related to the Malthusian parameter of population growth. Second, the coefficient of variation for population size is negligible for large homogeneous initial populations. But when the initial population is heterogeneous with respect to fertility or mortality, then substantial coefficients of variation, exceeding 0.4, are observed in some of the projections performed.

Limiting behavior of general branching processes

Limiting behavior of general branching processes

Branching Processes in Periodically Varying Environment

In order to model the diurnal variation of certain cell populations we study general one-type branching processes whose individuals reproduce in a manner that is influenced by time in a periodic fashion. A Malthusian growth rate is established as well as a periodically varying asymptotic composition, over ages e.g.

The stable pedigrees of critical branching populations

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

The stable pedigrees of critical branching populations

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

Asymptotics of the Extinction Probability for Critical General Branching Processes

Asymptotics of the Extinction Probability for Critical General Branching Processes

Critical Multitype Branching Processes

A general multitype branching process in which individuals are counted according to some possibly type-dependent characteristic may be defined along the lines laid out by Jagers (1969, 1974) for the single type process. In the critical case, the probability of nonextinction at time $t$ is shown to be $O(t^{-1})$, and, conditioned on nonextinction at time $t$, the totals of the characteristic counts, normalized by $t$, are shown to satisfy an exponential limit law, under weak (essentially, second moment) hypotheses.

Limit theorems for point processes generated in a general branching process

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit.Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

Limit theorems for point processes generated in a general branching process

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit. Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

On a general branching process

A branching process is described which contains some better-known simple branching processes as special cases. Variables examined are the rate of development, the number of individuals currently alive, and the duration of the process. The apparent dynamics of collapsing supercritical realisations are isolated.