#  Gaussian Approximation for Lag-Window Estimators and the Construction of
Confidence Bands for the Spectral Density

Jens-Peter Kreiß  Anne Leucht  Efstathios Paparoditis  Institut für
Mathematische Stochastik, TU Braunschweig, 38106 Braunschweig, Germany.
E-mail<EMAIL_ADDRESS>Institut für Statistik, Universität
Bamberg, 96052 Bamberg, Germany. E-mail<EMAIL_ADDRESS>Department of Mathematics and Statistics, University of Cyprus, P.O.Box 20537,
CY-1678 Nicosia, Cyprus. E-mail<EMAIL_ADDRESS>TU Braunschweig,
Universität Bamberg, University of Cyprus

###### Abstract

In this paper we consider the construction of simultaneous confidence bands
for the spectral density of a stationary time series using a Gaussian
approximation for classical lag-window spectral density estimators evaluated
at the set of all positive Fourier frequencies. The Gaussian approximation
opens up the possibility to verify asymptotic validity of a multiplier
bootstrap procedure and, even further, to derive the corresponding rate of
convergence. A small simulation study sheds light on the finite sample
properties of this bootstrap proposal.

62G20,

62G09, 62G15,

Bootstrap,

Confidence Bands,

Gaussian Approximation,

Sample Autocovariance,

Spectral Density,

###### keywords:

[class=MSC]

###### keywords:

##  1 Introduction

We consider the problem of constructing (simultaneous) confidence bands for
the spectral density of a stationary time series. Toward this goal we develop
Gaussian approximation results for the maximum deviation over all (positive)
Fourier frequencies of a lag-window spectral density estimator and for its
bootstrap counterpart. Based on observations $X_{1},X_{2},\ldots,X_{T}$
stemming from a strictly stationary and centered stochastic process
$\\{X_{t},t\in\mbox{$\mathbb{Z}$}\\}$, a lag-window estimator of the spectral
density $f(\lambda)$, $\lambda\in[0,\pi]$, is given by

$\widehat{f}_{T}(\lambda)=\frac{1}{2\pi}\sum_{|j|\leq
M_{T}}w(j/M_{T})\,\mathrm{e}^{-ij\lambda}\,\widehat{\gamma}(j),~{}\lambda\in[0,\pi].$
(1)

In (1) and for $j=0,1,2,\ldots,M_{T}<T$,

$\widehat{\gamma}(j)=\frac{1}{T}\sum_{t=j+1}^{T}X_{t}\,X_{t-j}$ (2)

are estimators of the autocovariances $\gamma(j)=\mathrm{Cov}(X_{0},X_{j})$ of
the process $\\{X_{t},t\in\mbox{$\mathbb{Z}$}\\}$ and
$\widehat{\gamma}(j)=\widehat{\gamma}(-j)$ for $j<0$. The function
$w:[-1,1]\rightarrow{\mathbb{R}}$ is a so-called lag-window, which assigns
weights to the $M_{T}$ sample autocovariances effectively used in the
calculation of the estimator for $f(\lambda)$ and which satisfies some
assumptions to be specified later.

It is well-known that for a fixed frequency $\lambda$, under suitable
assumptions on the dependence structure of the underlying time series and for
$M_{T}$ converging not too fast to infinity, asymptotic normality for lag-
window estimators can be shown. To elaborate and assuming sufficient
smoothness of $f$, which is equivalent to assuming a sufficiently fast decay
of the underlying autocovariances $\gamma(h)$ as $|h|\to\infty$, one typically
can show under certain conditions on the lag-window $w$ and for
$M_{T}\to\infty$ such that $T/M_{T}^{5}\to C^{2}\geq 0$ as $n\to\infty$, that

$\sqrt{\frac{T}{M_{T}}}\Big{(}\widehat{f}_{T}(\lambda)-f(\lambda)\Big{)}\xrightarrow{\mathcal{D}}{\mathcal{N}}\Big{(}C\,W\,f^{\prime\prime}(\lambda),\,f^{2}(\lambda)\int_{-1}^{1}w^{2}(u)\,du\Big{)},$
(3)

for $0<\lambda<\pi$, where $f^{\prime\prime}$ denotes the second derivative of
$f$ and $W=\lim_{u\rightarrow 0}(1-w(u))/u^{2}$, where the latter is assumed
to exist and to be positive.

Moreover, for the estimator $\widehat{f}_{T}(\lambda)$ with sophisticated
selected so-called flat-top lag-windows $w$ and under the additional
assumption that $\sum_{j\in\mathbb{Z}}|j|^{r}|\gamma(j)|<\infty$ for some
$r\geq 1$ convergence rates for the mean squared error
$\mbox{MSE}(\widehat{f}_{T}(\lambda))=O(T^{-2r/(2r+1)})$ can be achieved,
which corresponds to $M_{T}\sim T^{1/(2r+1)}$. See PolitisRomano87 and
BergPolitis2009 for details.

Instead of point-wise inference we aim in this paper for simultaneous
inference using the estimator $\widehat{f}_{T}$. More precisely, we aim for
confidence bands covering the spectral density uniformly at all positive
Fourier frequencies $\lambda_{k,T}=2\pi k/T,k=1,\ldots,\lfloor
T/2\rfloor=:N_{T}$ with a desired (high) probability. Hence, we keep all
information contained in $X_{1},\dots,X_{T}$ in the sense that the discrete
Fourier transform of any vector $x\in\mbox{$\mathbb{R}$}^{T}$ is a linear
combination of trigonometric polynomials at exactly these frequencies.

Note that so far simultaneous confidence bands for the spectral density had
only been derived for special cases. In particular, for autoregressive
processes relying on parametric spectral density estimates (see e.g. NP84 and
T87) while NP08 proposed a bootstrap-aided approach for Gaussian time series
using the integrated periodogram to construct simultaneous confidence bands
for (a smoothed version of) the spectral density. Generalizing results from
LiuWu2010, YZ22 derived simultaneous confidence bands for the spectral density
on a grid with mesh size wider than $2\pi/T$ and for locally stationary
Bernoulli shifts satisfying a geometric moment contraction condition. To this
end, they proved that the maximum deviation of a suitably standardized lag-
window spectral density estimator over such a grid asymptotically possesses a
Gumbel distribution. Motivated by the slow rate of convergence of the maximum
deviation to a Gumbel variable, they propose an asymptotically valid bootstrap
method to improve the finite sample behavior of their confidence regions.

As mentioned above, we aim to construct simultaneous confidence bands on the
finer grid of all (positive) Fourier frequencies. To achieve this goal, we
establish a Gaussian approximation result for the distribution of a properly
standardized statistic based on the random quanity

$\max_{1\leq k\leq N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|.$

In this context and in order to stabilize the asymptotic variance and to
construct confidence bands that automatically adapt to the local variability
of $f(\lambda)$, we are particularly interested in deriving a Gaussian
approximation for the normalized statistic

$\max_{1\leq k\leq
N_{T}}\frac{\displaystyle|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|}{\displaystyle\widehat{f}_{T}(\lambda_{k,T})}.$

While various Gaussian approximations of max-type statistics in the time
domain have been derived during the last decades, there are only few results
exists for frequency domain statistics. CCH22 investigated the maximum of
periodograms for functional data. A Gaussian approximation result for high
dimensional spectral density matrices of $\alpha$-mixing time series has been
derived in Changetal23. However, they require the dimension of the process to
increase with sample size making their results not applicable for our
purposes.

The paper is organized as follows. Section 2 summarizes key assumptions for
our results. The core material on Gaussian approximation for spectral density
estimators is contained in Section 3. In Section 4 we discuss application of
the Gaussian approximation results to the construction of simultaneous
confidence bands over the positive Fourier frequencies. Section 5 reports the
results of a small simulation study. Auxiliary lemmas and proofs are deferred
to Section 6.

##  2 Set up and Assumptions

We begin by imposing the following assumption on the dependence structure of
the stochastic process generating the observed time series
$X_{1},X_{2},\ldots,X_{T}$.

Assumption 1: $\\{X_{t},t\in\mbox{$\mathbb{Z}$}\\}$ is a strictly stationary,
centered process and $X_{t}=g(e_{t},e_{t-1},\ldots)$ for some measurable
function $g$ and an i.i.d. process $\\{e_{t},t\in\mbox{$\mathbb{Z}$}\\}$ with
mean zero and variance $0<\sigma^{2}_{e}<\infty$. For $s\geq 0$, denote by
$\mathcal{F}_{r,s}=\sigma(e_{r},e_{r-1},\dots,e_{r-s})$ the $\sigma$-algebra
generated by the random variables $\\{e_{r-j},0\leq j\leq s\\}$ and assume
that $E|X_{t}|^{m}<\infty$ for some $m>16$. For a random variable $X$ let
$\|X\|_{m}:=(E|X|^{m})^{1/m}$ and define for an independent copy
$\\{e_{t}^{\prime},t\in\mbox{$\mathbb{Z}$}\\}$ of
$\\{e_{t};t\in\mbox{$\mathbb{Z}$}\\}$,

$\delta_{m}(k):=\|X_{t}-g(e_{t},e_{t-1},\ldots,e_{t-k+1},e^{\prime}_{t-k},e_{t-k-1},\ldots)\|_{m}.$

The assumption is that

$\delta_{m}(k)\leq C\,(1+k)^{-\alpha}$ (4)

for some $\alpha>3$ and a constant $C<\infty$.

Assumption 1 implies that the autocovariance function
$\gamma(h)=\mathrm{Cov}(X_{0},X_{h}),$ $h\in\mbox{$\mathbb{Z}$},$ is absolute
summable, that is that the process $\\{X_{t},t\in\mbox{$\mathbb{Z}$}\\}$
possesses a continuous and bounded spectral density
$f(\lambda)=(2\pi)^{-1}\sum_{h\in\mbox{$\mathbb{Z}$}}\gamma(h)e^{-ih\lambda}$,
$\lambda\in(-\pi,\pi]$.
It can be even shown that (4) implies
$\sum_{j\in\mathbb{Z}}|j|^{r}|\gamma(j)|<\infty$ for $r<\alpha-1$, see Lemma 1
in Section 6. As has been mentioned in the Introduction this allows for the
option of convergence rates for $\widehat{f}_{T}(\lambda)$ of order
$O(T^{-r/(2r+1)})$, $r<\alpha-1$. Note that our assumptions on the decay of
the dependence coefficients are less restrictive than those in YZ22, where
exponentially decaying coefficients are presumed. We assume for simplicity
that $EX_{t}=0$. Our results can be generalized to non-centered time series
using the modified autocovariance estimator
$\widetilde{\gamma}(j)=\frac{1}{T}\sum_{t=j+1}^{T}(X_{t}-\overline{X})(X_{t-j}-\overline{X})\
\ \mbox{with}\ \ \overline{X}=\sum_{t=1}^{T}X_{t}/T$, instead of the estimator
$\widehat{\gamma}(j)$ defined in (2).

Additional to Assumption 1 we also require the following boundedness condition
for the spectral density $f$.

Assumption 2: The spectral density $f$ satisfies
$\inf_{\lambda\in(-\pi,\pi]}f(\lambda)>0$.

Finally, we impose the following conditions on the lag-window function $w$
used in obtaining the estimator $\widehat{f}_{T}$ and which are standard in
the literature; see Priestley81, Chapter 6.

Assumption 3: The lag-window $w\colon[-1,1]\to\mbox{$\mathbb{R}$}$ is assumed
to be a differentiable and symmetric function with $\int_{-1}^{1}w(u)\,du=1$
and $w(0)=1$.

