#  FR-SLAM: A SLAM Improvement Method Based on Floor Plan Registration
††thanks: ∗Xinde Li is corresponding author<EMAIL_ADDRESS>
Jiantao Feng  Suzhou Campus
Southeast University
Nanjing center for applied mathematics
Nanjing, China
<EMAIL_ADDRESS>Xinde Li  School of Automation
School of Cyber Science and Engineering
Shenzhen Research Institute
Southeast University
Nanjing center for applied mathematics
Nanjing, China
<EMAIL_ADDRESS>HyunCheol Park  Juan Liu  Samsung Electronics
South Korea
<EMAIL_ADDRESS>Samsung Electronics(China) R&D Center
Nanjing, China
<EMAIL_ADDRESS>Zhentong Zhang  School of Cyber Science and
Engineering
Southeast University
Nanjing center for applied mathematics
Nanjing, China
<EMAIL_ADDRESS>
###### Abstract

Simultaneous Localization and Mapping (SLAM) technology enables the
construction of environmental maps and localization, serving as a key
technique for indoor autonomous navigation of mobile robots. Traditional SLAM
methods typically require exhaustive traversal of all rooms during indoor
navigation to obtain a complete map, resulting in lengthy path planning times
and prolonged time to reach target points. Moreover, cumulative errors during
motion lead to inaccurate robot localization, impacting navigation efficiency.
This paper proposes an improved SLAM method, FR-SLAM, based on floor plan
registration, utilizing a morphology-based floor plan registration algorithm
to align and transform original floor plans. This approach facilitates the
rapid acquisition of comprehensive motion maps and efficient path planning,
enabling swift navigation to target positions within a shorter timeframe. To
enhance registration and robot motion localization accuracy, a real-time
update strategy is employed, comparing the current position’s building
structure with the map and dynamically updating floor plan registration
results for precise localization. Comparative tests conducted on real and
simulated datasets demonstrate that, compared to other benchmark algorithms,
this method achieves higher floor plan registration accuracy and shorter time
consumption to reach target positions.

###### Index Terms:

SLAM, Image registration, Floor plan, Morphological transformation

##  I Introduction

Figure 1: Schematic diagram of FR-SLAM algorithm Figure 2: The overall
workflow of FR-SLAM based on floor plans registration, including LiDAR map
generation, original floor plans registration, and map update strategy.

In recent years, with the accelerated process of urbanization, the rapid
expansion of urban areas, and the increasing accident rates in residential
areas, the use of publicly available floor plans as references for search and
rescue tasks by firefighters has become common. However, these floor plans
often suffer from anomalies in aspect ratios, as well as issues related to
rotation and flipping, which can adversely affect planning efforts.
Concurrently, the advancement in miniaturization and intelligence of unmanned
robotic systems has significantly contributed to tasks such as firefighting
rescue operations [1] and post-disaster search and rescue missions. Equipped
with laser rangefinders and utilizing ultrasonic sensors for high-precision
ranging, indoor localization, and complex environmental perception, these
robots play a vital role in map construction and path planning. Traditional
SLAM methods typically involve scanning the entire indoor structure to
complete global mapping, followed by loop closure detection to generate a
complete map. However, obstacles such as closed doors and debris blocking
pathways may prevent robots from accessing certain rooms, resulting in missing
map information. These factors render conventional SLAM methods inadequate for
real-time tasks such as firefighting rescue and post-disaster search and
rescue operations.

This paper presents an improved SLAM method based on floor plan registration.
The approach utilizes partial room LiDAR scans obtained by the robot as a
reference, employing morphological transformations for image registration of
the original floor plans (MFR). This enables the rapid acquisition of an
authentic indoor map and facilitates path planning within a short timeframe,
as depicted in Fig. 1. Furthermore, it enhances the accuracy of robot motion
localization through a real-time matching update strategy. The contribution of
this work are as follows:

  * •

This paper presents a SLAM improvement method based on floor plan
registration, which integrates the original floor plan with LiDAR maps through
image fusion to address the issues of accumulated errors during mapping and
motion localization of the robot.

  * •

In this method, a registration algorithm for floor plans and LiDAR maps is
proposed. The algorithm only requires contour information from images, and
based on the maximum intersection over union (IoU) of floor plans and LiDAR
maps under different morphological transformations, image registration can be
accomplished, irrespective of environmental constraints. The accuracy of
registration achieved by this algorithm on the testing dataset reaches 85$\%$,
surpassing common image matching algorithms.

  * •

To address the problem of robot motion localization, a real-time matching
update strategy is proposed. This strategy continuously matches the current
LiDAR map with the floor plan, eliminating motion errors and achieving real-
time localization.

##  II RELATED WORK

Image registration is widely used in various fields such as robot visual
navigation [2], motion planning [3], medical image processing [4], and remote
sensing. However, due to the diverse types of images applicable for image
registration, there is currently no universal optimization method that can
satisfy all purposes [5]. Commonly used image registration methods include
intensity-based matching algorithms and feature-based matching algorithms.
Intensity-based image registration methods are sensitive to changes in image
intensity and require images to have rich intensity information. On the other
hand, feature-based matching algorithms require the extraction of image
features such as closed area areas, image contours, key intersections, and
corners, which necessitates images with abundant and easily extractable
relevant features.

The SLAM method encompasses two primary tasks: localization and mapping. A
robot begins from an unknown location in an unknown environment and uses
repeated observations of environmental features during movement to determine
its position and orientation. Subsequently, it incrementally constructs a map
of the surrounding environment based on its location, thereby achieving the
objectives of simultaneous localization and mapping [6]. Traditional indoor
SLAM methods require the robot to perform a complete scan of the entire scene,
which is time-consuming and yields poor localization accuracy.

##  III METHOD

This section presents the complete workflow of FR-SLAM (illustrated in Fig.
2), which includes: image registration algorithm comprising image
preprocessing and registration, aligning the original floor plan with LiDAR
map, applying morphological transformations to the floor plan based on
registration results, thereby obtaining a floor plan that describes the
structure of the house as the robot’s motion map; real-time update strategy,
involving updating LiDAR information to refine the alignment of the floor plan
with the actual house structure, mitigating obstacles to robot motion.
Additionally, it compares the current location’s house structure with the
motion map to enhance the accuracy of robot motion localization.

###  III-A Floor Plan Registration

####  III-A1 Image Preprocessing

Image preprocessing consists of two parts: floor plan preprocessing and LiDAR
map preprocessing.

The purpose of floor plan preprocessing is to generate multiple filled images
covering different room arrangements, including living rooms and various other
rooms, to address the missing areas of different rooms in the LiDAR map. The
formula is as follows:

$\begin{array}[]{ll}F=\\{f(S):S\subseteq H\\}\end{array}$ (1)

where $\mathit{H}$ represents all room types and $\mathit{F}$ represents the
final filled image. The specific steps are shown in Alg. 1.

