Abstract In this paper, we consider a variant of a discrete time Galton–Watson Branching Process in which an individual is allowed to survive for more than one (but finite) number of generations and may also give birth to offsprings more than once. We model the process using multitype branching process and derive conditions on the mean matrix that determines the long-run behavior of the process. Next, we analyze the distribution of the number of forefathers in a given generation. Here, number of forefathers of an individual is defined as all the individuals since zeroth generation who have contributed to the birth of the individual under consideration. We derive an exact expression for expected number of individuals in a given generation having a specified number of forefathers. Using this exact expression, we provide a detailed analysis for a simple illustrative case. Some interesting insights and possible applications are also discussed.

中文翻译：

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Galton-Watson 分支过程的一种变体中的祖先分布

摘要在本文中，我们考虑了离散时间高尔顿-沃森分支过程的一种变体，其中允许个体存活多于一个（但有限）的世代，并且也可能生育不止一次的后代。我们使用多类型分支过程对过程进行建模，并在确定过程长期行为的均值矩阵上推导出条件。接下来，我们分析给定世代中祖先数量的分布。这里，个体的祖先数量定义为自第 0 代以来对所考虑个体的诞生做出贡献的所有个体。我们推导出具有指定数量祖先的给定世代中预期个体数量的精确表达式。使用这个确切的表达，我们为一个简单的说明性案例提供了详细的分析。还讨论了一些有趣的见解和可能的应用。