arXiv 2106.014262021-06-02math.PR

参数化分枝过程：Kesten \& Stigum定理的泛函形式

Parametrised branching processes: a functional version of Kesten \& Stigum theorem

Cécile Mailler, Jean-François Marckert

摘要：

设$(Z_n,n\geq 0)$是一个超临界Galton-Watson过程，其后代分布$μ$具有平均$λ>1$，且满足$\int x(\log(x))_+ dμ(x)<+\infty$ .根据著名的Kesten \& Stigum定理，$(Z_n/λ^n)$几乎必然收敛，如$n\to+\infty$ .极限随机变量的平均值为~1，其分布特征为一个不动点方程的解。在本文中，我们考虑一族Galton-Watson过程$(Z_n(λ), n\geq 0)$，定义为~ $λ$在区间$I\subset (1, \infty)$中的范围，并且其中我们将$λ$解释为时间（当$n$是生成时）。一个个体在时间~ $λ$的子代数由$X(λ)$给出，其中$(X(λ))_{λ\in I}$是一个càdlàg整数值过程，假设它几乎必然非减，并且使得对所有$λ\in I$有$\mathbb E(X(λ))=λ>1$。这允许我们定义$Z_n(λ)$，即在$λ$时，$n$第代中的元素数。为所有$n\geq 0$和$λ\in I$设置$W_n(λ)= Z_n(λ)/λ^n$。证明了在过程~ $X$上的矩条件下，过程序列$(W_n(λ), λ\in I)_{n\geq 0}$在具有Skorokhod拓扑的càdlàg过程空间中，当~ $n$趋于无穷大时依概率收敛于一个过程，并将其归结为一个不动点方程的解.

英文摘要：

Let $(Z_n,n\geq 0)$ be a supercritical Galton-Watson process whose offspring distribution $μ$ has mean $λ>1$ and is such that $\int x(\log(x))_+ dμ(x)<+\infty$. According to the famous Kesten \& Stigum theorem, $(Z_n/λ^n)$ converges almost surely, as $n\to+\infty$. 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 $(Z_n(λ), n\geq 0)$ defined for~$λ$ ranging in an interval $I\subset (1, \infty)$, and where we interpret $λ$ as the time (when $n$ is the generation). The number of children of an individual at time~$λ$ is given by $X(λ)$, where $(X(λ))_{λ\in I}$ is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that $\mathbb E(X(λ))=λ>1$ for all $λ\in I$. This allows us to define $Z_n(λ)$ the number of elements in the $n$th generation at time $λ$. Set $W_n(λ)= Z_n(λ)/λ^n$ for all $n\geq 0$ and $λ\in I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(λ), λ\in I)_{n\geq 0}$ converges in probability as~$n$ tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文