##  3 Gaussian Approximation for Spectral Density Estimators

For the following considerations and in order to separate any bias related
problems, we consider the centered sequence

$\displaystyle\sqrt{\frac{T}{M_{T}}}\Big{(}\widehat{f}_{T}(\lambda_{k,T})-E\,\widehat{f}_{T}(\lambda_{k,T})\Big{)}$
$\displaystyle=\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\sum_{j=0}^{M_{T}}a_{k,j}\big{(}X_{t}X_{t-j}-\gamma(j)\big{)}\,\mathbbm{1}_{{t>j}}$
$\displaystyle=:\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,k},$ (5)

with an obvious abbreviation for $Z_{t,k}$ and where for $\ k=1,\ldots,N_{T}$,

$a_{k,j}:=\begin{cases}\frac{\displaystyle 1}{\displaystyle
2\pi}\cdot\frac{\displaystyle 1}{\displaystyle\sqrt{M_{T}}}&,\;j=0\\\
\frac{\displaystyle 1}{\displaystyle\pi}\cdot\frac{\displaystyle
1}{\displaystyle\sqrt{M_{T}}}w(j/M_{T})\cos(j\lambda_{k,T})&,\;j\geq
1.\end{cases}$

Due to the differentiability of the lag-window $w$ (cf. Assumption 3), we have

$\displaystyle\max_{k=1,\ldots,N_{T}}\sum_{j=0}^{M_{T}}a_{k,j}^{2}$
$\displaystyle\leq\frac{1}{4\pi^{2}}\frac{1}{M_{T}}+\frac{1}{\pi^{2}M_{T}}\sum_{j=1}^{M_{T}}w^{2}(j/M_{T})$
$\displaystyle=\frac{1}{\pi^{2}}\int_{0}^{1}w^{2}(u)\,du+{\mathcal{O}}(M_{T}^{-1}).$

The following theorem is our first result and establishes a valid Gaussian
approximation for the maximum over all positive Fourier frequencies of the
centered estimator given in (1).

######  Theorem 1.

Suppose that $\\{X_{t},t\in\mathbb{Z}\\}$ fulfils Assumption 1 and 2. Let
$\widehat{f}_{T}$ be a lag-window estimator of $f$ as given in (1), where the
lag-window $w$ satisfies Assumption 3 and let $M_{T}\rightarrow\infty$,
$M_{T}=o(T)$ as $T\rightarrow\infty$ and $M_{T}\sim T^{a_{s}}$. Assume further
that

$\displaystyle\Big{\|}\,\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,k}\,\Big{\|}_{2}^{2}>c>0.$
(6)

Let $\xi_{k},k=1,\ldots,N_{T}$, be jointly normally distributed random
variables with zero mean and covariance $E\xi_{k_{1}}\xi_{k_{2}}$ equal to

$\frac{1}{T}E\Big{[}\Big{(}\sum_{j=0}^{M_{T}}\sum_{t=j+1}^{T}a_{k_{1},j}\big{(}X_{t}X_{t-j}-\gamma(j)\big{)}\Big{)}\Big{(}\sum_{j=0}^{M_{T}}\sum_{t=j+1}^{T}a_{k_{2},j}\big{(}X_{t}X_{t-j}-\gamma(j)\big{)}\Big{)}\Big{]}.$

Let $\lambda$, $a_{s}$, and $a_{l}$ be positive constants such that

$\alpha>\min\Big{\\{}1+\frac{a_{l}}{2a_{s}}\,,\,\frac{1-a_{l}-\frac{12}{m}-4\lambda-\max\\{\frac{4}{m},\,2\lambda\\}}{2a_{s}}+\frac{3}{2}\Big{\\}}$
(7)

and

$0<a_{s}+\max\\{4\lambda,2\lambda+4/m\\}<a_{l}<1-6\lambda-12/m.$ (8)

Then,

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\Big{|}\widehat{f}_{T}(\lambda_{k,T})-E\,\widehat{f}_{T}(\lambda_{k,T})\Big{|}\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}|\xi_{k}|\leq x\big{)}\Big{|}$
$\displaystyle={\mathcal{O}}\Big{(}T^{-\kappa}+T^{-\lambda}\big{(}\log(N_{T})\big{)}^{3/2}\Big{)},$
(9)

where $\kappa=\min\\{\kappa_{1},\kappa_{2}\\}$ with

$\kappa_{1}=a_{l}/2-a_{s}/2-\lambda-\max\\{2/m\,,\,\lambda\\}\ \mbox{and}\ \
\kappa_{2}=1/2-a_{l}/2-6/m-3\lambda.$

Some remarks regarding Theorem 1 are in order.

######  Remark 1.

The lower bound in (6) is valid for sufficiently large $T$. This results from
our assumption that $f$ is bounded away from zero (see Assumption 2), together
with

$\displaystyle\sup_{k\in\\{1,\dots,N_{T}\\}}\Big{|}\big{\|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,k}\big{\|}_{2}^{2}-f^{2}(\lambda_{k,T})\,\frac{1}{M_{T}}\sum_{j=-M_{T}}^{M_{T}}w^{2}\big{(}\frac{j}{M_{T}}\big{)}\,(1+\cos(2j\,\lambda_{k,T}))\Big{|}$
$\displaystyle=o(1),$

see Lemma 2 in Section 6.

######  Remark 2.

  * (i)

Notice first that the rate in the Gaussian approximation (1) does not depend
on the rate of decay of the dependence coefficients. Moreover, as can be seen
by an inspection of the proof of Theorem 1, even if the underlying time series
$\\{X_{t},t\in\mbox{$\mathbb{Z}$}\\}$ consists of i.i.d. observations, i.e.
$X_{t}=e_{t}$ for all $t$, we do not achieve better approximation rates using
our method of proof.

  * (ii)

If we assume that the number $m$ of moments assumed to exist and the
convergence rate for the underlying lag-window estimator
$a_{s}=1/(2r+1),\;r\in\mathbb{N},$ are fixed, then we need to optimize the
rate in the Gaussian approximation depending on $\lambda$ and $\alpha_{l}$. To
do so we first ignore log-terms and secondly we divide the consideration into
two cases according to the parameter $m$, namely Case I: $2/m\geq\lambda$
(moderate values of $m$) and Case II: $2/m\leq\lambda$ (large values of $m$).
For Case I we can find the resulting rate, which is the minimum of
$\max\\{-\lambda,\lambda+2/m+(a_{s}-a_{l})/2,3\lambda+6/m+(a_{l}-1)/2\\}$ by
balancing the three terms. This leads to $a_{l}=1/3+2/3\,a_{s}-4/(3m)$,
$\lambda=1/12(1-a_{s})-4/(3m)$ and the resulting rate
$T^{-r/(6(2r+1))+4/(3m)}$.
Exactly along the same lines we obtain for Case II that
$\lambda=1/14(1-a_{s})-6/(7m)$ and the resulting rate
$T^{-r/(7(2r+1))+6/(7m)}$. Both choices are in line with (8).
Hence, for usual lag-window estimators, i.e. $a_{s}=1/5$, and if sufficiently
high moments of the time series are assumed to exist, then we can achieve
rates up to $T^{-2/35}$ for the Gaussian approximation.
In the limit $a_{s}\to 0$, we achieve the rate $T^{-1/14+6/(7m)}$, which
reaches, if sufficiently high moments exist, almost the rate $T^{-1/14}$.

  * (iii)

If we would instead of sample autocovariances consider sample means of i.i.d.
observations, which would make the small blocks/large blocks considerations in
the proof of Theorem 1 superfluous, we could achieve with the presented method
of proof a Gaussian approximation with rate $T^{-1/8+3/(2m)}$.

##  4 Confidence Bands for the Spectral Density

To construct confidence bands that appropriately take into account the local
variability of the lag-window estimator, we make use of the following Gaussian
approximation result for the properly standardized lag-window estimators.

######  Theorem 2.

Suppose that the assumptions of Theorem 1 are satisfied. Further, assume that
$M_{T}^{3}/T\to 0$, where additionally to the conditions given in (7),

$\lambda+\kappa\leq\min\\{1/2-a_{s}/2-4/m\,,\,2a_{s}-4/m\\}$ (10)

holds true. Moreover, the lag-window is assumed to fulfil $\lim_{u\to
0}(1-w(u))/u^{2}>0$. Then,

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\Big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\Big{)}-P\Big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\Big{)}\Big{|}$ $\displaystyle=$
$\displaystyle\,{\mathcal{O}}\Big{(}T^{-\kappa}+T^{-\lambda}\big{(}\log(N_{T})\big{)}^{3/2}\Big{)},$

where $\widetilde{\xi}_{k},k=1,\ldots,N_{T}$, are jointly normally distributed
random variables with zero mean and covariance
$E\widetilde{\xi}_{k_{1}}\widetilde{\xi}_{k_{2}}$ equal to

$\displaystyle
C_{T}(k_{1},k_{2})=\frac{1}{f(\lambda_{k_{1},T})f(\lambda_{k_{2},T})}$
$\displaystyle\sum_{j_{1},j_{2}=0}^{M_{T}}a_{k_{1},j_{1}}a_{k_{2},j_{2}}$
$\displaystyle\times\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}$
$\displaystyle
E\Big{[}\big{(}X_{t}X_{t-j_{1}}-\gamma(j_{1})\big{)}\big{(}X_{s}X_{s-j_{2}}-\gamma(j_{2})\big{)}\Big{]}.$
(11)

######  Remark 3.

Choosing $a_{s}$ MSE optimal, that is $a_{s}=1/5$, the corresponding optimal
$\lambda=\kappa=2/35$ (see Remark 2) satisfy (10).

Theorem 2 motivates the following multiplier bootstrap procedure to construct
a confidence band for the smoothed spectral density

$\widetilde{f}_{T}(\cdot):=E(\widehat{f}_{T}(\cdot)).$

Step 1.

For $k_{1},k_{2}\in\\{1,2,\ldots,N_{T}\\}$, let $\widehat{C}_{T}(k_{1},k_{2})$
be an estimator of the covariance $C_{T}(k_{1},k_{2})$ given in (2) which
ensures that the $N_{T}\times N_{T}$ matrix $\widehat{\Sigma}_{N_{T}}$ with
the $(i,j)$-th element equal to $\widehat{C}_{T}(i,j)$ is non-negative
definite.

Step 2.

Generate random variables
$\xi^{\ast}_{1},\xi^{\ast}_{2},\ldots,\xi^{\ast}_{N_{T}}$, where

$\big{(}\xi^{\ast}_{1},\xi^{\ast}_{2},\ldots,\xi^{\ast}_{N_{T}}\big{)}^{\top}\sim{\mathcal{N}}\Big{(}0_{N_{T}},\widehat{\Sigma}_{N_{T}}\Big{)},$

with $0_{N_{T}}$ a $N_{T}$-dimensional vector or zeros and covariance matrix
$\widehat{\Sigma}_{N_{T}}$. Let
$\xi^{\ast}_{\max}=\max\big{\\{}|\xi^{\ast}_{1}|,|\xi^{\ast}_{2}|,\ldots,|\xi^{\ast}_{N_{T}}|\big{\\}}$
and for $a\in(0,1)$ given, denote by $q^{\ast}_{1-a}$ the upper $(1-a)$
percentage point of the distribution of $\xi^{\ast}_{\max}$.

Step 3.

