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 generator $L$ of $T_t$, and $ϕ_0$ and $\hatϕ_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $λ_0$. Under some conditions on the spatial motion and the $ϕ_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$ \lim_{t\rightarrow\infty}e^{-λ_0 t}< f, X_t> = W_\infty\int_E\hatϕ_0(y)f(y)m(dy),\quad \mathbb{P}_μ{-a.s.}, $$ for any finite initial measure $μ$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-λ_0t}< ϕ_0, X_t>$. Moreover, the exceptional set in the above limit does not depend on the initial measure $μ$ and the function $f$.
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 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.
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…
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, …
Consider a locally finite Dawson-Watanabe superprocess $ξ=(ξ_t)$ in $\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $ξ$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $ξ_t$ by a stationary cluster $\tildeη$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditiona…
Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $ξ$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$ assuming that $\mathbb{P} (Z_1 = 0) =0$. We also determine the critical value for the existence of harmonic moments of the random variable $W=\lim_{n\to\infty}\frac{Z_n}{\mathbb E (Z_n|ξ)}$ under a simple moment condition.
A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein-Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment mea…
Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.]…
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 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…
Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $ξ$. We study the asymptotic of the harmonic moments $\mathbb{E}\left[Z_n^{-r} | Z_0=k \right]$ of order $r>0$ as $n \to \infty$. We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\mathbb E p_1^k = \mathbb E m_0^{-r_k},$ where $m_0=\sum_{k=0}^\infty k p_k$ with $p_k=\mathbb P(Z_1=k | ξ),$ assuming that $p_0=0.$ Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence of t…
We consider a supercritical general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process of such population is a known as binary homogeneous Crump-Jargers-Mode process. It is known that such processes converges almost surely when correctly renormalized. In this paper, we study the error of this convergence. To this end, we use classical renewal theory and recent works [17, 6, 5] on this model to obtain the moments of the e…
The study of discrete-time stochastic processes on the half-line with mean drift at $x$ given by $μ_1 (x) \to 0$ as $x \to \infty$ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where $μ_1 (x)$ is of order $x^{-β}$ for some $β\in (0,1)$. The bounds are of order $t^{1/(1+β)}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of $(2+2β+\varepsilon)$-moments for our main results, so 4th moments certainly suffice) and do not assume th…
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…
We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $ν_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $ν_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=ν_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n ρ_n {\mathbb E} D_n (1+o(1))$, wher…
Let $(Z_n)$ be a supercritical branching process in a random environment $% ζ$, and $W$ be the limit of the normalized population size $Z_n/\mathbb{E%}(Z_n|ζ)$. We show necessary and sufficient conditions for the existence of weighted moments of $W$ of the form $\E W^α\ell(W)$, where $α\geq 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve those of Bingham and Doney (1974).
Let $(Z_{n})$ be a supercritical branching process in a random environment $ξ$, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|ξ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (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 $\log Z_n$ are also established.
We consider a supercritical branching Lévy process on the real line. Under mild moment assumptions on the number of offspring and their displacements, we prove a second-order limit theorem on the empirical mean position.
We consider the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the probabilistic reasoning of Kyprianou (2004) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (1974). We give a pathwise explanation of Evans' immortal particle picture …
Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the p…
Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\su |W-W_n|$, $\su |W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\rP\{W_1>x\}$ regularly varies at $\infty$ it is proved that $\rP\{AW_n>x\}$ regularly varies at $\infty$ with the same exponent. All the proofs given …
Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $ \E X (\ln^+X )^{1+λ}< \infty$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work.The results are establi…
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| ξ]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \log W_1^{α+1} < \infty$ for some $α>0$, we have $W-W_n = o (n^{-α})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there…
We study a continuous-time branching random walk on the lattice $\mathbb{Z}^{d}$, $d\in \mathbb{N}$, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be homogeneous, symmetric and irreducible but, in contrast to previous investigations, the random walk transition intensities $a(x,y)$ decrease as $|y-x|^{-(d+α)}$ for $|y-x|\to \infty$, where $α\in(0,2)$, that leads to an infinite variance of the random walk jumps. The~mechanism of the birth and death of particles at the source is governed by a continuous…