Input: Original floor plan room list $room\\_list$

Output: Floor plan collection $floor\\_list$

1 Initialize bedroom dictionary $bedroom\\_map=\text{\\{ \\}}$;

2 Initialize $temp\\_list=\text{[ ]}$;

3 for  _$i=0$ to $length(room\\_list)$_ do

4 Add bedroom to $bedroom\\_map$;

5

6 end for

7for  _$i=0$ to $length(bedroom\\_map.keys())$_ do

8 Calculate different bedroom arrangement combinations $combin\\_list$;

9 for  _$j=0$ to $length(combin\\_list)$_ do

10 $temp\\_list$.add($combin\\_list[j]+living\\_room$);

11

12 end for

13

14 end for

15for  _$i=0$ to $length(temp\\_list)$_ do

16 $temp\\_floor\leftarrow$ draw floor plan($temp\\_list(i)$);

17 $floor\\_list$.add($temp\\_floor$);

18

19 end for

20return _$floor\\_list$_ ;

Algorithm 1 Floor plan preprocessing algorithm

The preprocessing of LiDAR images involves initially converting the original
grayscale image $\mathit{I}$ into a binary image $\mathit{B}$. The
binarization process can be represented by the following equation:

$\begin{array}[]{ll}B(x,y)=\begin{cases}1&I(x,y)\geqslant\theta\\\
0&I(x,y)\leqslant\theta\end{cases}\end{array}$ (2)

where $\mathit{\theta}$ represents the grayscale threshold value of the pixel.
Subsequently, connected component analysis is performed on the binary image
$\mathit{B}$ to remove contours with small areas, retaining the main structure
of the LiDAR image, represented using set notation as:

$\begin{array}[]{ll}B^{\prime}=\\{p\in
B:\operatorname{area}(p)\geq\alpha\\}\end{array}$ (3)

where $\mathit{\alpha}$ represents the contour threshold, and
$\mathit{area(p)}$ denotes the area of connected domain $\mathit{p}$.

Subsequently, polygon fitting is employed to smooth the outer contours,
thereby reducing computational complexity. Polygon fitting typically utilizes
the least squares method. Given a set containing $\mathit{area(n)}$ points
${\text{[}(x_{1},y_{1}),(x_{2},y_{2}),…,(x_{n},y_{n})\text{]}}$, the fitting
problem is transformed into solving a system of linear equations:

$\begin{array}[]{ll}\left[\begin{array}[]{ccccc}1&x_{1}&x_{1}^{2}&\cdots&x_{1}^{m}\\\
1&x_{2}&x_{2}^{2}&\cdots&x_{2}^{m}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
1&x_{n}&x_{n}^{2}&\cdots&x_{n}^{m}\end{array}\right]\left[\begin{array}[]{c}a_{0}\\\
a_{1}\\\ \vdots\\\ a_{m}\end{array}\right]=\left[\begin{array}[]{c}y_{1}\\\
y_{2}\\\ \vdots\\\ y_{n}\end{array}\right]\end{array}$ (4)

By solving this system, polynomial coefficients are obtained, which are then
used to derive the fitted polygon $\mathit{f(x)}$. Finally, the flood fill
algorithm is applied to fill the surrounding area, and the image is cropped,
completing the preprocessing of the LiDAR image.

####  III-A2 Image Registration

The preprocessed floor plan and LiDAR map are fused and registered in the
image, as depicted in Alg. 2, following a comprehensive process.

Input: LiDAR image $L$; Original floor plan image $F$

Output: Rotation result $R_{r}$, Flip result $R_{f}$, Horizontal scaling
result $R_{h}$, Vertical scaling result $R_{v}$

1 $L^{*}\leftarrow\text{resize}(L,200,200)$;

2 $F^{*}\leftarrow\text{resize}(F,200,200)$;

3 $list\\_F^{*}\leftarrow$ Morphological transformations on floor plan image
$F^{*}$;

4 Initialize $list\\_IoU=\text{[ ]}$;

5 for  _$i=0$ to $\text{length}(list\\_F^{*})$_ do

6 $S_{\text{inter}},S_{\text{union}}\leftarrow$ Fuse $R^{*}$ and
$list\\_F^{*}\text{[i]}$, calculate intersection area and union area according
to Formula (5);

7 $temp\\_IoU\leftarrow$ Calculate IoU($S_{\text{inter}}$, $S_{\text{union}}$)
according to Formula (6);

8 $list\\_IoU.\text{add}(temp\\_IoU)$;

9

10 end for

11$R_{r},R_{f}\leftarrow$ Transformation corresponding to maximum IoU
($list\\_IoU$);

12 $F^{**}\leftarrow$ Deform $F$ according to transformation parameters
($R_{r},R_{f}$);

13 $R_{h},R_{v}\leftarrow$ Calculate horizontal and vertical scaling ratios
according to Formula (7) ($L,F^{**}$);

14 return _$R_{r},R_{f},R_{h},R_{v}$_ ;

Algorithm 2 Floor plan registration algorithm

When computing the intersection and union image areas, the Harris operator is
employed to locate the contour vertices of the image, resulting in a vertex
set of polygons
$A={\text{[}(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\text{]}}$. The
Gaussian area formula is utilized to calculate the area of this irregular
polygon:

$\begin{array}[]{ll}S=\frac{1}{2}\left|\sum_{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\right|\end{array}$
(5)

After obtaining the intersection area $S_{inter}$ and union area $S_{union}$
of the images, the IoU is calculated as follows:

$\begin{array}[]{ll}\text{ IoU }=S_{\text{inter }}/S_{\text{union
}}\end{array}$ (6)

Once the maximum IoU is found, the transformation parameters for folding and
rotating the floor plan can be determined. Morphological transformations are
applied to the corresponding combinations of floor plans to obtain the maximum
width and height $(H_{1}$, $V_{1})$ of the floor plan and the maximum width
and height $(H_{2}$, $V_{2})$ of the original LiDAR map. The horizontal and
vertical scaling ratios are then calculated as follows:

$\begin{array}[]{ll}\left\\{\begin{array}[]{l}S_{\mathrm{H}}=H_{1}/H_{2}\\\
S_{\mathrm{V}}=V_{1}/V_{2}\end{array}\right.\end{array}$ (7)

where $S_{H}$ represents the horizontal scaling ratio and $S_{V}$ represents
the vertical scaling ratio.This completes the registration of the original
floor plan with reference to the LiDAR map.

###  III-B Real-time updating strategy

Upon completing the registration of the original floor plan, the real floor
plan can be obtained by applying image transformations based on the
registration parameters. This transformed result serves as the robot’s motion
map, utilized for executing path planning tasks.

