论文标题
在不同环境中分支树的合并点过程
Coalescent point process of branching trees in varying environment
论文作者
论文摘要
考虑目前的任意人口，该人口源于过去的一个未指定的任意时间，同一世代的个人独立地，时间向前繁殖，具有相同的后代分布，但有可能改变世代。换句话说，繁殖是由在不同环境中的加尔顿 - 瓦特森过程驱动的。当前一代的族谱是及时倒退的唯一确定的是融合点进程$（a_i，i \ geq 1）$，其中$ a_i $是个人$ i $和$ i $和$ i+1 $之间的合并时间。通常，这个过程不是马尔可夫。在恒定环境中，兰伯特和波波维奇（Lambert and Popovic，2013年）提出了马尔可夫的点措施过程，以重建合并点过程。我们提出了一个反例，我们表明他们的过程没有马尔可夫属性。这项工作的主要贡献是提出一个有价值的马尔可夫流程$（b_i，i \ geq 1）$，该$达到了重建家谱的目标，每$ $ $ $ $ $ $。此外，当后代分布是线性分数时，我们表明变量$（a_i，i \ geq 1）$是独立的，并且分布相同。
Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals in the same generation reproduce independently, forward in time, with the same offspring distribution but potentially changing among generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i, i\geq 1)$, where $A_i$ is the coalescent time between individuals $i$ and $i+1$. In general, this process is not Markov. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We present a counterexample where we show that their process does not have the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\geq 1)$, that reach the goal to reconstruct the genealogy, with finite information for every $i$. Additionally, when the offspring distributions are lineal fractional, we show that the variables $(A_i, i\geq 1)$ are independent and identically distributed.
