We discuss the finite sample theoretical properties of online predictions in non-stationary time series under model misspecification. To analyze the theoretical predictive properties of statistical methods under this setting, we first define the Kullback-Leibler risk, in order to place the problem within a decision theoretic framework. Under this framework, we show that a specific class of dynamic models -- random walk dynamic linear models -- produce exact minimax predictive densities. We first show this result under Gaussian assumptions, then relax this assumption using semi-martingale processes. This result provides a theoretical baseline, under both non-stationary and stationary time series data, for which other models can be compared against. We extend the result to the synthesis of multiple predictive densities. Three topical applications in epidemiology, climatology, and economics, confirm and highlight our theoretical results.

翻译：我们讨论了模型误设下非平稳时间序列在线预测的有限样本理论性质。为分析该设定下统计方法的理论预测特性，我们首先定义Kullback-Leibler风险，将问题置于决策理论框架中。在此框架下，我们证明特定类型的动态模型——随机游走动态线性模型——能生成精确的极小化极大预测密度。我们先在高斯假设下证明该结论，随后利用半鞅过程放宽这一假设。该结果为非平稳与平稳时间序列数据提供了理论基准，便于与其他模型进行比较。我们将该结论扩展至多个预测密度的合成。流行病学、气候学与经济学三个领域的应用实例验证并凸显了我们的理论成果。