However, in dealing with complex room structures, there may be instances where
several rooms share similar configurations. In such cases, the indoor map
obtained through floor plan registration may be erroneous. To address this
issue, it is necessary to employ a new LiDAR map covering a larger area of the
rooms and conduct floor plan registration again to update the indoor map.
Additionally, after the robot has traveled a certain distance, it needs to
reposition itself and update its location on the map to ensure route
alignment. To tackle these challenges, this paper introduces a real-time map
updating strategy, as outlined in Alg. 3

Input: LiDAR map $L$; Original floor plan $F$; Relocation distance $D$

Output: Motion map $M$; Current robot position $P$

1 Initialize robot travel distance $d=0$;

2 Initialize remaining travel distance $S$;

3 Initialize motion map $M$;

4 Initialize partial room structure $N\leftarrow$ Current position partial
room structure;

5 Initialize robot position $P$;

6 if  _$M$ is not available_ then

7 $M\leftarrow$ Floor plan registration as described in Section III-A ($L,F$);

8

9 end if

10$P\leftarrow$ Obtain current robot position ($N,M$);

11 $S\leftarrow$ Path planning, calculate remaining travel distance ($P,M$);

12 while  _$S >0$_ do

13 $N\leftarrow$ Current position partial room structure;

14 $L\leftarrow$ LiDAR map updating ($L,N$);

15 // When robot motion is obstructed or exceeds relocation distance;

16 if  _$(d\leq 0$ or $d\geq D)$_ then

17 $M\leftarrow$ Floor plan registration ($L,F$);

18 $P\leftarrow$ Obtain current robot position ($N,M$);

19 $S\leftarrow$ Path planning, calculate remaining travel distance ($P,M$);

20 $d\leftarrow 0$;

21

22 end if

23 $d\leftarrow d+$ robot’s previous second travel distance;

24 $S\leftarrow S-$ robot’s previous second travel distance;

25

26 end while

27return $M,P$;

Algorithm 3 Real-time Map Updating Algorithm

Upon obtaining the indoor map with the floor plan as reference, the robot
commences movement along the planned path. During motion, the robot continues
to scan the surrounding environment and update the LiDAR map. If the motion
path is obstructed or after traveling a certain distance, the current position
$x_{t}$ is estimated based on the motion model:

$\begin{array}[]{ll}x_{t}=g\left(u_{t},x_{t-1}\right)+\varepsilon_{t}\end{array}$
(8)

where $g$ is the robot’s motion model function, $u_{t}$ is the control input
at time $t$, $x_{t-1}$ is the robot’s position at the previous moment,
$\varepsilon_{t}$ represents the motion noise. Simultaneously, the indoor map
is updated by registering the current LiDAR map with the floor plan. Finally,
based on the current room structure, the corresponding position coordinates
are identified on the map to achieve precise localization, thereby allowing
for path planning again. This workflow iterates until the task is completed.

##  IV EXPERIMENTS

###  IV-A Dataset Construction

The LiDAR images within the dataset are derived from two sources: some are
generated via simulations in a GazeBo environment, while others are captured
by robots operating in real-world settings, as illustrated in Fig. 3.

Figure 3: (a) The LiDAR map obtained by the robot running in the actual
environment (b) Simulation environment generating LiDAR map

The floor plans in the dataset also originate from two distinct sources: a
portion is collected from commonly available floor plans on the internet, and
another portion consists of manually drawn representations of actual room
layouts. All floor plans are stored in JSON format, detailing the coordinates
of corner points of each room, as depicted in Fig. 4.

Figure 4: (a) Annotate the floor plan obtained from the network (b) Annotate
the manually drawn floor plan

###  IV-B Comparison of registration

This paper employs the average IoU ($IoU_{a}$) of the morphological
transformed floor plans as a crucial metric for evaluating the accuracy of the
registration algorithm. The IoU of each room is calculated according to (6),
followed by the computation of $IoU_{a}$:

$\begin{array}[]{ll}IoU_{a}=SUM_{IoU}/n\end{array}$ (9)

where $SUM_{IoU}$ is the sum of all room IoU, $n$ is the number of rooms.

Figure 5: The folding accuracy, rotation accuracy, and $IoU_{a}$ are evaluated
for different degrees of completeness in the LiDAR map

The performance of the proposed algorithm on the test dataset is depicted in
Fig. 5. Notably, under various degrees of completeness in LiDAR images, the
accuracy of rotation and flipping consistently exceeds 85$\%$. The primary
factor influencing the $IoU_{a}$ is the accuracy of scaling, exhibiting a
similar trend to that of scaling accuracy. The $IoU_{a}$ of the registered
floor plans gradually increases with the completeness of the LiDAR images,
ultimately surpassing 94$\%$, while the overall $IoU_{a}$ exceeds 85$\%$,
demonstrating substantial practical utility of the algorithm.

To compare the effectiveness of different image registration algorithms,
experiments were conducted using SURF [7], OBR [8], NBB [9], DIRNet [10], and
the MFR algorithm. The algorithms were applied for image registration, and the
runtime of each algorithm was recorded.

TABLE I: Comparing the performance of different matching algorithms Algorithms |  | Fold
---
Accuracy(%)
| Rotation
---
Accuracy(%)
$IoU_{a}$(%) |  | Average
---
Time(s)
SURF | 45.97 | 43.57 | 43.49 | 0.45
ORB | 53.75 | 59.65 | 53.18 | 0.38
NBB | 75.98 | 77.36 | 74.85 | 1.73
DIRNet | 68.37 | 66.38 | 63.76 | 1.52
MFR | 87.71 | 96.71 | 85.12 | 0.92

The results, as shown in Table I, indicate that the registration algorithm
proposed in this paper outperforms other commonly used image matching
algorithms in terms of folding accuracy, rotation accuracy, and $IoU_{a}$ on
the test dataset. However, this algorithm consumes a relatively longer time,
with an average runtime of 0.92 seconds per match, higher than traditional
image registration algorithms but lower than deep learning algorithms.
Considering the general requirement of real-world applications for runtime to
be within 1 second, this algorithm meets the requirement. Through
comprehensive analysis, it is concluded that this algorithm has significant
advantages in solving the image registration problem between the original
floor plan and LiDAR map.

###  IV-C Simulated Rescue Scenario Testing

Using GazeBo, multiple indoor simulated search and rescue scenarios were
constructed based on the HM3D dataset [11] to comprehensively evaluate the
effectiveness of the proposed improvement method. In this study, comparisons
were made with E2DG-SLAM [12], UV-SLAM [13], and TLIO-SLAM [14]. The simulated
rescue scenarios encompass three situations: the first involves a single
target indoors with a known orientation; the second involves multiple targets
indoors with known orientations; and the third involves an unknown number of
targets indoors with unknown orientations.