A simultaneous $(1-2a)$-confidence band for $\widetilde{f}(\lambda_{j,T})$,
$j=1,2,\ldots,N_{T}$, is then given by

$\Big{\\{}\Big{[}\widehat{f}(\lambda_{j,T})\Big{(}1-q^{\ast}_{1-a}\sqrt{\frac{M_{T}}{T}}\Big{)},\
\
\widehat{f}(\lambda_{j,T})\Big{(}1+q^{\ast}_{1-a}\sqrt{\frac{M_{T}}{T}}\Big{)}\Big{]},\
j=1,2,\ldots,N_{T}.\Big{\\}}$ (12)

Notice that the distribution of $\xi^{\ast}_{\max}$ in Step 2 can be estimated
via Monte-Carlo simulation. While $f(\lambda_{k_{i},T})$, $i=1,2$, appearing
in the expression for $C_{T}(k_{1},k_{2})$ can be replaced by the estimator
$\widehat{f}_{T}(\lambda_{k_{i},T})$, the important step in the bootstrap
algorithm proposed, is the estimation of the covariances

$\sigma_{T}(j_{1},j_{2})=\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}E\Big{[}\big{(}X_{t}X_{t-j_{1}}-\gamma(j_{1})\big{)}\big{(}X_{s}X_{s-j_{2}}-\gamma(j_{2})\big{)}\Big{]}$

appearing in $C_{T}(k_{1},k_{2})$. To obtain an estimator for this quantity
which has some desired consistency properties, we follow Zhang_etal2022. In
particular, let $K$ be a kernel function satisfying the following conditions.

Assumption 4: $K:\mbox{$\mathbb{R}$}\to[0,+\infty)$ is symmetric, continuously
differentiable and decreasing on $[0,+\infty)$ with $K(0)=1$ and $\int
K(x)\,dx<\infty$. The Fourier transform of $K$ is integrable and nonnegative
on $\mathbb{R}$.

Consider next the estimator

$\widehat{\sigma}_{T}(j_{1},j_{2})=\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\big{(}X_{t}X_{t-j_{1}}-\widehat{\gamma}(j_{1})\big{)}\big{(}X_{s}X_{s-j_{2}}-\widehat{\gamma}(j_{2})\big{)},$
(13)

of $\sigma_{T}(j_{1},j_{2})$, where $b_{T}>0$ is a bandwidth parameter
satisfying $b_{T}\rightarrow\infty$ as $T\rightarrow\infty$. Note that the
conditions on the Fourier transform of $K$ in Assumption 4 guarantee positive
semi-definiteness of $\widehat{\sigma}_{T}(j_{1},j_{2})$. This assumption is
satisfied if $K$ is, for instance, the Gaussian kernel.

The following consistency result for the estimator proposed in (13) can be
established.

######  Proposition 1.

Suppose that Assumption 1 and Assumption 4 hold true. Let $b_{T}\sim T^{c}$
with $c<1/2-8a_{s}/m$, where $a_{s}$ is as in Theorem 2. Then,

$\sup_{1\leq j_{1},j_{2}\leq
M_{T}}\big{|}\widehat{\sigma}_{T}(j_{1},j_{2})-\sigma_{T}(j_{1},j_{2})\big{|}=O_{P}\Big{(}\frac{1}{b_{T}}+\frac{b_{T}M_{T}^{8/m}}{\sqrt{T}}\Big{)}.$

Let, as usual, $P^{*}$ denote the conditional probability given the time
series $X_{1},\dots,$ $X_{T}$. We then have the following result which proves
consistency of the multiplier bootstrap procedure proposed.

######  Theorem 3.

Suppose that the conditions of Theorem 2 hold true and that $b_{T}\sim T^{c}$,
where

$0<a_{s}<c<\frac{1}{2}-a_{s}\big{(}1+\frac{8}{m}\big{)}.$

Let $\xi_{k}^{\ast}$, $k=1,2,\ldots,N_{T}$, be Gaussian random variables
generated as in Step 2 of the multiplier bootstrap algorithm with
$\widehat{\Sigma}_{N_{T}}$ the $N_{T}\times N_{T}$ matrix the
$(k_{1},k_{2})$-th element of which equals

$\displaystyle\widehat{C}_{T}(k_{1},k_{2})=\frac{1}{\widehat{f}(\lambda_{k_{1},T})\widehat{f}(\lambda_{k_{2},T})}$
$\displaystyle\sum_{j_{1},j_{2}=0}^{M_{T}}a_{k_{1},j_{1}}a_{k_{2},j_{2}}\widehat{\sigma}_{T}(j_{1},j_{2}),$
(14)

and $\widehat{\sigma}_{T}(j_{1},j_{2})$ given in (13). Then, as
$n\rightarrow\infty$,

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}$
$\displaystyle\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\big{)}-P^{\ast}\big{(}\max_{k=1,\ldots,N_{T}}|\xi^{\ast}_{k}|\leq
x\big{)}\Big{|}$ $\displaystyle=$
$\displaystyle\,{\mathcal{O}}\Big{(}T^{-\kappa}+T^{-\lambda}\big{(}\log(N_{T})\big{)}^{3/2}\Big{)}$
$\displaystyle+o_{P}\big{(}\\{\sup_{k=1,\ldots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|\\}^{1/6}+T^{-\rho/6}\big{)}$
(15)

for $\rho=\min\\{c-a_{s},1/2-c-a_{s}(1+8/m)\\}$.

######  Remark 4.

  * (i)

So far we have considered the construction of (simultaneous) confidence bands
for the (smoothed) spectral density
$\widetilde{f}(\lambda_{j,T})=\mathrm{E}(\widehat{f}_{T}(\lambda_{j,T}))$ over
the Fourier frequencies $\lambda_{j,T}$, $j=1,2,\ldots,N_{T}$. To extend the
procedure proposed to one that also delivers an asymptotically valid
simultaneous confidence band for the spectral density $f$ itself, a
(uniformly) consistent estimator of the (rescaled) bias term
$B_{T}(\lambda_{j,T})=\sqrt{T/M_{T}}\big{(}\mathrm{E}(\widehat{f}_{T}(\lambda_{j,T}))-f(\lambda_{j,T})\big{)}/\widehat{f}_{T}(\lambda_{j,T})$
for $j=1,2,\ldots,N_{T}$, is needed, provided $B_{T}(\lambda_{j,T})$ does not
vanish asymptotically; see (3). If fact, it can be easily seen that a
(theoretically) valid confidence band for $f$ which takes into account the
bias in estimating $f$, is given by

$\displaystyle\Big{\\{}\Big{[}\widehat{f}(\lambda_{j,T})\Big{(}1-\sqrt{\frac{M_{T}}{T}}\big{(}q^{\ast}_{1-a}+B_{T}(\lambda_{j,T}\big{)}\Big{)},$
$\displaystyle\ \
\widehat{f}(\lambda_{j,T})\Big{(}1+\sqrt{\frac{M_{T}}{T}}\big{(}q^{\ast}_{1-a}-B_{T}(\lambda_{j,T}\big{)}\Big{)}\Big{]},$
$\displaystyle j=1,2,\ldots,N_{T}\Big{\\}};$ (16)

compare to (12). As in many other nonparametric inference problems too,
different approaches can be considered for this purpose; see Calonicoetal2018
and the references therein for the cases of nonparametric density and
regression estimation in an i.i.d. set up. One approach is an explicit bias
correction that uses a plug-in type estimator of $B_{T}(\lambda_{j,T})$ based
on the fact that, under certain conditions, see (3),
$\sqrt{T/M_{T}}\big{(}\mathrm{E}(\widehat{f}_{T}(\lambda_{j,T}))-f(\lambda_{j,T})\big{)}=CWf^{\prime\prime}(\lambda_{j,T})+o(1)$.
This approach requires a consistent (nonparametric) estimator of the second
order derivative $f^{\prime\prime}$ of the spectral density. An alternative
approach proposed in the literature is the so-called ’undersmoothing’. The
idea here is to make the bias term
$\sqrt{T/M_{T}}\big{(}\mathrm{E}(\widehat{f}_{T}(\lambda_{j,T}))-f(\lambda_{j,T})\big{)}$
asymptotically negligible. This can be achieved by using a truncation lag
$M_{T}$ which increases to infinity at a rate faster than the (MSE optimal)
rate $T^{1/5}$. Despite the fact that the problem of properly incorporating
the (possible) bias in the construction of (simultaneous) confidence bands for
the spectral density $f$ is an interesting one, we do not further pursue this
problem in this paper.

  * (ii)

(3) contains as part of the convergence rate for the multiplier bootstrap the
sup-distance of the lag-window estimator to the spectral density over all
Fourier frequencies $\lambda_{k,T}$. Assuming a geometric rate for the decay
of the physical dependence coefficients $\delta_{m}(k)$, one obtains from
WuZaffaroni2018, Theorem 6, the uniform convergence rate
$(M_{T}\log(M_{T})/T)^{1/2}$ over all frequencies for lag-window estimators,
which dominates the uniform term over all Fourier frequencies in (3) and
usually converges to zero faster than the other parts of the bound therein.

##  5 Simulations

In this section we investigate by means of simulations, the finite sample
performance of the Gaussian approximation and of the corresponding multiplier
bootstrap procedure proposed for the construction of simultaneous confidence
bands. For this purpose, time series of length $n=128,\;256$ and $512$ have
been generated from the following three time series models:

  1. Model I: $X_{t}=0.8X_{t-1}+\varepsilon_{t}$,

  2. Model II: $X_{t}=1.3X_{t-1}-0.75X_{t-2}+u_{t}$ with $u_{t}=\varepsilon_{t}\sqrt{1+0.25u^{2}_{t-1}}$,

  3. Model III: $X_{t}=\big{(}0.4+0.4\varepsilon_{t-1}\big{)}X_{t-1}+\varepsilon_{t}$.

In all models the innovations $\varepsilon_{t}$ are chosen to be i.i.d.
standard Gaussian. The empirical coverage over $R=500$ repetitions of each
model has been calculated for two different nominal coverages, $90\%$ and
$95\%$. The lag-window estimator of the spectral density used has been
obtained using the Parzen lag-window and truncation lag $M_{T}$. The Gauss
kernel with different values of the parameter $b_{T}$ has been used for
obtaining the covariance estimators $\widehat{\sigma}_{T}(j_{1},j_{2})$ given
in (13) and which are used for the calculation of the covariance matrix
$\widehat{C}_{T}(k_{1},k_{2})$; see equation (14). All bootstrap
approximations are based on $B=1,000$ replications. Table 1 summarizes the
results obtained.

|  | Model I | Model II | Model III
---|---|---|---|---
n=128 |  |  |  |  |  |  |  |  |  |
$M_{T}=10$ |  $b_{T}$= | 3 | 5 | 7 | 18 | 20 | 22 | 18 | 20 | 22
| 90% | 91.2 | 88.0 | 86.4 | 92.2 | 91.6 | 89.6 | 87.2 | 87.6 | 84.0
| 95% | 93.4 | 91.0 | 89.0 | 94.2 | 93.0 | 92.0 | 90.8 | 91.4 | 89.8
n=256 |  |  |  |  |  |  |  |  |  |
$M_{T}=12$ |  $b_{T}$= | 4 | 6 | 8 | 22 | 24 | 26 | 22 | 24 | 26
| 90% | 93.8 | 93.0 | 88.8 | 93.4 | 91.2 | 89.6 | 92.6 | 91.8 | 91.4
| 95% | 95.0 | 95.2 | 90.6 | 95.2 | 94.0 | 94.2 | 95.6 | 94.0 | 94.2
n=512 |  |  |  |  |  |  |  |  |  |
$M_{T}=14$ |  $b_{T}$= | 8 | 10 | 12 | 36 | 38 | 40 | 36 | 38 | 40
| 90% | 91.8 | 91.4 | 91.6 | 92.0 | 93.4 | 92.6 | 95.4 | 97.4 | 97.4
| 95% | 93.2 | 93.6 | 94.4 | 94.4 | 96.0 | 95.2 | 97.6 | 98.0 | 98.0

Table 1. Empirical coverages of simultaneous confidence bands for different
sample sizes and different values of the parameters $M_{T}$ and $b_{T}$ for
the three time series models considered.

It can be seen from Table 1 that our confidence bands reach the desired
coverage levels. The procedure is robust against the choice of $M_{T}$
specifying the number of empirical autocovariances evaluated in
$\widehat{f}_{T}$ across all models. However, it is more sensitive with
respect to the choice of the bandwidth $b_{T}$ involved in the estimation of
the variance matrix of the approximating Gaussian variables. The bandwidth
should be chosen larger for process with a stronger dependence structure
compared to rather weakly dependent data. This is a standard observation in
the context of covariance estimation, see e.g. A91.

