Rate of convergence for ACF-based statistics under IID white noise

Determine the finite-sample convergence rate of the joint distribution of the sample autocorrelations (\hat{\rho}(1), …, \hat{\rho}(k)) and of derived statistics such as the L1 norm of the first five lags, to their asymptotic multivariate normal limit in the IID Gaussian white noise setting, so that analytical critical values can replace Monte Carlo calibration for the normality/whiteness diagnostics used in the paper.

Background

The paper assesses normality and independence of residuals using skewness, kurtosis, and the L1 norm of the first five lags of the autocorrelation function (ACF). While asymptotic normality of the sample ACF vector is standard, the authors rely on Monte Carlo to compute critical values because they lack guarantees on how quickly the finite-sample distribution approaches its asymptotic limit.

Establishing a concrete convergence rate for the ACF vector and for functionals like the L1 statistic would justify analytical critical values and potentially obviate extensive simulation when testing whiteness and normality, especially for moderate sample sizes.