N. M. Yanev

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.

LIMIT THEOREMS AND ESTIMATION THEORY FOR BRANCHING PROCESSES WITH AN INCREASING RANDOM NUMBER OF ANCESTORS

This paper deals with a Bienaym6-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

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

Critical Branching Processes with Random Migration

A new class of branching processes allowing a random migration component in every generation is considered: with probability p two types of emigration are possible — a random number of families and a random number of individuals, or with probability q there is not any migration (i.e. the process develops like a Bienayme-Galton-Watson process), or with probability r a state-dependent immigration of new individuals is available, p + q + r = 1. The coresponding processes stopped at zero are studied in the critical case and the asymptotic behaviour of the non-extinction probability is obtained (depending on the range of an extra critical parameter).

Statistical inference for branching processes with an increasing random number of ancestors

Abstract Our main concern in this paper is the estimation of m and σ2 from a branching process {Zt(n)} having a random number of ancestors Z0(n), as both n (and thus Z0(n) in some sense) and t→∞. While the first part review the literature on the topic and motivate the consideration of Z0(n) random, the second part contains new material, in particular concerning m.l.e. and a family of l.s.e. for σ2. Consistency and asymptotic normality of these estimators are obtained for all values of the mean m, 0

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.

Branching Processes with Two Types Emigration and State-Dependent Immigration

We consider branching processes allowing a random migration component. In each generation the following three situations are possible: (i) with probability p - family emigration and individual emigration (possibly dependent); (ii) with probability q - no migration, i.e. the reproduction is as in the classical BGW process; (iii) with probability r - state dependent immigration; p + q + r = 1. In the critical case an additional parameter of recurrence is obtained. The asymptotic behaviour of the hitting zero probability and of the first two moments is investigated. Limiting distributions are also obtained depending on the range of the recurrence parameter.

A Critical Branching Process with Stationary-Limiting Distribution

Abstract Seneta (Seneta, E. The stationary distribution of a branching process allowing immigration: A remark on the critical case. J. Royal Statistical Society, Series B 1968, 30, 176–179) shows that a critical branching process with pure immigration has a stationary-limiting distribution provided that its offspring variance is infinite. We obtain a stationary-limiting distribution keeping the variance finite but allowing an emigration–immigration component in each generation.

Limit Theorems for Branching Processes with Random Migration Stopped at Zero

In this paper, we study a generalization of the classical BienaymeGalton-Watson branching process by allowing a random migration component stopped at zero (i.e. the state zero is absorbing). In each generation for which the population size is positive, with probability p two types of emigration are available - a random number of offsprings and a random number of families (these two random variables can be dependent); with probability q there is not any migration; with probability r an immigration of new individuals is possible, p+q+r = 1. The critical case is investigated with an extension when the initial law is attracted to a stable (p) law, p ≤ 1. The asymptotic form of the probability of non-extinction is studied and conditional limit theorems for the population size are obtained, depending on the range of an additional parameter of criticality.