##  6 Auxiliary Lemmas and Proofs

Throughout this section, $C$ denotes a generic constant that may vary from
line to line. We first state the following useful lemmas. See also XiaoWu2014
for related results to Lemma 1.

######  Lemma 1.

Under Assumption 1, the following assertions hold true:

  1. (i)

$|\gamma(j)|\leq C(1+|j|)^{-\alpha}$,

  2. (ii)

$\sum_{j\in\mbox{$\mathbb{Z}$}}|j|^{r}|\gamma(j)|<\infty~{}\forall
r<\alpha-1$,

  3. (iii)

$\max_{j_{1},j_{2}\geq 0}|E(Y_{t,j_{1}}Y_{s,j_{2}})|\leq
C(1+|s-t|)^{-\alpha}$, where $Y_{t,j}=X_{t}X_{t+j}-\gamma(j)$.

###### Proof.

  * (i)

We make use of the bound

$\|P_{0}(X_{s})\|_{m}\leq\delta_{m}(s),s\geq 0,$ (17)

of the so-called projection operator $P_{j-s}(X_{j}):=E[X_{j}|{\cal
F}_{j-s}]-E[X_{j}|{\cal F}_{j-s-1}]$ for $s\geq 0$ and $j\in\mathbb{Z}$, where
${\cal F}_{i}:=\sigma(e_{i},e_{i-1},\ldots)$. For a proof of (17) we refer to
Wu2005, Theorem 1.
Note that $X_{t}=\sum_{s=0}^{\infty}P_{t-s}(X_{t})$ a.s. and in $L_{1}$. This
is the case because $E[X_{t}|{\cal F}_{t-s}]$ converges for $s\to\infty$ by
the backward martingale convergence theorem a.s. and in $L_{1}$ towards a
limit measurable with respect to ${\cal F}_{-\infty}$, which is trivial
because of the i.i.d. structure of $(e_{i})$. Therefore the limit is constant
and coincides with the mean of $X_{t}$ which is assumed to be zero. Then (i)
follows since,

$\displaystyle|\gamma(j)|=$
$\displaystyle|E(X_{0}\,X_{j})|=\Big{|}\sum_{s_{1},s_{2}=0}^{\infty}E(P_{-s_{1}}(X_{0})P_{j-s_{2}}(X_{j}))\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\sum_{s=0}^{\infty}E(P_{-s}(X_{0})P_{-s}(X_{j}))\Big{|},\
\text{because }E(P_{-s_{1}}(X_{0})P_{j-s_{2}}(X_{j}))=0,-s_{1}\neq j-s_{2}$
$\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}E|P_{0}(X_{s})P_{0}(X_{j+s})|,\ \text{because
of stationarity}$ $\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}\|P_{0}(X_{s})\|_{m^{\prime}}\|P_{0}(X_{j+s})\|_{m},\
\text{where }1/m+1/m^{\prime}=1$ $\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}\|P_{0}(X_{s})\|_{m}\|P_{0}(X_{j+s})\|_{m},\
\text{because }m^{\prime}\leq m\text{ for }m\geq 2$ $\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}\delta_{m}(s)\delta_{m}(j+s),\text{ by
}\eqref{B-13a}.$

From this bound we get for $j\geq 0$,

$\displaystyle|\gamma(j)|\leq
C\sum_{s=0}^{\infty}\frac{1}{(1+s)^{\alpha}(1+s+j)^{\alpha}}\leq
C\frac{1}{(1+j)^{\alpha}}\sum_{s=0}^{\infty}\frac{1}{(1+s)^{\alpha}}\leq
C(1+j)^{-\alpha}.$

  * (ii)

The assertion follows because

$\sum_{j=1}^{\infty}j^{r}|\gamma(j)|\leq
C\sum_{j=1}^{\infty}j^{r}(1+j)^{-\alpha}\leq
C\sum_{j=1}^{\infty}\frac{1}{(1+j)^{\alpha-r}}<\infty$

for $r<\alpha-1$.

  * (iii)

We assume without loss of generality that $t>s$ and define
$X_{r}^{(k)}\,:=\,E[X_{r}\mid\mathcal{F}_{r,k}]$. The following three cases
can then occur. If $j_{1}\leq t-s$, then

$\displaystyle|E(Y_{t,j_{1}}Y_{s,j_{2}})|$
$\displaystyle\leq|E(X_{t}X_{t-{j_{1}}}X_{s}X_{s-j_{2}}|+|\gamma(j_{1})\gamma(j_{2})|$
$\displaystyle=|E(\big{(}X_{t}-X_{t}^{(t-s-1)})X_{t-j_{1}}X_{s}X_{s-j_{2}}\big{)}|+|\gamma(j_{1})\gamma(j_{2})|$
$\displaystyle\leq\|X_{t-j_{1}}X_{s}X_{s-j_{2}}\|_{m/(m-1)}\|X_{t}-X_{t}^{(t-s-1)}\|_{m}+C(1+j_{1})^{-\alpha}$
$\displaystyle\leq C\,(1+t-s)^{-\alpha}.$

If $j_{1}\in[(t-s)/2,t-s]$, then

$\displaystyle|E(Y_{t,j_{1}}Y_{s,j_{2}})|$
$\displaystyle\leq|E(X_{t}X_{t-{j_{1}}}X_{s}X_{s-j_{2}}|+|\gamma(j_{1})\gamma(j_{2})|$
$\displaystyle=|E(\big{(}X_{t}-X_{t}^{(t-j_{1}-1)})X_{t-j_{1}}X_{s}X_{s-j_{2}}\big{)}|+|\gamma(j_{1})\gamma(j_{2})|$
$\displaystyle\leq\|X_{t-j_{1}}X_{s}X_{s-j_{2}}\|_{m/(m-1)}\|X_{t}-X_{t}^{(t-j_{1}-1)}\|_{m}+C(1+j_{1})^{-\alpha}$
$\displaystyle\leq C(1+j_{1})^{-\alpha}\leq C(1+(t-s)/2)^{-\alpha}\leq
C2^{\alpha}(1+t-s)^{-\alpha}.$

Finally, if $j_{1}\in[0,(t-s)/2]$, then

$\displaystyle|E(Y_{t,j_{1}}Y_{s,j_{2}})|$
$\displaystyle\leq|E((Y_{t,j_{1}}-Y_{t,j_{1}}^{(t-s-1)})Y_{s,j_{2}})|$
$\displaystyle\leq\|Y_{t,j_{1}}-Y_{t,j_{1}}^{(t-s-1)}\|_{m/(m-2)}\|Y_{s,j_{2}}\|_{m/2}$
$\displaystyle\leq C\sum_{k=t-s}^{\infty}\delta_{m}(k)+C\sum_{k=t-s-
j_{1}}^{\infty}\delta_{m}(k)$ $\displaystyle\leq
C(1+t-s)^{-\alpha}+C\sum_{k=(t-s)/2}^{\infty}\delta_{m}(k)$ $\displaystyle\leq
C\,2^{\alpha}(1+t-s)^{-\alpha}.$

∎

######  Lemma 2.

Suppose that Assumptions 1 to 3 hold. Then, we have

$\displaystyle\big{\|}\frac{1}{\sqrt{T}}\sum_{j=0}^{M_{T}}\sum_{t=j+1}^{T}a_{k,j}\big{(}X_{t}X_{t-j}-\gamma(j)\big{)}\big{\|}_{m/2}\leq\mbox{C}$
(18)

and

$\displaystyle\sup_{k\in\\{1,\dots,N_{T}\\}}\Big{|}\big{\|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,k}\big{\|}_{2}^{2}-\,\frac{f^{2}(\lambda_{k,T})}{M_{T}}\sum_{j=-M_{T}}^{M_{T}}w^{2}\big{(}\frac{j}{M_{T}}\big{)}\,(1+\cos(2j\,\lambda_{k,T}))\Big{|}$
$\displaystyle=o(1).$ (19)

Further

$\displaystyle\Big{\|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}(Z_{t,k}-\widetilde{Z}_{t,k}^{(s)})\Big{\|}_{m/2}\,\leq\,\mbox{C}\cdot
d_{s,m}$ (20)

with the $m$-dependent random variables ($m=2s$) $\widetilde{Z}_{t,k}^{(s)}$,
$k=1,2,\ldots,N_{T},$ defined as:

$\widetilde{Z}_{t,k}^{(s)}\,:=\,\sum_{j=0}^{M_{T}}a_{k,j}\,\left(X_{t}^{(s)}X_{t-j}^{(s)}\,-\,E[X_{t}^{(s)}X_{t-j}^{(s)}]\right)\,\mathbbm{1}_{t>j}\,,$
(21)

where $X_{r}^{(s)}\,:=\,E[X_{r}\mid\mathcal{F}_{r,s}]$ and $s\geq M_{T}$, and
with

$d_{s,m}\,:=\,\sum_{h=0}^{\infty}\min\Big{\\{}\delta_{m}(h)\,,\,\big{(}\sum_{j=s+1}^{\infty}\delta_{m}^{2}(j)\big{)}^{1/2}\Big{\\}}.$
(22)

###### Proof.

Inequality (18) is a direct consequence of (S8) from the supplement of
XiaoWu2014 since

$\displaystyle\big{\|}\sum_{j=0}^{M_{T}}\sum_{t=j+1}^{T}a_{k,j}\big{(}X_{t}X_{t-j}-\gamma(j)\big{)}\big{\|}_{m/2}$
$\displaystyle=$
$\displaystyle\big{\|}\sum_{i,j=1,i-j\in\\{0,\ldots,M_{T}\\}}^{T}a_{k,i-j}\big{(}X_{i}X_{j}-\gamma(i-j)\big{)}\big{\|}_{m/2}$
$\displaystyle\leq$
$\displaystyle\mbox{C}\cdot\Big{(}\sum_{h=0}^{\infty}\delta_{m}(h)\Big{)}^{2}\max_{k=1,\ldots,N_{T}}\left(\sum_{j=0}^{M_{T}}a_{k,j}^{2}\right)^{1/2}\,\sqrt{T}.$

For (2), first note that Theorem S.6 in the supplement of XiaoWu2014 assures
summability of the joint fourth order cumulants. Hence,

$\displaystyle\big{\|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,k}\big{\|}_{2}^{2}$
(23)
$\displaystyle=\frac{1}{4\pi^{2}TM_{T}}\sum_{j_{1},j_{2}=-M_{T}}^{M_{T}}\sum_{t_{1}=|j_{1}|+1}^{T}\sum_{t_{2}=|j_{2}|+2}^{T}w\big{(}\frac{j_{1}}{M_{T}}\big{)}w\big{(}\frac{j_{2}}{M_{T}}\big{)}\,\cos(j_{1}\lambda_{k,T})\,\cos(j_{2}\lambda_{k,T})$
$\displaystyle\qquad\times\gamma(t_{1}-t_{2})\gamma(t_{1}-t_{2}+|j_{2}|-|j_{1}|)$
$\displaystyle\quad+\frac{1}{4\pi^{2}TM_{T}}\sum_{j_{1},j_{2}=-M_{T}}^{M_{T}}\sum_{t_{1}=|j_{1}|+1}^{T}\sum_{t_{2}=|j_{2}|+1}^{T}w\big{(}\frac{j_{1}}{T}\big{)}w\big{(}\frac{j_{2}}{T}\big{)}\,\cos(j_{1}\lambda_{k,T})\,\cos(j_{2}\lambda_{k,T})$
$\displaystyle\qquad\times\gamma(t_{1}-t_{2}+|j_{2}|)\gamma(t_{1}-t_{2}-|j_{1}|)+R^{(1)}_{T,k}$

with $\sup_{k}|R_{T,k}^{(1)}|=o(1)$. For the first summand on the r.h.s. we
get from $\sum_{k\in\mbox{$\mathbb{N}$}}k|\gamma(k)|<\infty$ that

