R_{X,t}\quad\text{and}\quad\mathcal F_{Y,\lambda}=\frac{1}{2\pi}\sum_{t\in\Z} e^{-i\lambda t} \mathcal R_{Y,t},$$ where $\mathcal R_{X,t}$ and $\mathcal R_{Y,t}$ denote the autocovariance operators of $X$ and $Y$ at lag $t$, induced by right integration of $r_{X,t}$ and $r_{Y,t}$, respectively. Convergence holds in nuclear norm.

As it can be seen from the proof of Lemma \[l.ex-SDO\], it suffices to assume 4th order integrability of the innovations $(\varepsilon_t)_{t \in \Z}$ and $(e_t)_{t \in \Z}$ rather than Gaussianity here. The kernels $f_{X,\lambda}$ and $f_{Y,\lambda}$ are called the *spectral density kernels* (at frequency $\lambda$) and the operators $\mathcal F_{X,\lambda}$ and $\mathcal F_{Y, \lambda}$ are referred to as the corresponding [*spectral density operators*]{}.

Under the assumptions of Lemma \[l.ex-SDO\], we can now state the hypothesis testing problem of interest as follows $$\begin{aligned}
&{\mathcal H_0}\colon \mathcal F_{X,\lambda}=\mathcal F_{Y, \lambda} \quad\text{ for }\mu\text{-}\text{almost all } \lambda\in (-\pi,\pi], 
 \quad\text{versus}\\
&{\mathcal H_1}\colon \mathcal F_{X, \lambda}\neq\mathcal F_{Y, \lambda} \quad \forall\lambda\in A \text{ for some $A$ with }\mu(A)>0.
\end{aligned}
\label{eq:testM}$$

The Test Statistic and its Asymptotic Behavior {#S3}
==============================================

We first estimate the unknown spectral density operator $\mathcal F_{X, \lambda}$ by an integral operator $\widehat{\mathcal F}_{X,\lambda}$ induced by right-integration with the kernel $$\hat f_{X,\lambda}(\sigma,\tau)=\frac{1}{b T}\sum_{t=-N}^{N} W\left(\frac{\lambda-\lambda_t}{b}\right)\, \widehat p_{X, \lambda_t} (\sigma, \tau), \;\; \mbox{for all} \;\; \sigma,\tau \in [0,1],$$ 