Figure 6: Motion trajectory maps of different SLAM algorithms

Fig. 6 illustrates the trajectory results of different SLAM algorithms in
finding two targets with known orientations. It can be intuitively observed
that the FR-SLAM algorithm, after completing the registration of the floor
plan and obtaining the overall room map, can directly reach the position of
the second target. Table II presents the average rescue times of different
SLAM algorithms in three simulated rescue scenarios. It can be seen that the
FR-SLAM algorithm consumes less time in different simulated scenarios compared
to the other algorithms, especially when the number and orientation of targets
are known, where it is 6.7$\%$ lower than the second-lowest method. This is
because the other three algorithms, during the rescue task, do not know if the
current room is connected to other rooms, and the location of the door to
enter the next room is also unknown, thus requiring exploration of each room.
The advance knowledge of the target orientation only changes the order of
house search but does not allow skipping rooms. The search can only be halted
when the number of found targets equals the number of targets known in
advance.

TABLE II: The time consumption of different SLAM algorithms Algorithms | Direction known | Direction unknown
---|---|---
Single | Multiple
E2DG-SLAM | 16m35s | 28m49s | 36m47s
UV-SLAM | 18m45s | 27m15s | 39m27s
TLIO-SLAM | 17m15s | 29m38s | 38m19s
FR-SLAM | 16m12s | 25m24s | 34m53s

##  V CONCLUSIONS

This paper proposes an improved SLAM method, FRSLAM, based on floor plan
registration, addressing the issues of prolonged mapping duration and
inaccurate localization. Utilizing LiDAR images for aligning and deforming
floor plans, a comprehensive motion map is obtained. Concurrently, a real-time
update strategy is employed to enhance the accuracy of floor plan registration
and robot motion localization in complex environments. The MFR algorithm
proposed within the method achieve $IoU_{a}$ of 85$\%$ in comparative tests,
surpassing other benchmark algorithms. In simulated rescue scenario tests, the
FR-SLAM algorithm presented in this paper achieves a maximum time savings of
6.7$\%$ compared to other benchmark algorithms in completing rescue tasks.
However, this algorithm is limited to two-dimensional floor plan information
and does not incorporate the three-dimensional information generated by LiDAR
point clouds, resulting in limited available information for image
registration and robot localization. Future work aims to leverage point cloud
information for three-dimensional reconstruction and utilize three-dimensional
models for robot navigation to improve algorithm efficiency.

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Step 1 Recall that
$\delta_{1}=\sqrt{a_{nT}+\delta_{T}}J_{t}/\sqrt{n_{t}}+a_{nT}+\delta_{T}$,
$\delta_{2}=\delta_{T}+a_{nT}+\sqrt{\delta_{T}}T^{1/(2q)}n_{t}^{-1/2}J_{t}^{1/2}$.
We first show that the standardization with estimated variance is negligible.

$\displaystyle\sup_{\beta}{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\hat{\sigma}^{1/2}_{t}(\beta)-{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma^{1/2}_{t}(\beta)$
$\displaystyle\lesssim\sup_{\beta}\text{$c(\delta_{1}+\delta_{2})|{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma_{t}(\beta)^{1/2}|$}.$

By the Bernstein inequality.

Let $X_{1},\ldots,X_{n}$ be independent zero-mean random variables. Suppose
that $\left|X_{i}\right|\leq M$ ($M$ is a positive constant) almost surely,
for all $i$. Then, for all positive $t$.

$\mathbb{P}\left(\sum_{i=1}^{n}X_{i}\geq
t\right)\leq\exp\left(-\frac{\frac{1}{2}t^{2}}{\sum_{i=1}^{n}\mathbb{E}\left[X_{i}^{2}\right]+\frac{1}{3}Mt}\right)$
(C.25)

If we assume that the $q-$th moment of $\varepsilon_{it}^{2}$ is finite, then
we shall study the inequalities on the event
$\mathcal{A}=\\{\max_{i}\varepsilon_{it}^{2}\Phi_{i,j_{t},t}^{*}\leq M\\}$.
$P(\mathcal{A}^{c})\lesssim n_{t}/(M^{q}J_{t})$, by Assumption 17.

Let $M=c_{n_{t}}n_{t}^{1/q}/J_{t}^{1/q}$, we have $P(\mathcal{A}^{c})\leq
1/(c_{n_{t}}^{q}).$ Now apply Bernstein inequality, conditional on
$\mathcal{F}_{t-1}$, we have

$\displaystyle\max_{j_{t}}n_{t}^{-1}\sum_{i}(\Phi_{i,j_{t},t}^{*}-\tilde{q}_{j_{t}})$
$\displaystyle\lesssim_{p}c_{n_{t}}\sqrt{\log J_{t}/n_{t}J_{t}}.$
$\displaystyle\max_{j_{t}}n_{t}^{-1}\sum_{i}(\Phi_{i,j_{t},t}^{*}\varepsilon_{it}^{2}-\mathbb{E}_{t-1}(\Phi_{i,j_{t},t}^{*}\varepsilon_{it}^{2}))$
$\displaystyle\lesssim_{p}c_{n_{t}}(\sqrt{\log
J_{t}/n_{t}J_{t}}+n_{t}^{-1+1/q}J_{t}^{-1/q}\log J_{t}).$

It is not hard to see that for $q>4,$ if $J_{t}(\log J_{t})^{2}/n_{t}\lesssim
1$, by assumption 23, then we have $c_{n_{t}}(\sqrt{\log
J_{t}/n_{t}J_{t}}+n_{t}^{-1+1/q}J_{t}^{-1/q}\log J_{t})\leq
c_{n_{t}}\sqrt{\log J_{t}/n_{t}J_{t}}.$

By Lemma B.18, we have
$\hat{\sigma}_{j_{t}}^{2}-\sigma_{j_{t}}^{2}\lesssim_{p}\sqrt{J_{t}\log
J_{t}/n_{t}}$

$\displaystyle(1/\hat{\sigma}_{t}(\beta)^{1/2}-1/\sigma_{t}(\beta)^{1/2})=((\sigma_{t}(\beta)^{1/2}-\hat{\sigma}_{t}(\beta)^{1/2})/\hat{\sigma}_{t}(\beta)^{1/2}\sigma_{t}(\beta)^{1/2}).$

We take the following steps,

$\displaystyle{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\hat{\sigma}_{t}(\beta)^{1/2}-{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/{\sigma}_{t}(\beta)^{1/2},$
$\displaystyle=$
$\displaystyle{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma_{t}(\beta)^{1/2}(\sigma_{t}(\beta)^{1/2}/\hat{\sigma}_{t}(\beta)^{1/2}-1),$
$\displaystyle\leq$
$\displaystyle|{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma_{t}(\beta)^{1/2}||(\sigma_{t}(\beta)^{1/2}/\hat{\sigma}_{t}^{1/2}(\beta)-1)|.$

