We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of $\mathbb{R}$. An order-$1$ autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one-step conditional Fr\'echet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalises or extends certain existing models of transportation-based regression and autoregression, and in doing so also provides some additional insights on existing models.

翻译：我们考虑在$\mathbb{R}$的紧区间上定义并拟合概率分布的自回归时间序列模型的问题。在此背景下，一阶自回归模型被理解为马尔可夫链，其中针对一步条件弗雷歇均值（相对于某种自然概率度量）指定了特定结构（回归）。我们基于最优传输映射的迭代随机函数系统构建并探索了不同模型。尽管这些模型的性质和解释取决于它们与迭代传输系统的关联方式，但所有模型均可通过统一方式进行理论分析。我们呈现了此类理论分析（包括收敛速率），并通过实际数据与模拟数据验证了所提方法。我们的方法推广或扩展了某些现有的基于传输的回归与自回归模型，并在此过程中为现有模型提供了额外见解。