论文标题
亚临界连续分支种群中祖先谱系的渐近行为
Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations
论文作者
论文摘要
考虑与亚临界连续分支过程（CSBP）相关的无限规模的人口模型。个体根据相同的亚临界后代分布独立繁殖。随着时间的流逝，我们研究了祖先谱系的长期行为，并表明祖先谱系的流动适当重新归一化，几乎可以肯定地收敛到无漂移的下属的倒数，其laplace指数在分支机制方面是明确的。我们根据人口的家谱提供了一种解释。特别是，我们表明，逆下属正在将当前人口分为具有不同祖先的祖先家庭。当满足灰色的状况时，人口来自一组离散的祖先，祖先家族是I.I.D，并根据CSBP的准平台分布进行分配，以不可扩展为条件。如果不满足Gray的条件，人口将来自祖先的连续体，该祖先被描述为限制逆下属的增加点$ \ MATHSCR {S} $。给出了$ \ mathscr {s} $的Hausdorff尺寸。该证明是基于随机性单调的独立利益过程的一般结果，该过程将$θ$ - invariant度量与$θ$ -Invariant函数相关联，用于该过程及其siegmund dual。
Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we show that the inverse subordinator is partitioning the current population into ancestral families with distinct common ancestors. When Grey's condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families are i.i.d and distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey's condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points $\mathscr{S}$ of the limiting inverse subordinator. The Hausdorff dimension of $\mathscr{S}$ is given. The proof is based on a general result for stochastically monotone processes of independent interest, which relates $θ$-invariant measures and $θ$-invariant functions for a process and its Siegmund dual.