Thus

$\displaystyle\sup_{\beta}{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\hat{\sigma}_{t}(\beta)^{1/2}-{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma_{t}^{1/2}(\beta)$
$\displaystyle\leq$
$\displaystyle\sup_{\beta}|{\sqrt{n_{t}}/\sqrt{J_{t}}}\\{\hat{L}_{t}(\beta)-L_{t}(\beta)\\}/\sigma_{t}^{1/2}(\beta)|\sup_{\beta}|(\sigma^{1/2}_{t}(\beta)/\hat{\sigma}^{1/2}_{t}(\beta)-1)|.$

By assumption 28, we have,

$\displaystyle
P(\sup_{\beta}|(\sigma_{t}(\beta)^{1/2}/\hat{\sigma}_{t}^{1/2}(\beta)-1)|\geq
x),$ $\displaystyle=$ $\displaystyle
P(\max_{j_{t}}|(\sigma_{j_{t}}/\hat{\sigma}_{j_{t}}-1)|\geq x),$
$\displaystyle\leq$ $\displaystyle
P(\max_{j_{t}}|(\sigma_{j_{t}}/\hat{\sigma}_{j_{t}}-1)(\sigma_{j_{t}}/\hat{\sigma}_{j_{t}}+1)|\geq
x),$ $\displaystyle\leq$ $\displaystyle
P(\max_{j_{t}}|(\sigma_{j_{t}}^{2}/\hat{\sigma}_{j_{t}}^{2}-1)|\geq x).$

Since by Lemma B.18

$\max_{j_{t}}|\hat{\sigma}_{j_{t}}^{2}-\sigma_{j_{t}}^{2}|=\textit{O}_{p}((\delta_{1}+\delta_{2})).$

The event for the positive constant $c>0$ (defined in Assumption 28) happens
with probability approaching 1,

$\max_{j_{t}}|\hat{\sigma}_{j_{t}}^{2}-\sigma_{j_{t}}^{2}|\leq c/2.$

We have, by Assumption 28, with probability approaching $1$, $\forall$
$j_{t}$, we have, exist a positive constant $c$,

$\hat{\sigma}_{j_{t}}^{2}\geq\sigma_{j_{t}}^{2}-c/2\geq c/2.$

So we have,

$\displaystyle\mathbb{P}(\max_{j_{t}}|(\sigma_{j_{t}}^{2}/\hat{\sigma}_{j_{t}}^{2}-1)|\geq
x),$ $\displaystyle\leq$
$\displaystyle\mathbb{P}(\max_{j_{t}}|(\sigma_{j_{t}}^{2}-\hat{\sigma}_{j_{t}}^{2})|\geq
xc/2).$

Thus we have

$\sup_{\beta}|(\sigma_{t}(\beta)/\hat{\sigma}_{t}(\beta)-1)|\lesssim_{p}c(\delta_{1}+\delta_{2}).$

It is not hard to derive that under the conditions of Theorem B.12. Similar to
the derivation,

$(\hat{L}_{t}(\beta)-L_{t}(\beta))=\hat{p}_{t}^{\top}(\beta)\\{\mbox{diag}[\tilde{q}_{jt}]\\}^{-1}\\{n_{t}^{-1}\Phi_{t}^{*}\varepsilon_{t}\\}+\textit{O}_{p}(1/J_{t}\vee
h).$ (C.26)

Then

$\displaystyle\sup_{\beta}{\sqrt{n_{t}}/\sqrt{J_{t}}}(\hat{L}_{t}(\beta)-L_{t}(\beta))/\sigma_{t}(\beta)=\sup_{\beta}\sqrt{n_{t}}/\sqrt{J_{t}}\hat{p}_{t}^{\top}(\beta)\\{\mbox{diag}[\tilde{q}_{jt}]\sigma_{t}(\beta)^{1/2}\\}^{-1}\\{n_{t}^{-1}\Phi_{t}^{*}\varepsilon_{t}\\}$
$\displaystyle+\textit{O}_{p}(\sqrt{n_{t}}/(\sqrt{J_{t}}J_{t})\vee\sqrt{n_{t}}h/\sqrt{J_{t}}\vee
1/\sqrt{J_{t}}).$

So the first step is proved.

Step 2 We now show the steps of strong approximation. We couple the sequence
$\sum_{j_{t}}\hat{p}_{j_{t}}(\beta)\sqrt{n_{t}/J_{t}}\tilde{q}_{j_{t}}^{-1}\sum_{i}\Phi_{i,j_{t},t}^{*}\varepsilon_{it}/\sigma_{t}(\beta)^{1/2}$
by
$\sum_{j_{t}}\hat{p}_{j_{t}}(\beta)\sqrt{n_{t}/J_{t}}\tilde{q}_{j_{t}}^{-1}\sum_{i}\Phi_{i,j_{t},t}^{*}(\sigma_{t}\eta_{it})/{\sigma}_{t}(\beta)^{1/2}$
, where $\varepsilon_{it}=_{d}\sigma_{i}\eta_{it}$ conditional on
$\mathcal{F}_{t-1}$.

Now we cite a KMT type theorem to show that such a coupling exists.

######  Theorem C.1.

(Theorem 2.1, Berkes, Liu, and Wu (2014)) Assume that
$X_{i}\in\mathcal{L}^{p}$ ($p$ is an integer.) with mean $0,p>2$, and there
exists $\alpha>p$ such that

$\Theta_{\alpha,p}:=\sum_{j=-\infty}^{\infty}|j|^{1/2-1/\alpha}\delta_{j,p}^{p/\alpha}<\infty$

($\delta_{j,p}$ is $\delta_{j,p}(X_{i})$). Further assume that there exists a
positive integer sequence $\left(m_{k}\right)_{k=1}^{\infty}$ such that

$\displaystyle
M_{\alpha,p}:=\sum_{k=1}^{\infty}3^{k-k\alpha/p}m_{k}^{\alpha/2-1}<\infty,$
$\displaystyle\sum_{k=1}^{\infty}\frac{3^{kp/2}\Theta_{m_{k},p}^{p}}{3^{k}}<\infty,$

and

$\Theta_{m_{k},p}+\min_{l\geq
0}\left(\Theta_{l,p}+l3^{k(2/p-1)}\right)=\textit{o}\left(\frac{3^{k(1/p-1/2)}}{(\log
k)^{1/2}}\right).$

Then there exists a probability space
$\left(\Omega_{c},\mathcal{A}_{c},\mathrm{P}_{c}\right)$ on which we can
define random variables $X_{i}^{c}$ with the partial sum process
$S_{n}^{c}=\sum_{i=1}^{n}X_{i}^{c}$, and a standard Brownian motion
$\mathbb{B}_{c}(\cdot)$, such that
$\left(X_{i}^{c}\right)_{i\in\mathbb{Z}}\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\left(X_{i}\right)_{i\in\mathbb{Z}}$
and (2.7),

