Multitype Markov Branching Process Research Articles

The multiple birth properties of multi-type Markov branching processes

The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can reveal the multiple birth characteristics. Then the joint probability distribution of multiple birth of multi-type Markov branching process until any time t is obtained by using the new process. Furthermore, the probability distribution of multiple birth until the extinction of the process is also given.

Subcritical multitype Markov branching processes with immigration generated by Poisson random measures

We investigate multitype subcritical Markov branching processes with immigration driven by Poisson random measures. Limiting distributions are established for various rates of the Poisson measures when they are asymptotically equivalent to exponential or regularly varying functions. Results analogous to a strong LLN are proved, and limiting normal distributions are obtained when the local intensity of the Poisson measure increases with time. When it decreases, conditional limiting distributions are established. When the intensity converges to a constant, a stationary limiting distribution is obtained. The asymptotic behavior of the first and second moments of the processes is also investigated.

Poisson random measures and supercritical multitype Markov branching processes

We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

This article deals with quasi- and pseudo-likelihood estimation for a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.

Limiting Distributions for Multitype Branching Processes

In this article, the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The article also considers the relative frequencies of distinct types of individuals motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.

Supercritical multitype branching processes: the ancestral types of typical individuals

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Supercritical multitype branching processes: the ancestral types of typical individuals

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Multitype branching processes with disasters II: Total sojourn time and number of deaths

We consider a multitype Markov branching process under the influence of disasters occurring as a renewal process. For the branching particles in the system, the disasters may lead to possible death or mutation to other types of particles. For such a process, the total mean sojourn time of all the particles in the system is analysed using the regeneration point technique. Explicit expressions of the mean matrices of the total sojourn time are obtained for the decomposable process and the Markovian models. The analysis gains importance in studying the total duration of the toxic effects of drugs in cellular systems. Expression for the mean matrix of the number of particles that die without offspring during the time interval (0, t) is also obtained. An interesting relationship connecting the mean of the total sojourn time of all particles, the mean of the number of particles which die during the time interval (0, t) and the mean number of particles which are alive at time t, is established.

Multitype branching processes with disasters I: The number of particles in the system

A multitype Markov branching process under the influence of disasters (catastrophes) arriving as a renewal process leading to possible mutation of particles to other types is considered. The integral equations for the probability generating function of the number of particles of each type are derived incorporating time dependent mutation rates. The asymptotic behaviours of the mean number of particles have been discussed for the general case. Explicit expressions for the mean number of particles of each type alive at time t are obtained for the Markovian case. Exponentially distributed disaster and constant mutation functions has been dealt with as a special case. The mean number of particles and the covariance matrix are obtained in explicit form for the reversible birth and death process with two types of particles. The results corresponding to Markov branching process involving only three types of particles are deduced. These models involving mutation have applications in the theory of carcinogenesis.

Estimation of extinction probability of a multitype markov branching process

Maximum likelihood estimation of the probability of ultimate extinction of a multitype Markov branching process is studied when the process is observed completely over a fixed time interval [0,t]. The asymptotic properties of the estimator are discussed.

A note on the multitype Markov branching process

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).

A note on the multitype Markov branching process

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).

Reduced Branching Processes

Previous article Next article Reduced Branching ProcessesA. L. YakymivA. L. Yakymivhttps://doi.org/10.1137/1125069PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[2] V. A. Vatutin, Limit theorems for critical multitype Markov branching processes with infinite second moments, Mat. Sb. (N.S.), 103(145) (1977), 253–264, 319, (In Russian.) 56:1488 Google Scholar[3] B. A. Sewast'yanow, Verzweigungsprozesse, Akademie-Verlag, Berlin, 1974xi+326, (German translation), Branching Processes, Nauka, Moscow, 1971. (In Russian.) 53:11785 Google Scholar[4] Klaus Fleischmann and , Rainer Siegmund-Schultze, The structure of reduced critical Galton-Watson processes, Math. Nachr., 79 (1977), 233–241 57:1674 0299.60065 CrossrefGoogle Scholar[5] R. R. Rao, Relations between weak and uniform convergence of measures with applications, Ann. Math. Statist., 33 (1962), 659–680 25:1258 CrossrefGoogle Scholar[6] Eugene Seneta, Regularly varying functions, Springer-Verlag, Berlin, 1976v+112, Lecture Notes in Mathematics, 508 56:12189 0324.26002 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution16 March 2017 | Journal of Theoretical Probability, Vol. 31, No. 3 Cross Ref The Structure of the Decomposable Reduced Branching Processes. I. Finite-Dimensional Distributions19 November 2015 | Theory of Probability & Its Applications, Vol. 59, No. 4AbstractPDF (254 KB)The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distributionElectronic Journal of Probability, Vol. 19, No. none Cross Ref Reduced Branching Processes with Very Heavy Tails14 July 2016 | Journal of Applied Probability, Vol. 45, No. 1 Cross Ref Reduced Branching Processes in Random Environment: The Critical Case25 July 2006 | Theory of Probability & Its Applications, Vol. 47, No. 1AbstractPDF (180 KB)The general coalescent with asynchronous mergers of ancestral lines14 July 2016 | Journal of Applied Probability, Vol. 36, No. 4 Cross Ref The Total Number of Particles in a Reduced Bellman-Harris Branching Process17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 3AbstractPDF (407 KB)Reduced Critical Bellman–Harris Branching Process with Several Types of Particles28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 4AbstractPDF (868 KB)Asymptotic Properties of Subcritical and Supercritical Reduced Branching Processes28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (572 KB) Volume 25, Issue 3| 1981Theory of Probability & Its Applications History Submitted:11 May 1979Published online:28 July 2006 InformationCopyright © 1981 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1125069Article page range:pp. 584-588ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z ( k )( t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.

Some refinements in the theory of supercritical multitype Markov branching processes

Some refinements in the theory of supercritical multitype Markov branching processes