Kosto V. Mitov

BELLMAN-HARRIS BRANCHING PROCESSES WITH STATE-DEPENDENT IMMIGRATION

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.

BELLMAN-HARRIS BRANCHING PROCESSES WITH A SPECIAL TYPE OF STATE-DEPENDENT IMMIGRATION

We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved. PROBABILITY OF EXTINCTION; LIMIT DISTRIBUTIONS A model of a branching process with state-dependent immigration was first considered by Foster (1971) and Pakes (1971), (1975), (1978). They investigated a modification of the Galton-Watson process which admits an immigration component only in the state 0. The continuous-time analogue of this process in the Markov case was studied by Yamazato (1975). Mitov (1983) considered the Foster-Pakes model in the multitype case. Different types of limit theorems were obtained by Mitov and Yanev (1983), (1984) for Foster-Pakes processes with time-dependent immigration in the state 0 and Mitov et al. (1984) considered the continuous-time Markov version. In the present paper we continue investigation of Bellman-Harris processes with state-dependent immigration which were introduced by Mitov and Yanev (1985). Observe that in the previous paper the local characteristics (distributions of immigrants and stay at 0) of the processes had finite mathematical expectations. In the present paper these means can be infinite. More precisely, we consider Bellman-Harris processes with state-dependent immigration where the distribution of immigrants and the distribution of duration of the stay at 0 belong to stable laws with parameters between 2 and 1. Since these distributions can be interpreted as control functions in some real situations then it is interesting to investigate the asymptotic behaviour of the processes with different types of control. BellmanHarris processes with state-dependent immigration might be given as an example of such mathematical models which describe cell proliferation (e.g. of E. coli) in broth

LIMIT THEOREMS FOR ALTERNATING RENEWAL PROCESSES IN THE INFINITE MEAN CASE

The asymptotic behaviour of an occupation-time process associated with alternating renewal processes is investigated in the infinite mean cycle case. The limit theorems obtained extend some asymptotic results proved by Dynkin (1955), Lamperti (1958) and Erickson (1970) for the classical spent lifetime process. Some new phenomena are also presented.

Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration

The branching processes with state-dependent immigration are considered as alternating regenerative processes. The main purpose is to demonstrate some new “regenerative” methods. Critical Bellman–Harris branching processes with state-dependent immigration are investigated and new limit theorems are obtained in the case of an infinite offspring variance and possibly infinite mean of the immigrants.

A Branching Process with Immigration in Varying Environments

Bienaymé–Galton–Watson branching processes with varying offspring variance and an immigration component are studied in the critical case. The asymptotic formulas for the probability for non extinction are derived, in dependence of immigration component. A limit theorem is proved too.

Sevastyanov branching processes with non-homogeneous Poisson immigration

Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity r(t). The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of r(t). Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity.

Extensions and Applications

This chapter is devoted to some extensions and applications of renewal theory. First, we discuss the renewal theorems in the case where the underlying mean is infinite. We proceed by a short discussion of alternating renewal processes. In order to discuss renewal reward processes and superposed renewal processes, we need some basic properties of bivariate renewal theory. The chapter ends with some important applications of renewal theory in different areas of stochastic processes.

Discrete Time Renewal Processes

In this chapter, we give a review of discrete renewal theory and prove the basic theorems for renewal sequences. We provide two different proofs of the theorem of Erdos-Feller-Pollard. Using extensions of a theorem of Wiener, we also obtain several rate of convergence results in discrete renewal theory.

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\

Critical Branching Processes with Nonhomogeneous Migration

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