(1/4,1/2)$,

-   $W$ is bounded, symmetric, positive, and Lipschitz continuous, has bounded support on $(-\pi,\pi]$ and satisfies $\int_{-\pi}^{\pi} W(x)\, dx=2\pi$.

Then, under ${\mathcal H_0}$, $$\sqrt{b} T \, \mathcal U_T-b^{-1/2} \mu_0\stackrel{d}{\longrightarrow} Z\sim{\mathcal N}(0,\theta_0^2),$$ where $$\begin{aligned}
\mu_0&=\frac{1}{\pi}\, \int_{-\pi}^\pi\left\{\int_0^1 f_{X,\lambda}(\tau,\tau) \,d\tau\,\right\}^2 d\lambda\int_{-\pi}^\pi W^2\left(u\right)\, du\\
\theta_0^2&=\frac{2}{\pi^2}\int_{-2\pi}^{2\pi}\left\{\int_{-\pi}^\pi W(u)W(u-x)\, du\right\}^2dx\, \int_{-\pi}^\pi \left\{\int_0^1\int_0^1f_{X,\lambda}^2(\sigma, \tau)\,\,d\sigma\, d\tau\right\}^2\, d\lambda.
\end{aligned}$$

Note that the assumptions (i) and (ii) on the weight function $W$ and the bandwidth $b$, respectively, in Theorem \[t.asy-test\] are identical to the assumptions for multivariate time series used in Dette and Paparoditis (2009).

[A careful inspection of the proof of Theorem \[t.asy-test\] shows that the assumption of Gaussianity on the functional innovations $(\varepsilon_t)_{t \in \Z}$ and $(e_t)_{t \in \Z}$ in (\[eq.lin-proc\]) is not essential. In fact, this assumption is solely used to simplify somehow the technical arguments applied in proving asymptotic normality of finite-dimensional distributions associated with the proof of Theorem \[t.asy-test\]. Moreover, this assumption is not required in order to prove convergence of the mean and of the variance of $ \sqrt{b}T \,\mathcal U_T$ or to prove tightness. Consequently, this assumption can be replaced by other assumptions on the stochastic properties of the innovations $(\varepsilon_t)_{t \in \Z}$ and $(e_t)_{t \in \Z}$, which will allow for the use 