Reconstructing the ancestral state of a group of species helps answer many important questions in evolutionary biology. Therefore, it is crucial to understand when we can estimate the ancestral state accurately. Previous works provide a necessary and sufficient condition, called the big bang condition, for the existence of an accurate reconstruction method under discrete trait evolution models and the Brownian motion model. In this paper, we extend this result to a wide range of continuous trait evolution models. In particular, we consider a general setting where continuous traits evolve along the tree according to stochastic processes that satisfy some regularity conditions. We verify these conditions for popular continuous trait evolution models including Ornstein-Uhlenbeck, reflected Brownian Motion, and Cox-Ingersoll-Ross.

翻译：重建一组物种的祖先状态有助于回答进化生物学中的许多重要问题。因此，理解何时能够准确估计祖先状态至关重要。先前的研究提供了一种充分必要条件，即大爆炸条件，用于判断在离散性状进化模型和布朗运动模型下是否存在精确的重建方法。在本文中，我们将这一结果扩展到广泛的连续性状进化模型。具体而言，我们考虑了一个通用设置：连续性状沿系统发育树依据满足某些正则条件的随机过程进化。我们验证了这些条件在流行的连续性状进化模型中的适用性，包括奥恩斯坦-乌伦贝克过程、反射布朗运动以及考克斯-英格索尔-罗斯过程。