This lecture presents a unified framework for white noise and diagnostic checking in time series analysis, tracing its methodological evolution from the time domain to the frequency domain and from finite-dimensional Euclidean spaces to infinite-dimensional Hilbert spaces.
Starting from the time-domain portmanteau framework for weak vector autoregressive models, we show that conventional test statistics often suffer from substantial size distortions due to unknown error dependence structures and parameter estimation effects. To overcome these limitations, we develop a blockwise random-weighting bootstrap procedure, which accurately approximates the null distribution and its asymptotic validity is justified. Extending this idea to the frequency domain，a Cramér–von Mises type statistic is constructed based on the discrepancy between the residual periodogram and its theoretical constant, enabling the detection of long-range dependence beyond finite lags while maintaining robustness to estimation uncertainty. Finally, the framework is generalized to functional time series in a Hilbert space setting, where portmanteau-type statistics based on squared empirical autocorrelation operators are employed to assess functional white noise and model adequacy.
Collectively, these developments provide a coherent and theoretically justified approach that robustly mitigates the impact of unknown dependence and estimation uncertainty in time series models.
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