Related papers: Precise large deviation estimates for branching pr…
We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cram\'er's condition is not satisfied. More precisely we consider step size…
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…
Evolutionary branching is analysed in a stochastic, individual-based population model under mutation and selection. In such models, the common assumption is that individual reproduction and life career are characterised by values of a…
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. For the subcritical regime a kind of phase transition…
In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree,…
There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities' we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed picture of…
The large deviation principle on phase space is proved for a class of Markov processes known as random population dynamics with catastrophes. In the paper we study the process which corresponds to the random population dynamics with linear…
Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$, let $Z_n(A)$ be the number of particles located in interval $A$ at generation $n$. It is well known (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt…
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the…
We consider the branching random walk drifting to $-\infty$ and we investigate large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total…
Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for…
Let $(Z_n)$ be a supercritical branching process with immigration in a random environment. The small positive values and some lower deviation inequalities for $Z$ are investigated. Based on these results, the central limit theorem of $\log…
Birth-death processes form a natural class where ideas and results on large deviations can be tested. In this paper, we derive a large deviation principle under the assumption that the rate of a jump down (death) is growing asymptotically…
In this paper we consider two branching processes living in a joint random environment. Assuming that both processes are critical we address the following question: What is the probability that both populations survive up to a large time…
Consider a multi-dimensional supercritical branching process with offspring distribution in a parametric family. Here, each vector coordinate corresponds to the number of offspring of a given type. The process is observed under family-size…
The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this…
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…
Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random…
We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay…