Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

Almost critical branching processes and limit theorems

We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 127–133, January, 2009.     

Some scaled limit theorems for an immigration super-Brownian motion

In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.     

Some limit theorems for critical branching Bessel processes,and related semilinear differential equations

Summary For a critical binary branching Bessel process starting fromx1 and stopped atx=1, we prove some conditional limit laws of the number of particles arriving atx=1 before a scaled large time. Five regions of the dimensional index of a Bessel process: –<d<2,d=2, 2<d<4,d=4 and 4<d< are showed to have somewhat different behaviors. Our probabilistic results are proved by analyzing differential equations satisfied by generating functions. A salient theme is a comparison principle technique deliberately used to estimate solutions of inR ⁺×R ⁺ wherep is greater than 1. The casep=2 corresponds to the process considered.     

Second order limit theorems for the Markov branching process in random environments

In a Markov branching process with random environments, limiting fluctuations of the population size arise from the changing environment, which causes random variation of the ‘deterministic’ population prediction, and from the stochastic wobble around this ‘deterministic’ mean, which is apparent in the ordinary Markov branching process. If the random environment is generated by a suitable stationary process, the first variation typically swamps the second kind. In this paper, environmental processes are considered which, in contrast, lead to sampling and environmental fluctuation of comparable magnitude. The method makes little use either of stationarity or of the branching property, and is amenable to some generalization away from the Markov branching process.     

Limit behavior of a critical branching process with immigration

A generalization of the Sevast’yanov branching process with immigrationwhich is a Cox process is studied. The generating function of the number of particles of the process is obtained. For critical processes, the limit behavior of the number of particles at infinity is established.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

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.     

Some limit theorems for the empirical process indexed by functions

Summary Let,n1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,,P). Under weak metric entropy conditions on,n1, and under growth conditions on we show that there are non-zero numerical constantsC ₁ andC ₂ such that where (n) is a non-decreasing function ofn related to the metric entropy of. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany     

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^∗.     

Limit theorems for supercritical branching random fields with immigration

A measure-valued process which carries genealogical information is defined for a supercritical branching random field with immigration. This process counts the particles present at a final time whose ancestors had specified locations at given times in the past. A law of large numbers and a fluctuation limit theorem are proved for this process under a space-time scaling. The fluctuation limit is a nonstationary generalized Ornstein-Uhlenbeck process. An example of interest in transport theory and polymer chemistry is given.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

A conditional limit theorem for a critical Branching process with immigration

The life period of a branching process with immigration begins at the moment T and has length if the number of particles (T –0)=0, (t)>0 for all Ttt and T=0.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 727–736, May, 1977.The author thanks A. M. Zubkov for the formulation of the problem and valuable advice.