We consider tests of serial independence for a sequence of functional observations. The new methods are formulated as L2-type criteria based on empirical characteristic functions and are convenient from the computational point of view. We derive asymptotic normality of the proposed test statistics for both discretely and continuously observed functions. In a Monte Carlo study, we show that the new test is sensitive with respect to functional GARCH alternatives, investigate the choice of necessary tuning parameters, and demonstrate that critical values obtained by resampling lead to a test with good performance in any setup, whereas the asymptotic critical values may be recommended only for a sufficiently fine discretization grid. Finite-sample comparison with a distance (auto)covariance test criterion is also included, and the article concludes with application on a real data set.

我们考虑一系列功能观察的序列独立性测试。这些新方法基于经验特征函数被公式化为L2型准则，并且从计算角度来看很方便。我们得出离散和连续观测函数的拟议测试统计量的渐近正态性。在蒙特卡洛（Monte Carlo）的一项研究中，我们证明了新测试对功能性GARCH替代品很敏感，研究了必要的调节参数的选择，并证明了通过重采样获得的临界值可以使测试在任何设置下均具有良好的性能，而仅对于足够精细的离散化网格，才可能建议渐近临界值。还包括带有距离（自动）协方差检验标准的有限样本比较，