Estimation of the offspring mean in a supercritical branching process with non-stationary immigration

In this paper, we consider the conditional least squares estimator (CLSE) of the offspring mean of a branching process with non-stationary immigration based on the observation of population sizes. In the supercritical case, assuming that the immigration variables follow known distributions, conditions guaranteeing the strong consistency of the proposed estimator will be derived. The asymptotic normality of the estimator will also be proved. The proofs are based on direct probabilistic arguments, unlike the previous papers, where functional limit theorems for the process were used.     

Estimation of the offspring mean in a controlled branching process with a random control function

Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773] for branching processes with immigration and provide a unified limit theory of estimation.     

Weighted moments for a supercritical branching process in a varying or random environment

Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.     

A limit theorem for a supercritical branching process

The limit distribution if calculated for the time at which a supercritical branching process reaches the level x, when x.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 585–592, May, 1971.     

The supercritical multitype age-dependent branching process

Let X(t) = (X₁(t),…, X_p(t)) be a p-dimensional supercritical age-dependent branching process. For an appropriate α > 0, necessary and sufficient conditions are found for X(t) e^?αt to converge to a nondegenerate random vector W. Several properties of W are also determined.     

Estimation of the PCR efficiency based on a size-dependent modelling of the amplification process

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé–Galton–Watson transient phase and then having a saturation near-critical size-dependent phase. This model based on the concept of saturation allows one to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy. To cite this article: N. Lalam et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Z_n) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W_n=Z_n/E[Z_n|ξ] to its limit W_∞: we show a central limit theorem for W_∞−W_n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W_n+k−W_n for each fixed k∈N^∗.     

A critical measure-valued branching process with infinite mean

A critical spatially homogeneous measure-valued branching process in R^dis studied where the initial state has infinite asymptotic density. In low dimen-dimensions it explodes (locally), but in a critical dimension both effects are exhibited.     

Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

Norming constants for the finite mean supercritical Bellman-Harris process

Summary Let {Z(t)} be a supercritical Bellman-Harris process with offspring distribution {p _k} and lifetime distributionG. It is shown that the finiteness of the offspring mean guarantees the existence of norming constants {C(t)} such that a.s. for some nondegenerate random variableW. C(t) is the-quantile of the distribution function ofZ(t), whereq<<1,q being the extinction probability of the process. As a byproduct of the proof, {Z(t)/C(t)} is shown to be asymptotic Markov. The theory of weakly stable sums of i.i.d. is used to get characterizations ofW and {C(t)}.     

Strong and weak convergence of the population size in a supercritical catalytic branching process

A general model of a catalytic branching process (CBP) with any number of catalysis centers in a discrete space is studied. The asymptotic (in time) behavior of the total number of particles and of the local particle numbers is investigated. The problems of finding the global and local extinction probabilities are solved. Necessary and sufficient conditions are established for the phase of pure global survival and strong local survival. Under wide conditions, limit theorems for the normalized total and local particle numbers in supercritical CBP are proved in the sense of almost sure convergence, as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

Limit theorems for a supercritical branching process with immigration in a random environment

Let (Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.     

Estimation of the mean in partially observed branching processes with general immigration

In the paper we investigate asymptotic properties of the branching process with non-stationary immigration which are sufficient for a natural estimator of the offspring mean based on partial observations to be strongly consistent and asymptotically normal. The estimator uses only a binomially distributed subset of the population of each generation. This approach allows us to obtain results without conditions on the criticality of the process which makes possible to develop a unified estimation procedure without knowledge of the range of the offspring mean. These results are to be contrasted with the existing literature related to i.i.d. immigration case where the asymptotic normality depends on the criticality of the process and are new for the fully observed processes as well. Examples of applications in the process with immigration with regularly varying mean and variance and subcritical processes with i.i.d. immigration are also considered.

     

The stable doubly infinite pedigree process of supercritical branching populations

Summary An individual is sampled randomly from a supercritical general branching population and the pedigree process, which centers around this ego-individual, is studied. The process describes not only lineage backwards and forwards, but also the lives of all individuals involved. Under mild conditions and in several senses, the process is shown to stabilize, as time passes. The limit is a doubly infinite population process, which generalizes the stable age distribution of branching processes and demography. It displays a nice independence structure, and can easily be constructed from the original branching law. The results are applied to certain kin-number problems, the process of ego's ancestors' births, and to the FLM-curve of cell kinetics.     

Estimation of the variances in the branching process with immigration

Summary Estimation theory for the variances of the offspring and immigration distributions in a simple branching process with immigration is developed, analogous to the estimation theory for the means given by Wei and Winnicki (1990). Conditional and weighted conditional least squares estimators are considered and their asymptotic properties for the full range of parameters are studied. Nonexistence of consistent estimators in the critical case is established, which complements analogous result of Wei and Winnicki for the supercritical case.Research supported by the National Science Foundation under Grant NSF-DMS-8801496