Nikolay M. Yanev

Relative frequencies in multitype branching processes

This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.

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.

Limit theorems for critical randomly indexed branching processes

We investigate a BGW process subordinated by a renewal process for which the interarrival periods have a finite mean or heavy tails. The branching process is critical with finite or infinite offspring variance and started with a random number of ancestors with infinite mean. The asymptotic behavior of the probability for non-extinction is investigated and limiting distributions are obtained.

Asymptotic Behavior of Cell Populations Described by Two-Type Reducible Age-Dependent Branching Processes With Non-Homogeneous Immigration

Stem and precursor cells play a critical role in tissue development, maintenance, and repair throughout the life. Often, experimental limitations prevent direct observation of the stem cell compartment, thereby posing substantial challenges to the analysis of such cellular systems. Two-type age-dependent branching processes with immigration are proposed to model populations of progenitor cells and their differentiated progenies. Immigration of cells into the pool of progenitor cells is formulated as a non-homogeneous Poisson process. The asymptotic behavior of the process is governed by the largest of two Malthusian parameters associated with embedded Bellman-Harris processes. Asymptotic approximations to the expectations of the total cell counts are improved by Markov compensators.

Branching Processes as Models of Progenitor Cell Populations and Estimation of the Offspring Distributions

We consider two new models of reducible age-dependent branching processes with emigration in conjunction with estimation problems arising in cell biology. Methods of statistical inference are developed using the relevant embedded discrete branching structure. Based on observations of the branching process with emigration, estimators of the offspring probabilities are proposed for the hidden unobservable process without emigration, which is of prime interest to investigators. The problem under consideration is motivated by experimental data generated by time-lapse video recording of cultured cells, which provides abundant information on their individual evolutions and thus on the basic parameters of their life cycle in tissue culture. Some parameters, such as the mean and variance of the mitotic cycle time, can be estimated nonparametrically without resorting to any mathematical model of cell population kinetics. For other parameters, such as the offspring distribution, a model-based inference is needed. Age-dependent branching processes have proven to be useful models for that purpose. A special feature of the data generated by time-lapse experiments is the presence of censoring effects due to the migration of cells out of the field of observation. For the time-to-event observations, such as the mitotic cycle time, the effects of data censoring can be accounted for by standard methods of survival analysis. No methods are available to accommodate such effects in the statistical inference on the offspring distribution. Within the framework of branching processes, the loss of cells to follow-up can be modeled as a process of emigration. Incorporating the emigration process into a pertinent branching model of cell evolution provides the basis for the proposed estimation techniques. Statistical inference on the offspring distribution is illustrated with an application to the development of oligodendrocytes in cell culture.

Supercritical Sevastyanov Branching Processes with Non-homogeneous Poisson Immigration

Sevastyanov (Theory Probab Appl 2:339–348, 1957) introduced a class of Markov branching processes in which immigration of individuals in the population is allowed at random time points described by a time-homogeneous Poisson process. In the present paper, we study a model generalized this process along two directions: Sevastyanov’s (Theory Probab Appl 9:577–594, 1964) age-dependent branching process and time-nonhomogeneous Poisson immigration. The resulting process can be used to describe the dynamics of cell populations arising from differentiating stem cells. Limit theorems are proved in the supercritical case for various classes of immigration rates. Some of the limiting results offer generalizations of the classical result obtained in Sevastyanov (Theory Probab Appl 2:339–348, 1957). We also derive novel LLN and a CLT that arise from the fact that the process is time-inhomogeneous.

A test of homogeneity for age-dependent branching processes with immigration

We propose a novel procedure to test whether the immigration process of a discretely observed age-dependent branching process with immigration is time-homogeneous. The construction of the test is motivated by the behavior of the coefficient of variation of the population size. When immigration is time-homogeneous, we find that this coefficient converges to a constant, whereas when immigration is time-inhomogeneous we find that it is time-dependent, at least transiently. Thus, we test the assumption that the immigration process is time-homogeneous by verifying that the sample coefficient of variation does not vary significantly over time. The test is simple to implement and does not require specification or fitting any branching process to the data. Simulations and an application to real data on the progression of leukemia are presented to illustrate the approach.

LIMIT THEOREMS FOR SUPERCRITICAL MARKOV BRANCHING PROCESSES WITH NON-HOMOGENEOUS POISSON IMMIGRATION

This paper deals with Markov branching processes allowing immigration at random time points described by a non-homogeneous Poisson process. This class of processes generalizes a classical model proposed by Sevastyanov, which included a time-homogeneous Poisson immigration. The proposed model finds applications in cell kinetics studies. Limit theorems are obtained in the supercritical case. Some of these results extend the classical results derived by Sevastyanov, others offer novel insights as a result of the non-homogeneity of the immigration process.

Cell Proliferation Kinetics and Branching Stochastic Processes

The main purpose of this work is to present some new ideas and results obtained in modeling of cell proliferation kinetics.\ Recent advances in experimental techniques of flow cytometry have made it possible to collect a wealth of information about the status of individual cells isolated from dissociated tissues. When one is interested in modeling tissue development starting from the earliest embryonic stages it is reasonable to begin with\

Transient processes in cell proliferation kinetics

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