Related papers: Limit theorems for fragmentation processes with im…
The objective of this article is to create a framework to study asymptotic equilibria in human populations with a special focus on immigration. We present a new model, based on Resource Dependent Branching Processes, which is now broad…
It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper we provide a deeper understanding of these intricate limiting…
In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process $(X_{i\Delta_{n}})_{i=0}^{\lfloor nt\rfloor-1}$, under the assumption that as…
The paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration…
We study the estimation of two-type continuous-state branching processes with immigration (CBI-processes). The ergodicity of the processes is proved. We also establish the strong consistency and central limit theorems of the conditional…
A Galton-Watson branching process with immigration evolving in a random environment is considered. Its associated random walk is assumed to be oscillating. We prove a functional limit theorem in which the process under consideration is…
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cram\'er condition the stationary distribution has a power law tail. We…
We study a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. Each block waits an exponential amount of time with parameter given by its size to some power $\alpha$,…
Let $(X_1, \xi_1), (X_2,\xi_2),\ldots$ be i.i.d.~copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_k := \xi_1+\ldots+\xi_k$, $k \in…
We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a Galton-Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first…
This article is devoted to the investigation of limit theorems for mixed max-sum processes with renewal type stopping indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems.
We consider the setting of either a general non-local branching particle process or a general non-local superprocess, in both cases, with and without immigration. Under the assumption that the mean semigroup has a Perron-Frobenious type…
We prove weak convergence on the Skorokhod space of Galton-Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot…
We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of…
We study a repulsion-diffusion equation with immigration, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp…
The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming…
We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three…
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation…
We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled…