ﻻ يوجد ملخص باللغة العربية
Let $(Z_n,ngeq 0)$ be a supercritical Galton-Watson process whose offspring distribution $mu$ has mean $lambda>1$ and is such that $int x(log(x))_+ dmu(x)<+infty$. According to the famous Kesten & Stigum theorem, $(Z_n/lambda^n)$ converges almost surely, as $nto+infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. par In this paper, we consider a family of Galton-Watson processes $(Z_n(lambda), ngeq 0)$ defined for~$lambda$ ranging in an interval $Isubset (1, infty)$, and where we interpret $lambda$ as the time (when $n$ is the generation). The number of children of an individual at time~$lambda$ is given by $X(lambda)$, where $(X(lambda))_{lambdain I}$ is a c`adl`ag integer-valued process which is assumed to be almost surely non-decreasing and such that $mathbb E(X(lambda))=lambda >1$ for all $lambdain I$. This allows us to define $Z_n(lambda)$ the number of elements in the $n$th generation at time $lambda$. Set $W_n(lambda)= Z_n(lambda)/lambda^n$ for all $ngeq 0$ and $lambdain I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(lambda), lambdain I)_{ngeq 0}$ converges in probability as~$n$ tends to infinity in the space of c`adl`ag processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.
We are interested in the genealogical structure of alleles for a Bienayme-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall es
We introduce and study the dynamics of an emph{immortal} critical branching process. In the classic, critical branching process, particles give birth to a single offspring or die at the same rates. Even though the average population is constant in ti
Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The ma
T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton i
The decomposable branching processes are relatively less studied objects, particularly in the continuous time framework. In this paper, we consider various variants of decomposable continuous time branching processes. As usual practice in the theory