$S_{n}^{c}-\sigma\mathbb{B}_{c}(n)=\textit{o}_{a.s.}\left(n^{1/p}\right)\quad\text{
in }\left(\Omega_{c},\mathcal{A}_{c},\mathrm{P}_{c}\right).$

We can see that for iid data the dependence assumption naturally satisfies by
assuming that $\varepsilon_{it}$ has finite $q$th moment, with $q>$4, and by
Assumption 17, we have, conditional on $\mathcal{F}_{t-1},$

$\max_{1\leq l\leq
n_{t}-1}|\sum_{i}^{l}(\varepsilon_{it}-\sigma_{t}\eta_{it})|=\textit{O}_{p}(n_{t}^{1/(2q)}).$

We define

$K(\beta,\beta_{it})=\sum_{j_{t}}\text{$\hat{p}_{j_{t}}(\beta)\sqrt{1/(n_{t}J_{t})}\tilde{q}_{j_{t}}^{-1}$}\Phi_{i,j_{t},t}^{*}/\sigma_{t}(\beta)^{1/2}.$

Then conditional on $\mathcal{F}_{t-1},$by summation by part,

$\displaystyle\sup_{\beta}|\sum_{i}K(\beta,\beta_{it})(\varepsilon_{it}-\sigma_{it}\eta_{it})|$
$\displaystyle\leq\sup_{\beta}|\sum_{i}^{n_{t}-1}[K(\beta,\beta_{(i)t})-K(\beta,\beta_{(i-1)t})]|\max_{1\leq
l\leq
n_{t}-1}|\sum_{i}^{l}\varepsilon_{it}-\sigma_{it}\eta_{it}|+K(\beta,\beta_{(n_{t})t})|\sum_{i}^{n_{t}}\varepsilon_{it}-\sigma_{t}\eta_{it}|$
$\displaystyle\lesssim\max_{1\leq l\leq
n_{t}-1}|\sum_{i}^{l}(\varepsilon_{it}-\sigma_{t}\eta_{it})|\sqrt{J_{t}/n_{t}}$
$\displaystyle\lesssim_{p}n_{t}^{-1/2+1/(2q)}\sqrt{J_{t}}.$

Step 3,

We now show that the conditional normal process
$\sum_{i}K(\beta,\beta_{it})(\sigma_{t}\eta_{it})=_{d}\sqrt{\sum_{i}K(\beta,\beta_{it})^{2}}Z=Z_{n_{t}}$
($Z$ is a standard normal variable) is close enough to the Gaussian variable
$Z_{t}(\beta)$, which we show by Gaussian maximal inequalities. Recall that we
define $Z_{t}(\beta)=\sum_{j_{t}}\hat{p}_{j_{t}}(\beta)Z_{j_{t}}$
($Z_{j_{t}}$s are iid standard normal random variable) as a conditional
Gaussian process, conditional on $\mathcal{G}_{t-1}$, $Z_{t}(\beta)$ is
conditional Gaussian process. Recall
$\sigma_{j_{t}}^{2}=\tilde{q}_{j_{t}}^{-1}\sigma_{t}^{2}/J_{t}$, and
$\sigma_{t}(\beta)^{1/2}=\sum_{j_{t}}\hat{p}_{j_{t}}(\beta)\sigma_{j_{t}}$.
$\beta_{j_{t}}$ lies in the interior point of $\hat{P}_{j_{t}}$. Recall that
$\beta_{j_{t}}$ are some interior point for each partition $j_{t}$. Since we
know that by Gaussian Maximal inequality, conditional on $\mathcal{F}_{t-1}$,

$\displaystyle\mathbb{P}(\sup_{\beta}|\sum_{i}\\{K(\beta,\beta_{it})\sigma_{t}\eta_{it}\\}-Z_{t}(\beta)|\geq
x|\mathcal{F}_{t-1}),$ $\displaystyle\lesssim$
$\displaystyle\sqrt{\log{J_{t}}}\max_{j_{t}}|\sigma_{t}\sqrt{\sum_{i}K(\beta_{j_{t}},\beta_{it})^{2}}-1|,$
$\displaystyle\leq$
$\displaystyle\sqrt{\log{J_{t}}}\max_{j_{t}}|{\sum_{i}K(\beta_{j_{t}},\beta_{it})^{2}\sigma_{t}^{2}}-1|,$
$\displaystyle\leq$
$\displaystyle\sqrt{\log{J_{t}}}\max_{j_{t}}(n_{t})^{-1}|\sum_{i}(\tilde{q}_{j_{t}}^{-1}\Phi^{*}_{i,j_{t},t}-1)|,$
$\displaystyle\lesssim_{p}$
$\displaystyle\sqrt{\log{J_{t}}}\sqrt{J_{t}}\sqrt{\log J_{t}}/\sqrt{n_{t}}.$

Step 4, We shall look at

$\sup_{x}\mathbb{P}(\sup_{\beta}Z_{n_{t}}(\beta)\leq
x)-\mathbb{P}(\sup_{\beta}Z_{t}(\beta)\leq x).$ (C.27)

Define a positive constant $\xi>0$.

$\displaystyle\sup_{x}|\mathbb{P}(\sup_{\beta}Z_{n_{t}}(\beta)\leq
x)-\mathbb{P}(\sup_{\beta}Z_{t}(\beta)\leq x)|$ $\displaystyle\leq$
$\displaystyle|\mathbb{P}(\sup_{\beta}|Z_{n_{t}}(\beta)-Z_{t}(\beta)|\geq\xi)|+|\sup_{x}\mathbb{P}(x\leq\sup_{\beta}Z_{t}(\beta)\leq
x+\xi)|.$

Due to the anticoncentration property of $Z(\beta)$ conditional on
$\mathcal{F}_{t-1}$ as in Lemma C.2, and the dominant convergence theorem, we
have, for $\xi\lesssim\sqrt{J_{t}}^{-1}$

$\displaystyle|\sup_{x}\mathbb{P}(x\leq\sup_{\beta}Z_{t}(\beta)\leq x+\xi)|,$
$\displaystyle|\mathbb{E}(\sup_{x}\mathbb{P}(x\leq\sup_{\beta}Z_{t}(\beta)\leq
x+\xi|\mathcal{F}_{t-1}))|,$
$\displaystyle|\sup_{x}\mathbb{E}\mathbb{P}(x\leq\sup_{\beta}Z_{t}(\beta)\leq
x+\xi|\mathcal{F}_{t-1})|,$ $\displaystyle\lesssim_{p}\sqrt{\log
J_{t}}/\sqrt{{J_{t}}},$

Notice that $\sup_{\beta}Z_{t}(\beta)=\max_{j_{t}}Z_{j_{t}}$. The above
statement is derived based on Lemma C.2. In particular, we have
$X_{j}=Z_{j_{t}}$, $\sigma_{j}=\sigma_{j_{t}}$. From Assumption 28, we have
$\min_{j_{t}}\sigma_{j_{t}}>0$. Also $\epsilon=\xi>0$ as the correspondence
the Lemma. $\mathbb{E}[\max_{j_{t}}Z_{j_{t}}/\sigma_{j_{t}}]\lesssim\sqrt{\log
J_{t}}$ by Gaussian maximal inequality.

