Test of independence for functional data

We wish to test the null hypothesis that a collection of functional observations are independent and identically distributed. Our procedure is based on the sum of the L2 norms of the empirical correlation functions. The limit distribution of the proposed test statistic is established under the null hypothesis. Under the alternative the sample exhibits serial correlation, and consistency is shown when the sample size as well as the number of lags used in the test statistic tend to ∞. A Monte Carlo study illustrates the small sample behavior of the test and the procedure is applied to data sets, Eurodollar futures and magnetogram records.