Random variables in metric spaces indexed by time and observed at equally spaced time points are receiving increased attention due to their broad applicability. However, the absence of inherent structure in metric spaces has resulted in a literature that is predominantly non-parametric and model-free. To address this gap in models for time series of random objects, we introduce an adaptation of the classical linear autoregressive model tailored for data lying in a Hadamard space. The parameters of interest in this model are the Fr\'echet mean and a concentration parameter, both of which we prove can be consistently estimated from data. Additionally, we propose a test statistic and establish its asymptotic normality, thereby enabling hypothesis testing for the absence of serial dependence. Finally, we introduce a bootstrap procedure to obtain critical values for the test statistic under the null hypothesis. Theoretical results of our method, including the convergence of the estimators as well as the size and power of the test, are illustrated through simulations, and the utility of the model is demonstrated by an analysis of a time series of consumer inflation expectations.

翻译：在度量空间中由时间索引且以等间隔时间观测的随机变量因其广泛适用性而日益受到关注。然而，度量空间缺乏固有结构导致现有文献主要为非参数和无模型方法。为填补随机对象时间序列模型领域的这一空白，我们引入对经典线性自回归模型的改进，专门适配于哈达玛空间中的数据类型。该模型的感兴趣参数包括弗雷歇均值与集中参数，我们证明两者均可通过数据得到一致估计。此外，我们提出一个检验统计量并建立其渐近正态性，从而实现对序列无关性的假设检验。最后，我们引入自助法过程以获取原假设下检验统计量的临界值。通过模拟验证了包括估计量收敛性及检验的水平和功效在内的理论结果，并通过消费者通胀预期时间序列的分析展示了该模型的实用性。