of different technical arguments, for instance arguments based on the convergence of cumulants to the appropriate limits, in order to establish asymptotic normality of finite-dimensional distributions (fidis).]{}

Based on Theorem \[t.asy-test\], the procedure to test hypothesis (\[eq:testM\]) is then defined as follows: Reject ${\mathcal H_0}$ if and only if $$\label{eq.stud-test}
t_{\mathcal U} = \frac{\sqrt{b} T \, \mathcal U_T-b^{-1/2} \widehat{\mu}_0}{\widehat{\theta}_0} \geq z_{1-\alpha},$$ where $ z_{1-\alpha}$ is the upper $1-\alpha$ percentage point of the standard Gaussian distribution and $ \widehat{\mu}_0$ and $\widehat{\theta}_0$ are consistent estimators of $ \mu_0$ and $\theta_0$, respectively. Such estimators can be, for instance, obtained if the unknown spectral density kernel $ f_{X,\lambda}$ is replaced by the pooled estimator $ \widehat{f}_{\lambda}(\tau,\sigma) =\widehat{f}_{X,\lambda}(\tau,\sigma)/2 +  \widehat{f}_{Y,\lambda}(\tau,\sigma)/2$. Notice that, under ${\mathcal H_0}$, $ f_{X,\lambda}=f_{Y,\lambda}=f_{X,\lambda} /2+f_{Y,\lambda} /2$, that is (asymptotically), it makes no difference if $ f_{X,\lambda}$ in $\mu_0$ and $\theta_0$ is replaced by $ \widehat{f}_{X,\lambda}$ (or by $ \widehat{f}_{Y,\lambda}$) instead of the pooled estimator $\widehat{f}_\lambda$. However, under ${\mathcal H_1}$ it matters and, for this reason, we use the pooled estimator $\widehat{f}_\lambda(\tau,\sigma)$ in applying the studentized test statistic $t_{\mathcal U}$ defined in (\[eq.stud-test\]); see also Lemma \[le.power\] in Section 4. Under the assumption that the pooled estimator $\widehat{f}_\lambda$ is uniformly consistent, (see also 