H_0$, $ \mu_0=\widetilde{\mu}_0$ and $\theta^2_0=\widetilde{\theta}_0^2$ since $ \mathcal F_{X, \lambda}= \mathcal F_{Y, \lambda} $ (or, respectively, $ f_{X,\lambda}=f_{Y, \lambda}$). Thus, in this case, the asymptotic behavior of the test statistics $\mathcal U_T$ and $\mathcal U^\ast_T$ is identical, that is, the bootstrap procedure estimates consistently the distribution of the test statistic $ \mathcal U_T$ under $\mathcal H_0$. Furthermore, under $\mathcal  H_1$, the following holds true.

\[le.power\] Suppose that the conditions of Theorem \[t.boo-test\] are satisfied. Then, under $\mathcal H_1$, as $ T \rightarrow \infty$, $$\begin{aligned}
t_{\mathcal U} & =  
 \sqrt{b}T \int_{-\pi}^{\pi}\|{\mathcal F}_{X,\lambda} -{\mathcal F}_{Y,\lambda}\|^2_{HS} d\lambda + o_P(\sqrt{b} T) \ \rightarrow +\infty, \quad \mbox{in probability}.\end{aligned}$$

The above result, together with Theorem \[t.boo-test\] and Slusky’s theorem, imply that the power of the studentized test $t_{\mathcal U}$ based on the bootstrap critical values obtained from the distribution of the bootstrap studentized test $t^\ast_{\mathcal U}$ converges to unity as $ T\rightarrow \infty$, i.e., the studentized test $t_{\mathcal U}$ is consistent.

Numerical Results
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Choice of the Smoothing Parameter
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Implementing the studentized test $t_{\mathcal U}$ requires the choice of the smoothing bandwidth $b$. For univariate and multivariate time series, this issue has been investigated in the context of a cross-validation type criterion by Beltrão and 