Define $\mathcal{L}\left(\max_{1\leq j\leq
p}X_{j},\epsilon\right)=\sup_{t}\mathbb{P}(|\max_{j}X_{j}-t|\leq\epsilon).$

######  Theorem C.2.

(Anti-concentration: Chernozhukov, Chetverikov, Kato, et al. (2013)). Let
$\left(X_{1},\ldots,X_{p}\right)^{T}$ be a centered Gaussian random vector in
$\mathbb{R}^{p}$ with $\sigma_{j}^{2}:=\mathbb{E}\left[X_{j}^{2}\right]>0$ for
all $1\leq j\leq p.$ Moreover, let $\underline{\sigma}:=\min_{1\leq j\leq
p}\sigma_{j},\bar{\sigma}:=\max_{1\leq j\leq p}\sigma_{j}$, and
$a_{p}:=\mathbb{E}\left[\max_{1\leq j\leq
p}\left(X_{j}/\sigma_{j}\right)\right]$ (i) If the variances are all equal,
namely $\underline{\sigma}=\bar{\sigma}=\sigma$, then for every $\epsilon>0$

$\mathcal{L}\left(\max_{1\leq j\leq p}X_{j},\epsilon\right)\leq
4\epsilon\left(a_{p}+1\right)/\sigma$

(ii) If the variances are not equal, namely $\underline{\sigma}<\bar{\sigma}$,
then for every $\epsilon>0$,

$\mathcal{L}\left(\max_{1\leq j\leq p}X_{j},\epsilon\right)\leq
C\epsilon\left\\{a_{p}+\sqrt{1\vee\log(\underline{\sigma}/\epsilon)}\right\\}$

where $C>0$ depends only on $\underline{\sigma}$ and $\bar{\sigma}$.

Since
$\sup_{\xi}|\mathbb{P}(\sup_{\beta}|Z_{n_{t}}(\beta)-Z_{t}(\beta)|\geq\xi)-\mathbb{P}(\sup_{\beta}|Z_{n_{t}}(\beta)-Z_{t}(\beta)|\geq\xi|\mathcal{F}_{t-1})|\to_{p}0$,
which follows similar arguments as in the proof of Theorem SA-4.1, Cattaneo,
Crump, Farrell, and Feng (2022).

###  C.13 Proof of Corollary B.19.1 and B.19.2

The proof is similar to the previous paper and therefore omitted.

###  C.14 Proof of Lemma B.20

###### Proof.

Since by Theorem B.14, we have

$T^{-1/2}\sum_{t}(\hat{\mu}_{t}(\beta)-{\mu}_{t}(\beta))=T^{-1/2}\sum_{t}\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}+\textit{o}_{p}(1).$
(C.28)

Now we can use Theorem C.3 to conduct our uniform inference for beta sorting
estimators. The strong approximation results can be applied on the term
$T^{-1/2}\sum_{t}\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$.
From any $\mathbb{J}_{1},\mathbb{J}_{2}\in B_{\mathbb{J}}$.

Assume that $\tilde{Z}_{t,j}$ follows a normal distribution with
$N(0,\mbox{diag}(\tilde{\Sigma})^{-1/2}\tilde{\Sigma}\mbox{diag}(\tilde{\Sigma})^{-1/2})$,
$c\leq\lambda_{\min}(\tilde{\Sigma})\leq\lambda_{\max}(\tilde{\Sigma})\leq C$,

$\max_{\mathbb{J}\in
B_{\mathbb{J}}}|T^{-1/2}\sum_{t}C_{\mathbb{J},\mathbb{J}}^{-1/2}(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}-\tilde{Z}_{\mathbb{J}}|=\textit{O}_{p}(T^{-\varepsilon^{\prime}}),$
(C.29)

where $\varepsilon^{\prime}$ is a constant between $1/6$ to $0$.
$\bar{\Sigma}=\mbox{diag}(\tilde{\Sigma})^{-1/2}\tilde{\Sigma}\mbox{diag}(\tilde{\Sigma})^{-1/2}$.

∎

###  C.15 Proof of Corollary B.21

###### Proof.

For a constant $\delta>0$ (defined within the proof only).

Let
$\tilde{\mathbb{{U}}}_{T}(\beta)=T^{-1/2}\sum_{t}\hat{\sigma}(\beta)^{-1/2}(\hat{\mu}_{t}(\beta)-\hat{\mu}_{t}(\beta))$.

Denote

$\displaystyle{\mathbb{U}}_{T}(\beta)=T^{-1/2}\sum_{t}\hat{\sigma}_{FM}(\beta,\beta)^{-1/2}(\hat{\mu}_{t}(\beta)-\hat{\mu}(\beta)).$
(C.30)

We shall derive

$\displaystyle\mathbb{P}(\sup_{\beta}\tilde{\mathbb{U}}_{T}(\beta)\geq
x)-\mathbb{P}(\sup_{\beta}G_{T}(\beta)\geq x)$
$\displaystyle\leq\mathbb{P}(\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-G_{T}(\beta)|\geq\delta)+\sup_{x}\mathbb{P}(x\leq\sup_{\beta}G_{T}(\beta)\leq
x+\delta).$

Define
$\mathbb{U}_{T}^{*}(\beta)=T^{-1/2}\sum_{t}{\sigma}(\beta)^{-1/2}(\hat{\mu}_{t}(\beta)-{\mu}_{t}(\beta))$.
Next we shall study $\sup_{\beta}\tilde{\mathbb{U}}_{T}(\beta)$ with the
replaced true ${\sigma}(\beta)$. Since
$\mathbb{P}(\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-G_{T}(\beta)|\geq\delta)\leq\mathbb{P}(\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-\mathbb{U}^{*}_{T}(\beta)|\geq\delta/2)+\mathbb{P}(\sup_{\beta}|\mathbb{U}^{*}_{T}(\beta)-G_{T}(\beta)|\geq\delta/2)$.
From Assumption 29, we have that
$\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-\mathbb{U}_{T}^{*}(\beta)|\leq(\sigma(\beta)^{-1/2}-\hat{\sigma}(\beta)^{-1/2})(T^{-1/2}\sum_{t}(\hat{\mu}_{t}(\beta)-\mu_{t}(\beta)))\lesssim{\textit{O}_{p}(r_{\sigma})\sup_{\beta}|\mathbb{U}_{T}^{*}(\beta)|}\lesssim
r_{\sigma}T^{-1/2+1/2q}J_{a}^{1/2q}\lesssim T^{-\varepsilon^{\prime}}$.

