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 Cramer-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 two real data applications are conducted to illustrate the effectiveness and versatility of our proposed method.

翻译：我们提出了一种在度量空间中检验对象值时间序列序列独立性的新方法，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，无需调整参数，且能够捕捉所有非线性成对依赖关系。本文的核心概念是度量空间中的距离协方差，并将其扩展为适用于对象值时间序列的自动距离协方差。此外，我们提出了一种广义谱密度函数，用于刻画所有滞后阶数的成对依赖关系，并构建了Cramer-von Mises型检验统计量。通过建立新的理论论证，我们确定了该检验统计量的渐近行为。同时引入野自助法以获取非枢轴零分布极限的临界值。通过大量数值模拟和两个实际数据应用，验证了所提方法的有效性与普适性。