for all $\sigma, \tau \in [0,1]$. Notice that this distribution can be evaluated by Monte Carlo.

[It is worth mentioning that, by the definition of $ \widehat{\mu}^\ast_0$ and $ \widehat{\theta}^\ast_0$, the bootstrap studentized test $t^\ast_{\mathcal U}$ imitates correctly also the randomness in $t_{\mathcal U}$ which is introduced by replacing the unknown spectral density kernel $ f_{X,\lambda}$ appearing in $ \mu_0$ and $\theta_0$ by its pooled estimator $ \widehat{f}_\lambda$; see (\[eq.stud-test\]). A computationally simpler alternative will be to ignore this asymptotically negligible effect, that is, to use, instead of $t^\ast_{\mathcal U}$ given in (\[tst-fanis\]), the studentized version $t^+_{\mathcal U}=(\sqrt{b} T\,\mathcal U^\ast_T  -b^{-1/2}\widehat{\mu}_0)/\widehat{\theta}_0$ of the bootstrap-based test.]{}

Before describing the asymptotic behavior of the bootstrap test statistic $ \mathcal U^\ast_T$ defined in (\[eq:boot-test-stat\]), we state the following assumption which clarifies our requirements on the pooled spectral density kernel estimator $ \widehat{f}_\lambda$ used.

 : The pooled spectral density kernel estimator $ \widehat{f}_\lambda $ satisfies $$\sup_{\lambda_t\in [0,\pi]} \Big|\int_0^1 \int_0^1 \big(\widehat{f}_{\lambda_t}(\sigma,\tau) - f_{\lambda_t}(\sigma,\tau)\big)d\sigma d\tau \Big|  = o_P(\sqrt{b}), \quad \mbox{as} \;\; T \rightarrow \infty,$$ where $ f_\lambda$ is the spectral density kernel of the pooled spectral density operator $\mathcal F_\lambda=(1/2) \mathcal F_{X,\lambda} + (1/2) \mathcal F_{Y,\lambda} $.

Notice that the above assumption can be easily verified 