We consider a general branching population where the lifetimes of individuals are i.i.d.\ with arbitrary distribution and where each individual gives birth to new individuals at Poisson times independently from each other. In addition, we suppose that individuals experience mutations at Poissonian rate $\theta$ under the infinitely many alleles assumption assuming that types are transmitted from parents to offspring. This mechanism leads to a partition of the population by type, called the allelic partition. The main object of this work is the frequency spectrum $A(k,t)$ which counts the number of families of size $k$ in the population at time $t$. The process $(A(k,t),\ t\in\mathbb{R}_+)$ is an example of non-Markovian branching process belonging to the class of general branching processes counted by random characteristics. In this work, we propose methods of approximation to replace the frequency spectrum by simpler quantities. Our main goal is study the asymptotic error made during these approximations through central limit theorems. In a last section, we perform several numerical analysis using this model, in particular to analyze the behavior of one of these approximations with respect to Sabeti's Extended Haplotype Homozygosity [18].

中文翻译：

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一般超临界分支群中等位基因频谱的近似

我们考虑一个一般的分支种群，其中个体的寿命是随机分布的，并且每个个体在泊松时间彼此独立地生育新个体。此外，我们假设个体在无限多等位基因假设下以泊松率 $\theta$ 经历突变，假设类型从父母传给后代。这种机制导致按类型划分种群，称为等位基因划分。这项工作的主要对象是频谱 $A(k,t)$，它计算时间 $t$ 时人口中规模为 $k$ 的家庭的数量。过程$(A(k,t),\ t\in\mathbb{R}_+)$ 是非马尔可夫分支过程的一个例子，属于按随机特征计数的一般分支过程类。在这项工作中，我们提出了用更简单的量代替频谱的近似方法。我们的主要目标是通过中心极限定理研究在这些近似过程中产生的渐近误差。在最后一节中，我们使用该模型进行了多项数值分析，特别是分析这些近似值之一相对于 Sabeti 的扩展单倍型纯合性 [18] 的行为。