$\displaystyle\frac{1}{4\pi^{2}TM_{T}}\sum_{j_{1},j_{2}=-M_{T}}^{M_{T}}\sum_{t_{1}=|j_{1}|+1}^{T}\sum_{t_{2}=|j_{2}|+1}^{T}w\big{(}\frac{j_{1}}{T}\big{)}w\big{(}\frac{j_{2}}{T}\big{)}\,\cos(j_{1}\lambda_{k,T})\,\cos(j_{2}\lambda_{k,T})$
$\displaystyle\qquad\times\gamma(t_{1}-t_{2})\gamma(t_{1}-t_{2}+|j_{2}|-|j_{1}|)$
$\displaystyle=\frac{1}{\pi^{2}TM_{T}}\sum_{j_{1},j_{2}=1}^{M_{T}}\sum_{t_{1}=j_{1}+1}^{T}\sum_{t_{2}=j_{2}+1}^{T}w\big{(}\frac{j_{1}}{M_{T}}\big{)}w\big{(}\frac{j_{2}}{M_{T}}\big{)}\,\cos(j_{1}\lambda_{k,T})\,\cos(j_{2}\lambda_{k,T})$
$\displaystyle\qquad\times\gamma(t_{1}-t_{2})\gamma(t_{1}-t_{2}+j_{2}-j_{1})+R^{(2)}_{T,k}$
$\displaystyle=\frac{1}{\pi^{2}M_{T}}\sum_{t\in\mbox{$\mathbb{Z}$}}\sum_{j\in\mbox{$\mathbb{Z}$}}\gamma(t)\gamma(t+j)\sum_{j_{1}=1\vee(1-j)}^{M_{T}\wedge(M_{T}-j)}w\big{(}\frac{j_{1}}{M_{T}}\big{)}w\big{(}\frac{j+j_{1}}{M_{T}}\big{)}\,\cos(j_{1}\lambda_{k,T})$
$\displaystyle\qquad\times\,\cos((j+j_{1})\lambda_{k,T})+R^{(3)}_{T,k}$
$\displaystyle=\frac{1}{2\pi^{2}M_{T}}\sum_{t\in\mbox{$\mathbb{Z}$}}\sum_{j\in\mbox{$\mathbb{Z}$}}\gamma(t)\gamma(t+j)\sum_{j_{1}=1}^{M_{T}}w^{2}\big{(}\frac{j_{1}}{M_{T}}\big{)}\,$
$\displaystyle\qquad\times\,\big{[}\cos(j\lambda_{k,T})\,+\,\cos((j+2j_{1})\lambda_{k,T})\big{]}+R^{(4)}_{T,k}$
$\displaystyle=f^{2}(\lambda_{k,T})\,\frac{2}{M_{T}}\sum_{j_{1}=1}^{M_{T}}w^{2}\big{(}\frac{j_{1}}{M_{T}}\big{)}\,(1+\cos(2j_{1}\lambda_{k,T}))+R^{(5)}_{T,k},$

where $\sup_{k}|R_{T,k}^{(\ell)}|=o(1),~{}\ell=2,\dots,5.$ Similarly, we
obtain from $\sum_{k\in\mbox{$\mathbb{N}$}}k^{2}|\gamma(k)|<\infty$ for the
second summand on the r.h.s. of (23)

$\displaystyle\frac{1}{4\pi^{2}TM_{T}}\sum_{j_{1},j_{2}=-M_{T}}^{M_{T}}\sum_{t_{1}=|j_{1}|+1}^{T}\sum_{t_{2}>|j_{2}|+1}^{T}w\big{(}\frac{j_{1}}{T}\big{)}w\big{(}\frac{j_{2}}{T}\big{)}\,\cos(j_{1}\lambda_{k,T})\,\cos(j_{2}\lambda_{k,T})$
$\displaystyle\qquad\times\gamma(t_{1}-t_{2}+|j_{2}|)\gamma(t_{1}-t_{2}-|j_{1}|)$
$\displaystyle=\frac{1}{\pi^{2}M_{T}}\sum_{t\in\mbox{$\mathbb{Z}$}}\sum_{j=2}^{2M_{T}}\gamma(t)\gamma(t-j)\,\sum_{j_{1}=1\vee(j-M_{T})}^{M_{T}\wedge(j-1)}w\big{(}\frac{j_{1}}{T}\big{)}w\big{(}\frac{j-j_{1}}{T}\big{)}\,\cos(j_{1}\lambda_{k,T})\,$
$\displaystyle\qquad\times\,\cos((j-j_{1})\lambda_{k,T})\,+R_{T,k}^{(6)}$
$\displaystyle=\frac{1}{\pi^{2}M_{T}}\sum_{t\in\mbox{$\mathbb{Z}$}}\sum_{j=2}^{\sqrt{M_{T}}}\gamma(t)\gamma(t-j)\,\sum_{j_{1}=1}^{\sqrt{M_{T}}-1}w\big{(}\frac{j_{1}}{T}\big{)}w\big{(}\frac{j-j_{1}}{T}\big{)}\,\cos(j_{1}\lambda_{k,T})\,$
$\displaystyle\qquad\times\,\cos((j-j_{1})\lambda_{k,T})\,+R_{T,k}^{(7)}$
$\displaystyle=R_{T,k}^{(8)},$

where $\sup_{k}|R_{T,k}^{(\ell)}|=o(1),~{}\ell=6,\,7,\,8.$ This finishes the
proof of (2).
Inequality (20) in turn can be deduced from Proposition 1 in LiuWu2010 as
follows

$\Big{\|}\sum_{t=1}^{T}(Z_{t,k}-\widetilde{Z}_{t,k}^{(s)})\Big{\|}_{m/2}\leq\mbox{C}\,\sqrt{T}\,d_{s,m}\,\max_{k=1,\ldots,N_{T}}\left(\sum_{j=0}^{M_{T}}a_{k,j}^{2}\right)^{1/2}\,\sum_{h=0}^{\infty}\delta_{m}(h).$

∎

###### Proof of Theorem 1.

We mainly follow the strategy of the proof of Theorem 1 in Zhang_etal2022
although some of the conditions used there are not fulfilled in our case. In
particular, a different $m$-dependent approximation
$\widetilde{Z}^{(s)}_{t,k}$, i.e. $m=2s$, is used in our proof which then
leads to an improved rate of convergence. Using the notation $h_{\tau,\tau,x}$
as in Zhang_etal2022, we get the bound

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\Big{\\{}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\Big{|}\widehat{f}_{T}(\lambda_{k,T})-E\,\widehat{f}_{T}(\lambda_{k,T})\Big{|}\leq
x\Big{\\}}$
$\displaystyle\quad\quad-P\Big{\\{}\max_{k=1,\ldots,N_{T}}|\xi_{k}|\leq
x\Big{\\}}\Big{|}$ $\displaystyle\leq$
$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\,\Big{|}Eh_{\tau,\tau,x}\Big{(}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,1},\dots,\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,N_{T}}\Big{)}-Eh_{\tau,\tau,x}\left(\xi_{1},\dots,\xi_{N_{T}}\right)\Big{|}$
$\displaystyle\quad\quad+C\,t\,\left(1+\sqrt{\log(N_{T})}+\sqrt{|\log(t)|}\right),$
(24)

in view of (6) and (18), where $t=(1+\log(2N_{T}))/\tau$ for some $\tau>0$ to
be specified later. For the $m$-dependent random variables (with $m=2s$)
$\widetilde{Z}_{t,k}^{(s)}$, $k=1,2,\ldots,N_{T}$ , defined in (21), by the
properties of the function $h_{\tau,\tau,x}$ and from Lemma 2, we get

$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\,\Big{|}Eh_{\tau,\tau,x}\Big{(}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,1},\dots,\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z_{t,N_{T}}\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-Eh_{\tau,\tau,x}\Big{(}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\widetilde{Z}^{(s)}_{t,1},\dots,\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z^{(s)}_{t,N_{T}}\Big{)}\Big{|}$
$\displaystyle\leq
C\,\tau\,E\Big{[}\max_{k=1,\dots,N_{T}}\Big{|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}(Z_{t,k}-\widetilde{Z}_{t,k}^{(s)})\Big{|}\Big{]}$
$\displaystyle\leq
C\,\tau\,N_{T}^{2/m}\,\max_{k=1,\dots,N_{T}}\Big{\|}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}(Z_{t,k}-\widetilde{Z}_{t,k}^{(s)})\Big{\|}_{m/2}$
$\displaystyle\leq C\,\tau\,N_{T}^{2/m}\,d_{s,m}$ (25)

with $d_{s,m}$ given in (22). The introduction of the random variables
$\widetilde{Z}^{(s)}_{t,k}$ allows us to proceed with the classical big
block/small block technique. Toward this, we define for any integer $l>2s$ big
and small blocks of the $\widetilde{Z}_{t,k}^{(s)}$ as

$S_{j,k}\,:=\,\frac{1}{\sqrt{T}}\sum_{t=2(j-1)(s+l)+1}^{2(j-1)(s+l)+2l}\widetilde{Z}_{t,k}^{(s)}\quad\text{and}\quad
U_{j,k}\,:=\,\frac{1}{\sqrt{T}}\sum_{t=2(j-1)(s+l)+2l+1}^{(2j(s+l))\wedge
T}\widetilde{Z}_{t,k}^{(s)}$

for $j=1,\dots,\lceil\frac{T}{2(s+l)}\rceil=:V_{T}$. Smoothness of
$h_{x,x,\tau}$, Rosenthal’s inequality and similar arguments as in (18) yield

$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\,\Big{|}Eh_{\tau,\tau,x}\Big{(}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\widetilde{Z}^{(s)}_{t,1},\dots,\frac{1}{\sqrt{T}}\sum_{t=1}^{T}Z^{(s)}_{t,N_{T}}\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-Eh_{\tau,\tau,x}\Big{(}\sum_{j=1}^{V_{T}}S_{j,1},\dots,\sum_{j=1}^{V_{T}}S_{j,N_{T}}\Big{)}\Big{|}$
$\displaystyle\leq
C\,\tau\,N_{T}^{2/m}\,\max_{k=1,\dots,N_{T}}\Big{\|}\sum_{j=1}^{V_{T}}U_{j,k}\Big{\|}_{m/2}$
$\displaystyle\leq
C\,\tau\,N_{T}^{2/m}\,\sqrt{V_{T}}\,\max_{k=1,\dots,N_{T}}\max_{j=1,\dots,V_{T}}\|U_{j,k}\|_{m/2}$
$\displaystyle\leq C\,\tau\,N_{T}^{2/m}\,\sqrt{\frac{V_{T}\,s}{T}}$
$\displaystyle\leq C\,\tau\,N_{T}^{2/m}\,\sqrt{\frac{s}{l}}.$ (26)

Next, we define centred, joint normal random variables
$(S^{*}_{j,k})_{j=1,\dots,V_{T},k=1,\dots,N_{T}}$ with
$E[S^{*}_{j,k_{1}}S^{*}_{j,k_{2}}]=E[S_{j,k_{1}}S_{j,k_{2}}]$ and such that
$(S^{*}_{j,1},\dots,S^{*}_{j,N_{T}}),$ $j=1,\dots,V_{T}$ are independent.
Further these variables are constructed such that they are independent of
$(S_{j,k})_{j=1,\dots,V_{T},k=1,\dots,N_{T}}$. We obtain from Lindeberg’s
method and using

