Recovering a unique causal graph from observational data is an ill-posed problem because multiple generating mechanisms can lead to the same observational distribution. This problem becomes solvable only by exploiting specific structural or distributional assumptions. While recent work has separately utilized time-series dynamics or multi-environment heterogeneity to constrain this problem, we integrate both as complementary sources of heterogeneity. This integration yields unified necessary identifiability conditions and enables a rigorous analysis of the statistical limits of recovery under thin versus heavy-tailed noise. In particular, temporal structure is shown to effectively substitute for missing environmental diversity, possibly achieving identifiability even under insufficient heterogeneity. Extending this analysis to heavy-tailed (Student's t) distributions, we demonstrate that while geometric identifiability conditions remain invariant, the sample complexity diverges significantly from the Gaussian baseline. Explicit information-theoretic bounds quantify this cost of robustness, establishing the fundamental limits of covariance-based causal graph recovery methods in realistic non-stationary systems. This work shifts the focus from whether causal structure is identifiable to whether it is statistically recoverable in practice.

翻译：从观测数据中恢复唯一的因果图是一个不适定问题，因为多种生成机制可能导致相同的观测分布。该问题仅能通过利用特定的结构或分布假设才可求解。尽管近期研究已分别利用时间序列动态或多环境异质性来约束此问题，但我们将二者整合为互补的异质性来源。这种整合产生了统一的必要可识别性条件，并使得在薄尾与厚尾噪声下恢复的统计极限得以严格分析。特别地，时间结构被证明能有效替代缺失的环境多样性，甚至在异质性不足时仍可能实现可识别性。将此分析扩展至厚尾（学生t）分布，我们证明尽管几何可识别性条件保持不变，但样本复杂度显著偏离高斯基线。显式的信息论界量化了这种鲁棒性代价，从而确立了现实非平稳系统中基于协方差的因果图恢复方法的基本极限。本研究将关注点从因果结构是否可识别转向其在实践中是否具有统计可恢复性。