Related papers: Parametrised branching processes: a functional ver…

This article is an essay, both expository and argumentative, on the Galton-Watson process as a tool in the domain of Branching Processes. It is at the same time the author's ways to honour two distinguished scientists in this domain, both…

Let $a$ be a finite signed measure on $[-r, 0]$ with $r \in (0, \infty)$. Consider a stochastic process $(X^{(\vartheta)}(t))_{t\in[-r,\infty)}$ given by a linear stochastic delay differential equation \[ \mathrm{d} X^{(\vartheta)}(t) =…

Scaling limits for continuous-time branching processes with discrete state space are provided as the initial state tends to infinity. Depending on the finiteness or non-finiteness of the mean and/or the variance of the offspring…

We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d.~$\mathrm{Poiss}(\lambda)$ many inactive particles are placed at each non-root vertex. Active…

A branching process $Z$ is said to be non conservative if it hits $\infty$ in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism $\varphi$ of $Z$ satisfies…

We study the exploration (or height) process of a continuous time non-binary Galton-Watson random tree, in the subcritical, critical and supercritical cases. Thus we consider the branching process in continuous time (Z_{t})_{t\geq 0}, which…

We consider random walks $\lambda$-biased towards the root on a Galton-Watson tree, whose offspring distribution $(p_k)_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda<m$, the loop-erased…

The Ewens-Pitman model refers to a distribution for random partitions of $[n]=\{1,\ldots,n\}$, which is indexed by a pair of parameters $\alpha \in [0,1)$ and $\theta>-\alpha$, with $\alpha=0$ corresponding to the Ewens model in population…

This paper provides a detailed analysis of the lower deviation probability properties for a $d$-type ($d>1$) Galton--Watson (GW) process $\{\textbf{Z}_n=(Z_n^{(i)})_{1\le i\le d};n\ge0\}$ in both Schr\"{o}der and B\"{o}ttcher cases. We…

Given a discrete spatial structure $X$, we define continuous-time branching processes that model a population breeding and dying on $X$. These processes are usually called branching random walks. They are characterized by breeding rates…

We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for…

We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally…

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process…

We present a new algorithm for computing the quasi-stationary distribution of subcritical Galton--Watson branching processes. This algorithm is based on a particular discretization of a well-known functional equation that characterizes the…

Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cram\'{e}r moderate deviation results for the Lotka-Nagaev…

We study the typical behavior of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution has finite variance. The harmonic measure considered here refers to the hitting distribution of height $n$ by…

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus…

By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random variables…

Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong…