Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

中文翻译：

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Wald's martingale 和 Moran 过程中吸收时间的条件分布

许多进化模型是随机过程，其中某些兴趣量随时间随机波动。一个经典的例子是 Moranbirth-death 过程，其中的数量是种群中突变体的数量。在这样的过程中，我们经常对它们的吸收（即固定）概率和吸收时间的条件分布感兴趣。这些条件时间分布可能非常难以计算，即使对于像莫兰生死模型这样相对简单的过程也是如此。我们不考虑吸收时间，而是考虑一个密切相关的数量：吸收前突变种群大小变化的数量。我们使用 Wald 鞅来获得 Moran 过程中该量的条件特征函数。我们的表达新颖、分析和准确，并且它们的参数依赖性是明确的。我们使用我们的结果来近似吸收时间的条件特征函数。我们陈述了该近似特别准确的条件。Martingales 是一个优雅的框架，用于解决进化随机过程的主要问题。它们不要求我们评估递归关系，因此当它们适用时，我们可以快速、轻松地获得进化模型的吸收概率和次数。