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.     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.     

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

Long-term behaviour of a cyclic catalytic branching system

We investigate the long-term behaviour of a system of SDEs for d≥2 types, involving catalytic branching and mutation between types. In particular, we show that the overall sum of masses converges to zero but does not hit zero in finite time a.s. We shall then focus on the relative behaviour of types in the limit. We prove weak convergence to a unique stationary distribution that does not put mass on the set where at least one of the coordinates is zero. Finally, we provide a complete analysis of the case d=2.     

The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

The (Ξ,A)$(\Xi ,A)$-Fleming–Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming–Viot process except that Kingman’s coalescent is replaced by the Ξ$\Xi$-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A$A$. We then investigate its reversibility. We also study both the weak and strong uniqueness of the solution to the associated stochastic partial differential equation.     

Asymptotic distribution of the CLSE in a critical process with immigration

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n^2/3$n^{2/3}$. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α$\alpha$ and β$\beta$, respectively. We prove that if β<1+2α$\beta <1+2\alpha$, the CLSE is asymptotically normal with two different normalization factors and if β>1+2α$\beta >1+2\alpha$, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β=1+2α$\beta =1+2\alpha$ the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.     

Extinction probability for a critical general branching process

A general branching process begins with a single individual born at time t=0. At random ages during its random lifespan L it gives birth to offspring, N(t) being the number born in the age interval [0,t]. Each offspring behaves as a probabilistically independent copy of the initial individual. Let Z(t) be the population at time t, and let N=N(∞). Theorem: If a general branching process is critical, i. e E{N}=1, and if $\text{σ}{}^2\text{=E {N(N?1)}<∞, 0<a≡}\text{∫}\text{0}\text{∞}\text{tdE{N(t)}}$,and as t → ∞ both $\text{t}{}^2\text{(1?}\text{E}\text{{N(t)})→0}$ and $\text{t}{}^2\text{P}\text{[L>t]→0}$, then $\text{tP[Z(t)>0]→}\text{2a}\text{σ}{}^2$ as t→∞.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion and α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d<α$d<\alpha$, which involves fractional Brownian motion. We use a space–time random field approach.     

Rate of convergence in the law of large numbers for supercritical general multi-type branching processes

We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman’s martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the processes in focus.     

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

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^∗.     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

Pathwise construction of additive H-transforms of super-Brownian motion

Summary The pathwise construction of additiveH-transforms of the super-Brownian motion is carried out as a modification of Le Gall's construction of superprocesses. It provides then the explicit conditioning of the super-Brownian motion on its exit behaviour at its Martin boundary, which yields an additiveH-transforms of the super-Brownian motion. The condition turns out to be that the space-time point of death of the super-Brownian motion converges in the Martin topology of the Brownian motion.Supported by an EC-Individual-Fellowship under Contract No. ERBCHBICT930682 and the SFB 256 of the University of Bonn, Germany     

A new approach to the single point catalytic super-Brownian motion

Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst _c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure ^c ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX _t is determined from ^c by an explicit representation formula. Heuristically, a mass ^c (ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure ^c . In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure ^c .     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.