$||S_{j,k}^{\ast}\|_{m/2}\leq
C_{m}\|S^{\ast}_{j,k}\|_{2}=C_{m}\|S_{j,k}\|_{2}\leq C_{m}\|S_{j,k}\|_{m/2},$

that,

$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\,\Big{|}Eh_{\tau,\tau,x}\Big{(}\sum_{j=1}^{V_{T}}S_{j,1},\dots,\sum_{j=1}^{V_{T}}S_{j,N_{T}}\Big{)}-Eh_{\tau,\tau,x}\Big{(}\sum_{j=1}^{V_{T}}S_{j,1}^{*},\dots,\sum_{j=1}^{V_{T}}S_{j,N_{T}}^{*}\Big{)}\Big{|}$
(27) $\displaystyle\leq
C\,(1+\tau^{3})\,\sum_{j=1}^{V_{T}}E\max_{k=1,\dots,N_{T}}[|S_{j,k}|^{3}\,+\,|S_{j,k}^{*}|^{3}]$
$\displaystyle\leq
C^{\prime}_{m}\,(1+\tau^{3})\,\,N_{T}^{6/m}\,V_{T}\,\left(\frac{l}{T}\right)^{3/2}.$

Using the bounds

$\|\frac{1}{\sqrt{T}}\sum_{i=1}^{T}(Z_{i,k}-\widetilde{Z}_{i,k}^{(s)})\|_{m/2}\leq
Cd_{s,m},\ \ \|\frac{1}{\sqrt{T}}\sum_{i=1}^{T}Z_{i,k}\|_{m/2}\leq C,$

derived in Lemma 2,

$\displaystyle|\sum_{j_{1}=1}^{V_{T}}\sum_{j_{2}=1}^{V_{T}}ES_{j_{1},k_{1}}U_{j_{2},k_{2}}|$
$\displaystyle\leq\|\sum_{j=1}^{V_{T}}S_{j,k_{1}}\|_{m/2}\|\sum_{j=1}^{V_{T}}U_{j,k_{2}}\|_{m/2}$
$\displaystyle\leq
V_{T}\max_{j=1,\ldots,V_{t}}\|S_{j,k_{1}}\|_{m/2}\max_{j=1,\ldots,V_{t}}\|U_{j,k_{2}}\|_{m/2}$
$\displaystyle\leq C\,V_{T}\,\frac{\sqrt{ls}}{T}$

and

$|\sum_{j_{1}=1}^{V_{T}}\sum_{j_{2}=1}^{V_{T}}EU_{j_{1},k_{1}}U_{j_{2},k_{2}}|\leq
V_{T}\max_{j=1,\ldots,V_{T}}\|U_{j,k_{1}}\|_{m/2}\max_{j=1,\ldots,V_{T}}\|U_{j,k_{2}}\|_{m/2}\leq
CV_{T}\frac{s}{T},$

we get

$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\,\Big{|}Eh_{\tau,\tau,x}\Big{(}\sum_{j=1}^{V_{T}}S^{\ast}_{j,1},\dots,\sum_{j=1}^{V_{T}}S^{\ast}_{j,N_{T}}\Big{)}$
$\displaystyle-
Eh_{\tau,\tau,x}\Big{(}\xi_{1},\ldots,\xi_{N_{T}}\Big{)}\Big{|}$ (28)
$\displaystyle\leq
C\,\tau^{2}\,\big{(}d_{s,m}+V_{T}\frac{\sqrt{sl}}{T}\big{)}.$

Now, let $\tau\sim T^{\lambda}$, $s\sim T^{a_{s}}$ and $l\sim T^{a_{l}}$ for
some positive numbers $\lambda$, $a_{s}$ and $a_{l}$. For (6), (6), (6), (27)
and (28) to vanish asymptotically, the following conditions are sufficient:

  1. (i)

$t\big{(}1+\sqrt{\log(N_{T})}+\sqrt{|\log(t)|}\big{)}\rightarrow 0$,

  2. (ii)

$2\lambda+4/m+a_{s}<a_{l}$, $6\lambda+12/m+a_{l}<1$, $4\lambda+a_{s}<a_{l}$
and

  3. (iii)

$\big{(}T^{\lambda+2/m}+T^{2\lambda}\big{)}d_{s,m}\rightarrow 0$.

For (i) and because $t=(1+\log(2N_{T}))/\tau$, it is easily seen that

$t\big{(}1+\sqrt{\log(N_{T})}+\sqrt{|\log(t)|}\big{)}\leq
CT^{-\lambda}\big{(}\log(N_{T})\big{)}^{3/2},$

which converges to zero provided $\lambda>0$. Furthermore, since $m>16$, (ii)
is satisfied if $a_{s}+\max\\{4\lambda,2\lambda+4/m\\}<a_{l}<1-6\lambda-12/m$.
Finally, for (iii) notice first that for $0<\delta<\alpha-1$ and because
$s\sim T^{a_{s}}$,

$\displaystyle\big{(}\sum_{j=s+1}^{\infty}\delta^{2}_{m}(j)\big{)}^{1/2}$
$\displaystyle=\big{(}\sum_{j=s+1}^{\infty}j^{-2\alpha+1+2\delta}\cdot
j^{-(1+2\delta)}\big{)}^{1/2}$ $\displaystyle\leq Cs^{-\alpha+\delta+1/2}\leq
CT^{-a_{s}(\alpha-\delta-1/2)}.$

Now, for $r>0$ we have,

$\displaystyle\sum_{h=0}^{\infty}\min\\{(1+h)^{-\alpha},T^{-a_{s}(\alpha-1/2-\delta)}\\}$
$\displaystyle\leq\sum_{h=0}^{T^{r}}T^{-a_{s}(\alpha-1/2-\delta)}+\sum_{h=T^{r}}^{\infty}(1+h)^{-\alpha+1+\delta}h^{-1-\delta}$
$\displaystyle\leq T^{r-a_{s}(\alpha-1/2-\delta)}+CT^{-r(\alpha-1-\delta)}.$

Balancing both terms in the last bound above yields,

$r-a_{s}(\alpha-1/2-\delta)=-r(\alpha-1-\delta)\ \Longleftrightarrow
r=\frac{a_{s}(\alpha-1/2-\delta)}{\alpha-\delta}$

and, therefore,

$\sum_{h=0}^{\infty}\min\big{\\{}\delta_{m}(h),\big{(}\sum_{j=s+1}^{\infty}\delta^{2}_{m}(j)\big{)}^{1/2}\big{\\}}=T^{-\frac{\displaystyle
a_{s}(\alpha-1/2-\delta)(\alpha-1-\delta)}{\displaystyle\alpha-\delta}}\rightarrow
0.$

Hence for (iii) to be satisfied,

$\frac{\displaystyle
a_{s}(\alpha-1/2-\delta)(\alpha-1-\delta)}{\displaystyle\alpha-\delta}-\max\\{\lambda+2/m\,,\,2\lambda\\}>0$
(29)

should hold true. Notice that for $\alpha$ satisfying condition (7), (29)
holds true and at the same time the rate in (29) is larger or equal to
$\min\\{\kappa_{1},\kappa_{2}\\}$. ∎

###### Proof of Theorem 2.

Let

$R_{T}(\lambda_{k,T})=\sqrt{\frac{T}{M_{T}}}\,\frac{\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})}{f(\lambda_{k,T})}\,\left(\frac{f(\lambda_{k,T})}{\widehat{f}_{T}(\lambda_{k,T})}-1\right)$

and $t=(1+\log(2N_{T}))/\tau$ with $\tau\sim T^{\lambda}$. Then we can split
up

$\displaystyle P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\big{)}$ $\displaystyle\leq
P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq x-t\big{)}$
$\displaystyle\quad+\,P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x-t\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{f(\lambda_{k,T})}\leq
x-t\big{)}$
$\displaystyle\quad+\,P\big{(}\max_{k=1,\ldots,N_{T}}|R_{T}(\lambda_{k,T})|>t\big{)}.$

One can proceed analogously to obtain a lower bound which then results in

$\displaystyle\sup_{x\in\mbox{$\mathbb{R}$}}\big{|}\,P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\big{)}\,\big{|}$
$\displaystyle\leq\sup_{x\in\mbox{$\mathbb{R}$}}\big{|}P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x-t\big{)}\big{|}$
$\displaystyle\quad+\,\sup_{x\in\mbox{$\mathbb{R}$}}\big{|}P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{f(\lambda_{k,T})}\leq
x\big{)}|$
$\displaystyle\quad+\,\sup_{x\in\mbox{$\mathbb{R}$}}\big{|}P\big{(}\max_{k=1,\ldots,N_{T}}|R_{T}(\lambda_{k,T})|>t\big{)}\big{|}$
$\displaystyle=P_{1}+P_{2}+P_{3}$

with obvious abbreviations for $P_{1},~{}P_{2},$ and $P_{3}$. Lemma A.1 in
Zhang_etal2022 gives

$P_{1}\,\leq\,C\,t\,\left(1+\sqrt{\log(N_{T})}+\sqrt{|\log(t)|}\right).$

The desired rate for $P_{2}$ can be derived with exactly the same arguments as
in the proof of Theorem 1 since the spectral density is assumed to be
uniformly bounded from below.

Finally, for $P_{3}$ we use that

$\sup_{\lambda}|E\widehat{f}_{T}(\lambda)-f(\lambda)|=O(M^{-2}(T)),$

see [Theorem 9.4.3]A71. This allows to bound $P_{3}$ as follows. We have

$\displaystyle|R_{T}(\lambda_{k,T})|$
$\displaystyle=\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{f(\lambda_{k,T})}\,\Big{|}\frac{f(\lambda_{k,T})}{\widehat{f}_{T}(\lambda_{k,T})}-1\Big{|}$
$\displaystyle=\sqrt{\frac{T}{M_{T}}}\,\big{|}\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})\big{|}\frac{|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})\,f(\lambda_{k,T})}\,.$

A division of the considerations depending on whether
$\max_{k}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|$ is less or equal
or larger than $\inf_{\lambda}f(\lambda)/2$ results in

$\displaystyle P(\max_{k=1,\dots,N_{T}}|R_{T}(\lambda_{k,T})|>t)$
$\displaystyle\leq
P\Big{(}\sqrt{\frac{T}{M_{T}}}\max_{k=1,\dots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|\frac{2}{\inf_{\lambda}f(\lambda)^{2}}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|>t\Big{)}$
$\displaystyle\quad+P\big{(}\max_{k=1,\dots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|>\inf_{\lambda}f(\lambda)/2\big{)}\,.$

The first summand is bounded through

$\displaystyle
Ct^{-1}\sqrt{\frac{T}{M_{T}}}\Big{(}E\max_{k=1,\dots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|^{2}$
$\displaystyle\hskip
44.9554pt+E\max_{k=1,\dots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|\cdot
M_{T}^{-2}\Big{)}$
$\displaystyle\leq\,Ct^{-1}\Big{(}N_{T}^{4/m}\sqrt{\frac{M_{T}}{T}}+N_{T}^{2/m}M_{T}^{-2}\Big{)}\,,$

where for arbitrary random variables $U_{k}$ we make use of
$E\max_{k=1,\dots,N_{T}}|U_{k}|\leq(E\sum_{k=1}^{N_{T}}|U_{k}|^{r})^{1/r}\leq
N_{T}^{1/r}\max_{k=1,\dots,N_{T}}\|U_{k}\|_{r}$, $r\geq 1,$ and the last
inequality follows from Lemma 2.
A similar consideration for the second summand finally leads to

$P(\max_{k=1,\dots,N_{T}}|R_{T}(\lambda_{k,T})|>t)\leq\,C(t^{-1}\,N_{T}^{4/m}+N_{T}^{2/m})\,\Big{(}\sqrt{\frac{M_{T}}{T}}+M_{T}^{-2}\Big{)}\,.$

Using $M_{T}\sim T^{a_{s}}$ and $t^{-1}=T^{\lambda}\big{/}(1+\log(2N_{T}))$,
the above bound for $P(\max_{k=1,\dots,N_{T}}|R_{T}(\lambda_{k,T})|>t)$
implies the order

