In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren, Song and Zhang[2015]. We first give law of large numbers for supercritical superprocesses with immigration since there is few convergence result on immigration superprocesses, then based on these results, we establish the central limit theorem.
A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time-space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637-686] and Abraha…
In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.
In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state $E$ is a finite set and the underline motion is an irreducible Markov chain on $E$, our results are superprocess analogs of the functional central limit theorems of \cite{Janson} for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in \cite{RSZ3}.
In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Miłoś (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is that it allows us to characterize the limit Gaussian field. In the case of supe…
The goal of this paper has two-folds. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space $E$ and the branching mechanism takes the form $$ψ_β(x,λ)=-β(x)λ+α(x)λ^{2}+\int_{(0,{\infty})}(e^{-λy}-1+λy)π(x,dy)$$ with $β\in\mathcal{B}_{b}(E)$, $α\in \mathcal{B}^{+}_{b}(E)$ and $π$ being a kernel from $E$ to $(0,{\i…
Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically `thinner' Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the ori…
Suppose $X=\{X_t, t\ge 0\}$ is a supercritical superprocess. Let $ϕ$ be the non-negative eigenfunction of the mean semigroup of $X$ corresponding to the principal eigenvalue $λ>0$. Then $M_t(ϕ)=e^{-λt}\langleϕ, X_t\rangle, t\geq 0,$ is a non-negative martingale with almost sure limit $M_\infty(ϕ)$. In this paper we study the rate at which $M_t(ϕ)-M_\infty(ϕ)$ converges to $0$ as $t\to \infty$ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the conver…
We develop an idea of Evans and O'Connell, Englander and Pinsky and Duquesne and Winkel by giving a pathwise construction of the so called `backbone' decomposition for supercritical superprocesses. Our results also complement a related result for critical $(1+β)$-superprocesses given in Etheridge and Williams \cite{EW}. Our approach takes an analytical point of view which is more in the spirit of the original Evans and O'Connell paper.
It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritial superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is individuals who produce eternal genealogical lines of decent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess…
We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.
Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_μ\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $ϕ_0$ be a positive eigenfunction corresponding to the first eigenvalue $λ_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-λ_0t}\langleϕ_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We pr…
Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_μ\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $ϕ_0$ be a positive eigenfunction corresponding to the first eigenvalue $λ_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-λ_0t}\langleϕ_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. When the $L\log L$ condition may not be satisfied, we recently proved in (arXiv:1708.04422) that there exist a non-negative function $γ_t$ on $[0, …
Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we…
Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson--Watanabe superprocesses without or with immigration. The strong Feller property in the total variation distance is derived as a by-product. The key of the approach is a set of estimates for the variations of the transition probabilities. The estimates in Wasserstein distance are derived from an upper bound of the kernels induced by the first moment of the superprocess. Those in total variation distance are based on a comparison of the cumulant semigroup of t…
In this paper, we consider Galton-Watson processes with immigration. Pick $i(\ge2)$ individuals randomly without replacement from the $n$-th generation and trace their lines of descent back in time till they coalesce into $1$ individual in a certain generation, which we denote by $X_{i,1}^n$ and is called the coalescence time. Firstly, we give the probability distribution of $X_{i,1}^n$ in terms of the probability generating functions of both the offspring distribution and the immigration law. Then by studying the limit behaviors of various functionals of the Galton-Watson process with immigra…
Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration (CBI process) converges almost surely. If an $x \log(x)$ moment condition on the branching mechanism does not hold, then the limit is zero. If this $x \log(x)$ moment condition holds, then we prove $L_1$ convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing. If, in addition, a suitable extra power moment condi…
Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$ ψ(x,λ)=-a(x)λ+b(x)λ^2+\int_{(0,+\infty)}(e^{-λy}-1+λy)n(x,dy), \quad x\in E, \quadλ> 0, $$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$ \sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty. $$ Put $T_tf(x)=\mathbb{P}_{δ_x}< f,X_t>$. Let $λ_0>0$ be the largest eigenvalue of the g…
We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: …
We prove and extend some results stated by Mark Pinsky: Limit theorems for continuous state branching processes with immigration [Bull. Amer. Math. Soc. 78(1972), 242--244]. Consider a continuous-state branching process with immigration $(Y_t,t\geq 0)$ with branching mechanism $Ψ$ and immigration mechanism $Φ$ (CBI$(Ψ,Φ)$ for short). We shed some light on two different asymptotic regimes occurring when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$ or $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$. We first observe that when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$, supercritical CBIs have a growth rate dictate…
In this paper we study supercritical super-OU processes with general branching mechanisms satisfying a second moment condition. We establish central limit theorems for the super-OU processes. In the small and crtical branching rate cases, our central limit theorems sharpen the corresponding results in the recent preprint of Milos in that the limit normal random variables in our central limit theorems are non-degenerate. Our central limit theorems in the large branching rate case are completely new. The main tool of the paper is the so called "backbone decomposition" of superprocesses.
