We propose a novel method for testing serial independence of object-valued time series in metric spaces, which is more general than Euclidean or Hilbert spaces. The proposed method is fully nonparametric, free of tuning parameters, and can capture all nonlinear pairwise dependence. The key concept used in this paper is the distance covariance in metric spaces, which is extended to auto distance covariance for object-valued time series. Furthermore, we propose a generalized spectral density function to account for pairwise dependence at all lags and construct a Cram\'er-von Mises type test statistic. New theoretical arguments are developed to establish the asymptotic behavior of the test statistic. A wild bootstrap is also introduced to obtain the critical values of the non-pivotal limiting null distribution. Extensive numerical simulations and three real data applications are conducted to illustrate the effectiveness and versatility of our proposed method.

翻译：本文提出了一种新颖的方法，用于检验度量空间中对象值时间序列的序列独立性，其适用范围比欧几里得空间或希尔伯特空间更为广泛。该方法完全非参数化、无需调节参数，并能捕捉所有非线性成对依赖关系。本文核心概念采用度量空间中的距离协方差，并将其扩展为对象值时间序列的自动距离协方差。进一步，我们提出了一种广义谱密度函数，用以衡量所有滞后阶数下的成对依赖关系，并构建了克拉默-冯·米塞斯型检验统计量。本文发展了新的理论论证，以建立该检验统计量的渐近行为。同时引入野生自助法，用于获取非枢轴极限零分布的关键值。通过大量数值模拟及三项真实数据应用，验证了所提方法的有效性与通用性。