$\frac{\displaystyle T^{\lambda+a_{s}/2}T^{4/m}}{\displaystyle
T^{1/2}(1+\log(2N_{T}))}+\frac{\displaystyle T^{\lambda}T^{4/m}}{\displaystyle
T^{2a_{s}}(1+\log(2N_{T}))},$

for $P_{3}$, which for
$\lambda+\kappa\leq\min\\{1/2-a_{s}/2-4/m\,,\,2a_{s}-4/m\\}$ leads to
$P_{3}=O(T^{-\kappa}/\log(2N_{T}))$. ∎

###### Proof of Proposition 1.

First, we split up

$\displaystyle\big{|}\widehat{\sigma}_{T}(j_{1},j_{2})\,-\,\sigma_{T}(j_{1},j_{2})\big{|}$
$\displaystyle\leq\ $
$\displaystyle\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\big{(}X_{t}X_{t-j_{1}}-{\gamma}(j_{1})\big{)}\big{(}X_{s}X_{s-j_{2}}-{\gamma}(j_{2})\big{)}\,-\,\sigma(j_{1},j_{2})\Big{|}$
$\displaystyle+\,|\widehat{\gamma}(j_{1})-\gamma(j_{1})|\,\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\big{(}X_{s}X_{s-j_{2}}-{\gamma}(j_{2})\big{)}\Big{|}$
$\displaystyle+\,|\widehat{\gamma}(j_{2})-\gamma(j_{2})|\,\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\big{(}X_{t}X_{t-j_{1}}-{\gamma}(j_{1})\big{)}\Big{|}$
$\displaystyle+|\widehat{\gamma}(j_{1})-\gamma(j_{1})|\,|\widehat{\gamma}(j_{2})-\gamma(j_{2})|\,\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{|}.$
(30)

While the last summand is of order $O(b_{T}\,T^{-1})$ (see (E.4) in
[Zhang_etal2022]), the two middle terms can be bounded from above by
$O(b_{T}\,T^{-1/2})$ using Cauchy-Schwarz inequality. Both bounds hold
uniformly in $j_{1}$ and $j_{2}$. Let $Y_{t,j}=X_{t}X_{t-j}-\gamma(j)$ and

$\widetilde{\sigma}_{T}(j_{1},j_{2})=\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}K\Big{(}\frac{t-s}{b_{T}}\Big{)}Y_{t,j_{1}}Y_{s,j_{2}}.$

Then the first term on the right hand side of the bound given in (6) equals
$|\widetilde{\sigma}_{T}(j_{1},j_{2})-\sigma(j_{1},j_{2})|$, and for this we
have

$\displaystyle|\widetilde{\sigma}(j_{1},j_{2})-\sigma(j_{1},j_{2})|$
$\displaystyle\leq\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}K\Big{(}\frac{t-s}{b_{T}}\Big{)}-1\Big{)}E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{|}$
$\displaystyle+\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}Y_{t,j_{1}}Y_{s,j_{2}}-E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{)}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{|}.$
(31)

Using Lemma 1(iii) the first term of the right hand side of (6) can be bounded
by

$\displaystyle C\frac{1}{T}\sum_{t=1}^{T}\sum_{s=1}^{T}$
$\displaystyle\Big{(}1-K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{)}\frac{1}{(1+|t-s|)^{\alpha}}\leq
2C\sum_{s=0}^{\infty}\Big{(}1-K\Big{(}\frac{s}{b_{T}}\Big{)}\Big{)}\frac{1}{(1+s)^{\alpha}}.$

Let $S=b_{T}$. Using
$|1-K(s/b_{T})|\leq\sup_{u\in[0,1]}|K^{\prime}(u)|s/b_{T}$ for $0\leq s\leq S$
and $|1-K(s/b_{T})|\leq 1$ for $s\geq S+1$, we get

$\displaystyle\sum_{s=0}^{\infty}\Big{(}1-$ $\displaystyle
K\Big{(}\frac{s}{b_{T}}\Big{)}\Big{)}\frac{1}{(1+s)^{\alpha}}$
$\displaystyle\leq\frac{\sup_{u\in[0,1]}|K^{\prime}(u)|}{b_{T}}\sum_{s=0}^{S}\frac{1}{(1+s)^{\alpha-1}}+\sum_{s=S+1}^{\infty}\frac{1}{(1+s)^{\alpha}},$

where the first term is $O(b_{T}^{-1})$ because $\alpha>2$. For the second
term we get

$\displaystyle\sum_{s=S+1}^{\infty}\frac{1}{(1+s)^{\alpha}}$
$\displaystyle\leq\int_{S+1}^{\infty}\frac{1}{x^{\alpha}}dx=\frac{1}{(\alpha-1)(S+1)^{\alpha-1}}=O(b_{T}^{-1}).$

Hence

$\max_{1\leq j_{1},j_{2}\leq
M_{T}}\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}K\Big{(}\frac{t-s}{b_{T}}\Big{)}-1\Big{)}E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{|}=O(b_{T}^{-1}).$

Consider next the second term of the bound given in (6) and observe that this
term can be bounded by

$M_{T}^{8/m}\max_{1\leq j_{1},j_{2}\leq
M_{T}}\Big{\|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}Y_{t,j_{1}}Y_{s,j_{2}}-E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{)}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{\|}_{m/4},$
(32)

where

$\displaystyle\Big{\|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}$
$\displaystyle
Y_{t,j_{1}}Y_{s,j_{2}}-E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{)}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{\|}_{m/4}$
$\displaystyle\leq$
$\displaystyle\frac{1}{T}\sum_{\ell=0}^{T-1}K(\ell/b_{T})\big{\|}\sum_{t=1}^{T-\ell}(Y_{t,j_{2}}Y_{t+\ell,j_{1}}-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}))\big{\|}_{m/4}$
$\displaystyle\ \
+\frac{1}{T}\sum_{\ell=1}^{T-1}K(\ell/b_{T})\big{\|}\sum_{s=1}^{T-\ell}(Y_{s,j_{1}}Y_{s+\ell,j_{2}}-E(Y_{s,j_{1}}Y_{s+\ell,j_{2}}))\big{\|}_{m/4}.$

Recall that for $s\geq 0$, ${\mathcal{F}}_{r,s}$ denotes the $\sigma$-algebra
generated by the set of random variables $\\{e_{r},e_{r-1},\ldots,e_{r-s}\\}$.
Note, that by Assumption A.1 for some measurable function $H$,
$Y_{t,j_{2}}Y_{t+\ell,j_{1}}=H(e_{t+\ell},e_{t+\ell-1},\ldots)$ and that the
$e_{t}$’s are i.i.d.. Then

$\displaystyle Y_{t,j_{2}}Y_{t+\ell,j_{1}}$
$\displaystyle-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})=E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,\infty})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})$
$\displaystyle=\lim_{q\rightarrow\infty}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,q})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})$
$\displaystyle=E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})$
$\displaystyle\ \ \ \
+\sum_{r=1}^{\infty}\big{\\{}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r-1})\big{\\}},$

a.s. Therefore,

$\displaystyle\big{\|}\sum_{t=1}^{T-\ell}($ $\displaystyle
Y_{t,j_{2}}Y_{t+\ell,j_{1}}-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}))\big{\|}_{m/4}$
$\displaystyle\leq$
$\displaystyle\big{\|}\sum_{t=1}^{T-\ell}\big{\\{}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})\big{\\}}\big{\|}_{m/4}$
$\displaystyle+\sum_{r=1}^{\infty}\big{\|}\sum_{t=1}^{T-\ell}\big{\\{}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r-1})\big{\\}}\big{\|}_{m/4}$
$\displaystyle=$ $\displaystyle S_{1,n}+S_{2,n},$

with an obvious notation for $S_{1,n}$ and $S_{2,n}$. Consider $S_{1,n}$ and
observe that
$E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})=\widetilde{g}(e_{t+\ell})$
and therefore, $E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})$ and
$E(Y_{s,j_{2}}Y_{s+\ell,j_{1}}|{\mathcal{F}}_{s+\ell,0})$ are independent for
$t\neq s$. Hence,

$\displaystyle S_{1,n}$ $\displaystyle\leq
C\sqrt{\sum_{t=1}^{T-\ell}\big{\|}\big{\\{}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})\big{\\}}\big{\|}_{m/4}}$
$\displaystyle\leq
C\sqrt{T}\big{\|}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,0})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})\big{\|}_{m/4}.$

To bound the term $S_{2,n}$ define first

$W_{u}=\sum_{t=T-\ell-u+1}^{T-\ell}\big{\\{}E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r-1})\big{\\}}$

and denote by ${\mathcal{A}}_{u}$ the $\sigma$-algebra generated by the set
$\\{e_{T},e_{T-1},\ldots,e_{T-u+1-r}\\}$. Notice that $W_{u}$ is measurable
with respect to ${\mathcal{A}}_{u}$ and that
${\mathcal{A}}_{u}\subset{\mathcal{A}}_{u+1}$. Furthermore,

$\displaystyle E($ $\displaystyle W_{u+1}-W_{u}|{\mathcal{A}}_{u})$
$\displaystyle=E\Big{[}E\big{(}Y_{T-\ell-u,j_{2}}Y_{T-u,j_{1}}|{\mathcal{F}}_{T-u,r}\big{)}-E\big{(}Y_{T-\ell-u,j_{2}}Y_{T-u,j_{1}}|{\mathcal{F}}_{T-u,r-1}\big{)}\Big{|}{\mathcal{A}}_{u}\Big{]}$
$\displaystyle=E\big{(}Y_{T-\ell-u,j_{2}}Y_{T-u,j_{1}}|{\mathcal{F}}_{T-u,r-1}\big{)}-E\big{(}Y_{T-\ell-u,j_{2}}Y_{T-u,j_{1}}|{\mathcal{F}}_{T-u,r-1}\big{)}$
$\displaystyle=0.$

Since $W_{u}$ forms a martingale we get

$\|W_{T-\ell}\|_{m/4}\leq
C\sqrt{\sum_{t=1}^{T-\ell}\|E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r-1})\|^{2}_{m/4}}.$

Recall that $Y_{t,j_{2}}Y_{t+\ell,j_{1}}=H(e_{t+\ell},e_{t+\ell-1},\ldots)$
for some measurable function $H$ and let

$Y_{t,j_{2}}(r-\ell)Y_{t+\ell,j_{1}}(r)=H(e_{t+\ell},e_{t+\ell-1},\ldots,e_{t+\ell-r+1},e^{\prime}_{t+\ell-r},e_{t+l-r-1},\ldots),$

where $\\{e_{t}^{\prime},t\in\mbox{$\mathbb{Z}$}\\}$ is an independent copy of
$\\{e_{t},t\in\mbox{$\mathbb{Z}$}\\}$. Then,

$\displaystyle\|E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})$
$\displaystyle-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r-1})\|_{m/4}$
$\displaystyle=\|E(Y_{t,j_{2}}Y_{t+\ell,j_{1}}|{\mathcal{F}}_{t+\ell,r})-E(Y_{t,j_{2}}(r-\ell)Y_{t+\ell,j_{1}}(r)|{\mathcal{F}}_{t+\ell,r})\|_{m/4}$
$\displaystyle\leq\|Y_{t,j_{2}}Y_{t+\ell,j_{1}}-Y_{t,j_{2}}(r-\ell)Y_{t+\ell,j_{1}}(r)\|_{m/4},$

that is,

$\|W_{T-\ell}\|_{m/4}\leq
C\sqrt{T}\|Y_{t,j_{2}}Y_{t+\ell,j_{1}}-Y_{t,j_{2}}(r-\ell)Y_{t+\ell,j_{1}}(r)\|_{m/4}.$

