Critical Galton-Watson processes: The maximum of total progenies within a large window

Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m is such that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_{infinity}(j).Comment: 28 page

Multi-scale clustering for a non-Markovian spatial branching process

Consider a system of particles which move in R^d according to a symmetric alpha-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+beta. In case of the critical dimension d=alpha/beta, the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+beta. Our result generalizes the case alpha=2 of Brownian particles of Klenke (1998), where pde methods had been used which are not available in the present setting

Local limit theorems for ladder moments

Let $S_{0}=0,\{S_{n}\}_{n\geq1}$ be a random walk generated by a sequence of i.i.d. random variables $X_{1},X_{2},...$ and let $\tau^{-}:=\min\left\{ n\geq1:\ S_{n}\leq0\right\}$ and $\tau^{+}:=\min\left\{ n\geq 1:\ S_{n}>0\right\}$. Assuming that the distribution of $X_{1}$ belongs to the domain of attraction of an $\alpha$-stable law$,\alpha\neq1,$ we study the asymptotic behavior of $\mathbb{P}(\tau^{\pm}=n)$ as $n\rightarrow\infty.

Some limit theorems for a particle system of single point catalytic branching random walks

We study the scaling limit for a catalytic branching particle system whose particles performs random walks on \ZZ and can branch at 0 only. Varying the initial (finite) number of particles we get for this system different limiting distributions. To be more specific, suppose that initially there are n^{\be} particles and consider the scaled process $Z^n_t(\bullet)=Z_{nt}(\sqrt{n}\, \bullet)$ where $Z_t$ is the measure-valued process representing the original particle system. We prove that $Z^n_t$ converges to 0 when \be<\frac{1}{4} and to a nondegenerate discrete distribution when \be=\frac{1}{4}. In addition, if \frac{1}{4}<\be<\frac{1}{2} then n^{-(2\be-\frac{1}{2})}Z^n_t converges to a random limit while if \be>\frac{1}{2} then n^{-\be}Z^n_t converges to a deterministic limit. Note that according to Kaj and Sagitov \cite{KS} $n^{-\frac{1}{2}}Z^n_t$ converges to a random limit if $\be=\frac{1}{2}.

Branching systems with long living particles at the critical dimension

A spatial branching process is considered in which particles have a life time law with a tail index smaller than one. Other than in classical branching particle systems, at the critical dimension the system does not suffer local extinction when started from a spatially homogenous initial population. In fact, persistent convergence to a mixed Poissonian system is shown. The random limiting intensity is characterized in law by the random density in a space point of a related age-dependent superprocess at a fixed time. The proof relies on a refined study of the system starting from asymptotically large but finite initial populations

Stochasticity in the adaptive dynamics of evolution: The bare bones

First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied. Subsequently the results are extended to such a population with two competing morphs and are applied to a simple model, where morphs arise through mutation. The movement in the trait space of a monomorphic population and its possible branching into polymorphism are discussed. This is a first report. It purports to display the basic conceptual structure of a simple exact probabilistic formulation of adaptive dynamics