An important issue in functional time series analysis is whether an observed series comes from a purely random process. We extend the BDS test, a widely-used nonlinear independence test, to the functional time series. Like the BDS test in the univariate case, the functional BDS test can act as the model specification test to evaluate the adequacy of various prediction models and as a nonlinearity test to detect the existence of nonlinear structures in a functional time series after removing the linear structure exhibited. We show that the test statistic from the functional BDS test has the same asymptotic properties as those in the univariate case and provides the recommended range of its hyperparameters. Additionally, empirical data analysis features its applications in evaluating the adequacy of the fAR(1) and fGARCH(1,1) models in fitting the daily curves of cumulative intraday returns (CIDR) of the VIX index. We showed that the functional BDS test remedies the weakness of the existing independence test in the literature, as the latter is restricted in detecting linear structures, thus, can neglect nonlinear temporal structures.

翻译：函数型时间序列分析中的一个重要问题是，观测到的序列是否来自纯随机过程。我们将广泛使用的非线性独立性检验——BDS检验，扩展至函数型时间序列。与单变量情况下的BDS检验类似，函数型BDS检验既可作为模型设定检验，用于评估各种预测模型的充分性；也可作为非线性检验，用于检测函数型时间序列在移除线性结构后是否存在非线性结构。我们证明函数型BDS检验的统计量具有与单变量情况相同的渐近性质，并给出了其超参数的推荐范围。此外，实证数据分析展示了该检验在评估fAR(1)和fGARCH(1,1)模型拟合VIX指数日内累积收益（CIDR）日曲线的充分性方面的应用。我们表明，函数型BDS检验弥补了现有独立性检验的不足，因为后者仅限于检测线性结构，从而可能忽视非线性时间结构。