When can we reconstruct the ancestral state? Beyond Brownian motion

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, bounded Brownian Motion, and Cox-Ingersoll-Ross.