|f^*(u,\lambda)-f^*(u,0)|\leq C\;f^*(u,0)\;|\lambda|^\beta,\;\lambda\in[-\pi,\pi] \;,$$ where $C>0$ and $\beta\in(0,2]$.

The assumption on the model is summarized hereafter.

\[ass:model\_lin\] The array $\{X_{t,T}\}$ of real-valued random variables has local memory parameter $d(u)\in(-\infty,p+1/2)$ at time $u\in [0,1]$. Moreover $\{\varepsilon_t\}$ in (\[eq:epsZ\]) is a weak white noise such that $\E[\varepsilon_0]=0$, $\mathrm{Var}(\varepsilon_0)=1$, $\E[\varepsilon_t^4]$ is finite for all $t\in\Z$ and the fourth-order cumulants of its spectral representation $\rmd Z(\lambda)$ satisfy $$\label{eq:cum-spectral}
    \mathrm{Cum}\left(\rmd
      Z(\lambda_k),\,1\leq
      k\leq4\right)=\hat{\kappa}_4(\lambda)\;\rmd\mu(\lambda),
    \quad\lambda=(\lambda_k)_{1\leq k\leq 4}\in[-\pi,\pi]^4\;,$$ where $\hat{\kappa}_4(\lambda)=\hat{\kappa}_4(\lambda_1,\lambda_2,\lambda_3)$ is a bounded function defined on $[-\pi,\pi]^3$, and $\mu$ is defined as the measure on $[-\pi,\pi]^4$ such that, for any $(2\pi)$-periodic functions $A_k$, $1\leq k\leq 4$, $$\label{eq:muDef}
    \int_{[-\pi,\pi]^4} \prod_{k=1}^4A_k(\lambda_k)\;\rmd\mu(\lambda)=
\int_{[-\pi,\pi]^3}A_4(-\lambda_1-\lambda_2-\lambda_3)\,
\prod_{k=1}^3A_k(\lambda_k)\;\rmd\lambda\;.$$

Assumption (\[eq:cum-spectral\]) is standard for linear representations of time series and was used by Dahlhaus (for cumulants of all orders) in the original definition of locally stationary processes in [@dahlhaus:1997]. The measure $\mu$ defined by (\[eq:muDef\]) is sometimes denoted as $\rmd\mu(\lambda)=\eta(\lambda_1+\dots+\lambda_4)\rmd\lambda_1\dots\rmd\lambda_4$, where $\eta$ is the $(2\pi)$-periodic Dirac comb, see *e.g.* [@dahlhaus:1997; @dahlhaus-neumann-2001]. An immediate consequence of (\[eq:cum-spectral\]) is the following bound of fourth-order cumulants, for any set of $(2\pi)$-periodic functions $A_k$, $1\leq k\leq
4$, $$\label{eq:cumBound}
  \left|\mathrm{Cum}\left(\int_{-\pi}^\pi A_k(\lambda)\rmd Z(\lambda),\,1\leq k\leq4\right)\right|
  \leq c_4 \;\|A_1\|_2\,\|A_4\|_2\,\|A_2\|_1\|A_3\|_1\;,$$ where $c_4$ is a positive constant and $\|A_k\|_p=(\int_{-\pi}^\pi|A_k(\lambda)|^p\rmd\lambda)^{1/p}$.

We now give a small series of examples, 