Related papers: Central limit theorem for supercritical binary hom…
Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^{\varphi})_{t \geq 0}$ counted with a random characteristic $\varphi$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that,…
For supercritical multitype branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population…
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…
Consider a critical branching random walk on $\mathbb{R}$. Let $Z^{(n)}(A)$ be the number of individuals in the $n$-th generation located in $A\in \mathcal{B}(\mathbb{R})$ and $Z_{n}:=Z^{(n)}(\mathbb{R})$ denote the population of the $n$-th…
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…
A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…
We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may…
For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the…
We consider Gaussian approximation in a variant of the classical Johnson--Mehl birth-growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a…
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d \ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit…
Let $(Z_{n})$ be a supercritical branching process in a random environment $\xi $, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. We show large and moderate deviation principles for the sequence $\log…
A famous result in renewal theory is the Central Limit Theorem for renewal processes. As in applications usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is…
In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it…
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…
We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms…
A controlled branching process (CBP) is a modification of the standard Bienaym\'e-Galton-Watson process in which the number of progenitors in each generation is determined by a random mechanism. We consider a CBP starting from a random…
Splitting trees are those random trees where individuals give birth at constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous…
Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk $\eta_1$, $\xi_1+\eta_2$, $\xi_1+\xi_2+\eta_3,\ldots$. Here, $(\xi_1,\eta_1)$, $(\xi_2, \eta_2),\ldots$ are independent identically…
We study a universal object for the genealogy of a sample in populations with mutations: the critical birth-death process with Poissonian mutations, conditioned on its population size at a fixed time horizon. We show how this process arises…
In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The…