of the winter and summer periods, i.e., the curves in each group are transformed by subtracting the corresponding group sample mean functions.

Using the cross-validation algorithm described in Section 5.1, the choice of bandwidth is $b_{CV}=0.075$ (see Figure \[fig.fre-plot\](a)) and the resulting bootstrap $p$-value is equal to 0.030, leading to the rejection of the null hypothesis for almost all commonly used $\alpha$-levels. This implies that dependence properties as measured by autocovariances of the temperature measurements of the winter period differ significantly from those of the summer period.

\[fig.fre-plot\]

In order to understand the reasons leading to this rejection, we decompose the standardized test $ t_{\mathcal U}$ after ignoring the centering sequence $b^{-1/2} \mu_0$ and approximating the integral of the (squared) Hilbert-Schmidt norm by the corresponding Riemann sum over the Fourier frequencies $ \lambda_j=2\pi j/T$, as follows: $$\begin{aligned}
 \label{eq.dec}
\sqrt{b}T \, \mathcal U_T\Big/\theta_0 & \approx  2\pi \sqrt{b}\sum_{j=-N}^{N}  \|\widehat{\mathcal F}_{X,\lambda_j} - \widehat{\mathcal F}_{Y,\lambda_j}\|^2_{HS}\Big/\theta_0 \nonumber \\
& = \sum_{j=-[(T-1)/2]}^{[(T-1)/2]} Q_{T,\lambda_j},\end{aligned}$$ where $$Q_{T,\lambda_j} = 2\pi \sqrt{b} \, \|\widehat{\mathcal F}_{X,\lambda_j} - \widehat{\mathcal F}_{Y,\lambda_j}\|^2_{HS}\Big/\theta_0  \geq 0.$$ Expression (\[eq.dec\]) shows the contributions of the differences $  \|\widehat{\mathcal F}_{X,\lambda_j} - \widehat{\mathcal F}_{Y,\lambda_j}\|^2_{HS} $ for each frequency $ \lambda_j$ to the total value of the test statistic $ \mathcal 