Hence,

$\displaystyle S_{2,n}=$
$\displaystyle\sum_{r=1}^{\infty}\|W_{T-\ell}\|_{m/4}$ $\displaystyle\leq$
$\displaystyle\
C\sqrt{T}\sum_{r=1}^{\infty}\|Y_{t,j_{2}}Y_{t+\ell,j_{1}}-Y_{t,j_{2}}(r-\ell)Y_{t+\ell,j_{1}}(r)\|_{m/4}$
$\displaystyle\leq$ $\displaystyle
C\sqrt{T}\Big{\\{}\sum_{r=1}^{\infty}\|Y_{t+\ell,j_{1}}\|_{m/2}\|Y_{t,j_{2}}(r-\ell)-Y_{t,j_{2}}\|_{m/2}$
$\displaystyle+\sum_{r=1}^{\infty}\|Y_{t,j_{2}}(r-\ell)\|_{m/2}\|Y_{t+\ell,j_{1}}(r)-Y_{t+\ell,j_{1}}\|_{m/2}\Big{\\}}\
\leq\ C\sqrt{T},$

since

$\displaystyle\|Y_{t,j_{2}}(r-\ell)-Y_{t,j_{2}}\|_{m/2}$
$\displaystyle=\|X_{t}(r-\ell)X_{t-j_{2}}(r-\ell-
j_{2})-X_{t}X_{t-j_{2}}\|_{m/2}$
$\displaystyle\leq\|X_{t}(r-\ell)-X_{t}\|_{m}\|X_{t-j_{2}}(r-\ell-
j_{2})\|_{m}$ $\displaystyle\ \ +\|X_{t}\|_{m}\|X_{t-j_{2}}(r-\ell-
j_{2})-X_{t-j_{2}}\|_{m}$ $\displaystyle\leq C(r-\ell)^{\alpha}$

and an analogue argument for $\|Y_{t+\ell,j_{1}}(r)-Y_{t+\ell,j_{1}}\|_{m/2}$.
Therefore,

$\displaystyle\Big{\|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}Y_{t,j_{1}}Y_{s,j_{2}}-E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{)}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{\|}_{m/4}$
$\displaystyle\leq$ $\displaystyle\
\frac{1}{T}\sum_{\ell=0}^{T-1}K(\ell/b_{T})\|\sum_{t=1}^{T-\ell}Y_{t,j_{2}}Y_{t+\ell,j_{1}}-E(Y_{t,j_{2}}Y_{t+\ell,j_{1}})\|_{m/4}$
$\displaystyle+\frac{1}{T}\sum_{\ell=0}^{T-1}K(\ell/b_{T})\|\sum_{t=1}^{T-\ell}Y_{t,j_{1}}Y_{t+\ell,j_{2}}-E(Y_{t,j_{2}}Y_{t+\ell,j_{2}})\|_{m/4}$
$\displaystyle\leq$ $\displaystyle\ CT^{-1/2}\sum_{\ell=0}^{T-1}K(\ell/b_{T})$
$\displaystyle\leq$ $\displaystyle\
C(T^{-1/2}\big{(}K(0)+\sum_{\ell=1}^{\infty}K(\ell/b_{T})\big{)}$
$\displaystyle\leq$ $\displaystyle\
CT^{-/2}\big{(}K(0)+\int_{0}^{\infty}K(x/b_{T})dx\big{)}$ $\displaystyle\leq$
$\displaystyle\ Cb_{T}/\sqrt{T}.$

From this and recalling (32) we conclude that

$\Big{|}\frac{1}{T}\sum_{t=j_{1}+1}^{T}\sum_{s=j_{2}+1}^{T}\Big{(}Y_{t,j_{1}}Y_{s,j_{2}}-E(Y_{t,j_{1}}Y_{s,j_{2}})\Big{)}K\Big{(}\frac{t-s}{b_{T}}\Big{)}\Big{|}=O_{P}\big{(}M_{T}^{8/m}b_{T}/\sqrt{T}\big{)}.$

∎

###### Proof of Theorem 3.

Let $\widetilde{\xi}_{k}$, $k=1,2,\ldots,N_{T},$ be Gaussian random variables
as in Theorem 2. By the triangular inequality,

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\big{)}-P^{\ast}\big{(}\max_{k=1,\ldots,N_{T}}|\xi^{\ast}_{k}|\leq
x\big{)}\Big{|}$
$\displaystyle\leq\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}\sqrt{\frac{T}{M_{T}}}\frac{|\widehat{f}_{T}(\lambda_{k,T})-E\widehat{f}_{T}(\lambda_{k,T})|}{\widehat{f}_{T}(\lambda_{k,T})}\leq
x\big{)}-P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}\Big{|}$
$\displaystyle+\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-P^{\ast}\big{(}\max_{k=1,\ldots,N_{T}}|\xi^{\ast}_{k}|\leq
x\big{)}\Big{|}.$ (33)

The first term is bounded by Theorem 2. To handle the second term, we first
show that

$\displaystyle\Delta_{T}:=\max_{1\leq k_{1},k_{2}\leq
N_{T}}\big{|}\widehat{C}_{T}(k_{1},k_{2})-C_{T}(k_{1},k_{2})\big{|}$
$\displaystyle=$ $\displaystyle\
O_{P}\Big{(}\max_{k=1,\ldots,N_{T}}|\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})|+\frac{b_{T}M_{T}}{\sqrt{T}}+\frac{M_{T}}{b_{T}}\,+\,M_{T}b_{T}T^{8a_{s}/m-1/2}\Big{)}.$
(34)

We have

$\displaystyle\Delta_{T}\leq$ $\displaystyle\max_{1\leq k_{1},k_{2}\leq
N_{T}}\Big{|}\Big{(}\frac{1}{\widehat{f}_{T}(\lambda_{k_{1},T})\widehat{f}(\lambda_{k_{2},T})}-\frac{1}{f_{T}(\lambda_{k_{1},T})f(\lambda_{k_{2},T})}\Big{)}$
$\displaystyle\times\sum_{j_{1}=1}^{M_{T}}\sum_{j_{2}=1}^{M_{T}}a_{k_{1},j_{1}}a_{k_{2},j_{2}}\sigma_{T}(j_{1},j_{2})\Big{|}+\max_{1\leq
k_{1},k_{2}\leq
N_{T}}\Big{|}\frac{1}{\widehat{f}_{T}(\lambda_{k_{1},T})\widehat{f}(\lambda_{k_{2},T})}$
$\displaystyle\times\sum_{j_{1}=1}^{M_{T}}\sum_{j_{2}=1}^{M_{T}}a_{k_{1},j_{1}}a_{k_{2},j_{2}}\big{(}\widehat{\sigma}_{T}(j_{1},j_{2})-\sigma_{T}(j_{1},j_{2})\big{)}\Big{|}$
$\displaystyle=$ $\displaystyle\ \Delta_{1,T}+\Delta_{2,T},$

with an obvious notation for $\Delta_{1,T}$ and $\Delta_{2,T}$.

For $\Delta_{1,T}$ notice that by the boundedness of the spectral density $f$
and Assumption 2 we get that

$\displaystyle\max_{1\leq k_{1},k_{2}\leq
N_{T}}\Big{|}\frac{1}{\widehat{f}_{T}(\lambda_{k_{1},T})\widehat{f}(\lambda_{k_{2},T})}-\frac{1}{f_{T}(\lambda_{k_{1},T})f(\lambda_{k_{2},T})}\Big{|}$
$\displaystyle\leq\ $ $\displaystyle\Big{(}\frac{1}{\min_{1\leq k\leq
N_{T}}\widehat{f}_{T}(\lambda_{k,T})}\Big{)}^{2}\Big{(}\frac{1}{\min_{1\leq
k\leq N_{T}}f_{T}(\lambda_{k,T})}\Big{)}^{2}$ $\displaystyle\ \
\times\big{(}\max_{1\leq k\leq N_{T}}f(\lambda_{k,T})+\max_{1\leq k\leq
N_{T}}\widehat{f}(\lambda_{k,T})\big{)}\max_{1\leq k\leq
N_{T}}\big{|}\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})\big{|}$
$\displaystyle={\mathcal{O}}_{P}(\max_{1\leq k\leq
N_{T}}\big{|}\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})\big{|}).$

Furthermore, recalling the definition of the Gaussian random variables
$\xi_{k}$, we have

$\displaystyle\max_{1\leq k_{1},k_{2}\leq
N_{T}}\Big{|}\sum_{j_{1}=1}^{M_{T}}\sum_{j_{2}=1}^{M_{T}}a_{k_{1},j_{1}}a_{k_{2},j_{2}}\sigma_{T}(j_{1},j_{2})\Big{|}=\max_{1\leq
k_{1},k_{2}\leq N_{T}}|E(\xi_{k_{1}},\xi_{k_{2}})|$ $\displaystyle\leq$
$\displaystyle\ \big{(}\max_{1\leq k\leq
N_{T}}\|\xi_{k}\|_{2}\big{)}^{2}={\mathcal{O}}(1);$

see Remark 1. Hence $\Delta_{1,T}={\mathcal{O}}_{P}\big{(}\max_{1\leq k\leq
N_{T}}\big{|}\widehat{f}_{T}(\lambda_{k,T})-f(\lambda_{k,T})\big{|}\big{)}$.

For $\Delta_{2,T}$ we have

$\displaystyle\Delta_{2,T}$ $\displaystyle\leq\max_{1\leq j_{1},j_{2}\leq
M_{T}}\big{|}\widehat{\sigma}_{T}(j_{1},j_{2})-\sigma_{T}(j_{1},j_{2})\Big{|}\Big{(}\frac{1}{\min_{1\leq
k\leq N_{T}}\widehat{f}_{T}(\lambda_{k,T})}\Big{)}^{2}$ $\displaystyle\ \ \ \
\times\Big{(}\max_{1\leq k\leq N_{T}}\sum_{j=0}^{M_{T}}a_{k,j}\Big{)}^{2}$
$\displaystyle={\mathcal{O}}_{P}\Big{(}\frac{M_{T}b_{T}}{\sqrt{T}}+\frac{M_{T}}{b_{T}}+M_{T}b_{T}T^{8a_{s}/m-1/2}\Big{)},$

by Proposition 1 and the fact that $\sum_{j=0}^{M_{T}}a_{k,j}=O(\sqrt{M_{T}})$
uniformly in $1\leq k\leq N_{T}$. This establishes (6).

Recall next that for $1\leq k\leq N_{T}$,

$E|\xi_{k}|^{2}=f^{2}(\lambda_{k,T})\int_{-1}^{1}w^{2}(u)du+o(1),$

and that $E|\widetilde{\xi}_{k}|^{2}=E|\xi_{k}|^{2}/f^{2}(\lambda_{k,T})$.
From these expressions and by Assumption 2 and the boundedness of $f$ we get
that $\min_{1\leq k\leq N_{T}}C_{T}(k,k)$ and $\max_{1\leq k\leq
N_{T}}C_{T}(k,k)$ are, for $T$ large enough, bounded from below and from
above, respectively, by positive constants. Using Lemma A.1 of Zhang et al.
(2022) we then get

$\displaystyle\sup_{x\in\mathbb{R}}\Big{|}P\big{(}\max_{k=1,\ldots,N_{T}}|\widetilde{\xi}_{k}|\leq
x\big{)}-$ $\displaystyle
P^{\ast}\big{(}\max_{k=1,\ldots,N_{T}}|\xi^{\ast}_{k}|\leq x\big{)}\Big{|}$
$\displaystyle=$
$\displaystyle{\mathcal{O}}_{P}\Big{(}\frac{\Delta_{T}^{1/6}}{(1+\log(N_{T}))^{1/4}}+\Delta_{T}^{1/3}\log^{3}(N_{T})\Big{)}$
$\displaystyle=$ $\displaystyle o_{P}(\Delta_{T}^{1/6}).$

∎

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