Bootstrapping Portmanteau Tests for Functional White Noise under Unknown Dependence

We propose portmanteau tests for functional white noise utilizing the sum of squared empirical autocorrelation functions of functional time series.By applying a Hilbert space approach, we establish the limiting properties of the test under the null hypothesis of uncorrelated but not necessarily independent processes.The test is non-pivotal due to unknown dependence within the sequence.To address this issue, we employ the blockwise random weighting bootstrap to obtain critical values and justify its validity.Furthermore, we extend this method for diagnostics of functional autoregressive model and demonstrate its effectiveness through extensive Monte Carlo simulations and a real data application.An accompanying R package is provided to facilitate checks for general functional white noise.