For the term
$\mathbb{P}(\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-G_{T}(\beta)|\geq\delta)$
we analyze via a linearization following Theorem B.14 and strong approximation
to the leading term. For the term
$\sup_{x}\mathbb{P}(x\leq\sup_{\beta}G_{T}(\beta)\leq x+\delta)$ we study via
anticoncentration inequalities for Gaussian processes (c.f. Lemma C.2.)
Because of Assumption 29 and there are at most $J_{a}$ jumps in the limiting
process (recall $J_{a}$ is the number combined intervals over time), we have
for $\delta\lesssim\sqrt{J}^{-1}$,
$\sup_{x}\mathbb{P}(x\leq\sup_{\beta}G_{T}(\beta)\leq
x+\delta)\lesssim\sqrt{\log J_{a}}/\sqrt{J}$.

Next we study,
$\mathbb{P}(\sup_{\beta}|\tilde{\mathbb{U}}_{T}(\beta)-G_{T}(\beta)|\geq\delta)$.
For a fine enough grid on the interval $[\beta_{l},\beta_{u}]$. For a small
positive constant $\tilde{\delta}$, we denote the grids as
$B_{\tilde{\delta}}=[\beta_{1},\beta_{2},\cdots,\beta_{c/\tilde{\delta}}]$,
and $\pi(\beta)$ denotes the closest point of $\beta$ in terms of the absolute
norm $|.|$. We select the partition such that
$G_{T}\circ\pi_{\tilde{\delta}}(\beta)=Z_{\mathbb{J}}$. Denote
$\pi_{\tilde{\delta}}(\beta)$ as the closest point of $\beta$ in terms of
metric $|.|$ in $B_{\delta}$.

Recall that we let
$\tilde{\mathbb{U}}_{T}(\beta)={\text{$\sqrt{{T}}$}}\\{\hat{\mu}(\beta)-T^{-1}\sum_{t}\mu_{t}(\beta)\\}/\hat{\sigma}^{1/2}(\beta)$.
We define $\sigma(\beta)^{-1/2}G_{T}(\beta)$ as a Gaussian process on a proper
probability space with covariance
$\sigma(\beta_{1})^{-1/2}\sigma(\beta_{1},\beta_{2})\sigma(\beta_{2})^{-1/2}$.

Define $\tilde{\mathbb{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta)$ as the process
$\tilde{\mathbb{U}}_{T}$ evaluated at the grid points $B_{\tilde{\delta}}$.

The following relations will be proved below for some
$1/6>\varepsilon^{\prime}>0$ and some $J\rightarrow\infty$ :

  * •

(Coupling with a Normal Vector)let the limit variance covariance matrix be
$\Sigma(\beta_{\mathbb{J}})=\lim_{T\to\infty}\mathbb{C}\mathrm{ov}(\tilde{\mathbb{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta_{\mathbb{J}})$,
denote $\mathbb{J}$ as the number of elements within $B_{\tilde{\delta}}$,
there exists $G_{T}\circ\pi_{\delta}(\beta_{\mathbb{J}})\sim
N\left(0,\Sigma(\beta_{\mathbb{J}})\right)$ such that

$r_{2}=|\tilde{\mathbb{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta_{\mathbb{J}})-Z_{\mathbb{J}}|_{\infty}=\textit{O}_{p}(T^{-\varepsilon^{\prime}}\vee\sqrt{T}(J^{-1}\vee
h))=\textit{o}\left(T^{-\varepsilon^{\prime}}\right);$

The follow steps proves the statement (1). Denote for a small enough positive
constant $\varepsilon$

$\displaystyle\hat{\beta}_{(j)_{,}t}$
$\displaystyle=F_{\hat{\beta},t}^{-1}(j/J_{t}),$
$\displaystyle\hat{\beta}_{(j)_{,}t}^{-}$
$\displaystyle=F_{\hat{\beta},t}^{-1}(j/J_{t})-\varepsilon.$

We denote $B_{\mathbb{J}}$ as the collection $\hat{\beta}_{(j)_{,}t}$ for all
$j_{t}$ and $t$. Thus we have,

$\displaystyle\sup_{\beta\in[\beta_{l},\beta_{u}]}|\sqrt{T}\sum_{t}\\{\hat{\mu}_{t}(\beta)^{1/2}-\mu_{t}(\beta)\\}/\hat{\sigma}(\beta)^{1/2}-\sqrt{T}\\{\hat{\mu}_{t}(\pi(\beta))-\mu_{t}(\pi(\beta))\\}/\hat{\sigma}(\pi(\beta))^{1/2}|$
$\displaystyle\leq$
$\displaystyle\sup_{\beta\in[\beta_{l},\beta_{u}]}|\sqrt{T}^{-1}\\{\sum_{t}\sum_{j_{t}}\\{\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\hat{\sigma}(\beta)^{1/2}-\hat{p}_{j_{t},t}\text{($\pi$($\beta$))}E_{n_{t},j_{t}}/\hat{\sigma}(\pi(\beta))^{1/2}\\}\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle+\textit{o}_{p}(T^{-1/2+1/2q}).$

Moreover for the leading term we have,

$\displaystyle\sqrt{T}^{-1}\max_{\beta\in
B_{\mathbb{J}}}|\sum_{t=l}^{T-1}(\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\hat{\sigma}(\beta)^{1/2}-\sum_{j_{t}}\hat{p}_{j_{t},t}\text{($\beta^{-}$)}E_{n_{t},(j_{t}-1)}/\hat{\sigma}(\beta^{-})^{1/2})\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle\leq$ $\displaystyle\sqrt{T}^{-1}\max_{\beta\in
B_{\mathbb{J}}}|\sum_{t}(\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\hat{\sigma}_{j_{t}}-\sum_{j_{t}}\hat{p}_{j_{t},t}\text{($\beta^{-}$)}E_{n_{t},(j_{t}-1)}/\hat{\sigma}_{j_{t}-1})\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle\leq$ $\displaystyle\sqrt{T}^{-1}\max_{\beta\in
B_{\mathbb{J}}}|\sum_{t}[(\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\hat{\sigma}_{j_{t}}-\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\sigma_{j_{t}}-\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta^{-})E_{n_{t},(j_{t}-1)}/\hat{\sigma}_{j_{t}-1}$
$\displaystyle+$
$\displaystyle\sum_{j_{t}}\hat{p}_{j_{t},t}\text{($\beta^{-}$)}E_{n_{t},(j_{t}-1)}/\sigma_{j_{t}-1})\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}]|$
$\displaystyle+$ $\displaystyle\sqrt{T}^{-1}\max_{\beta\in
B_{\mathbb{J}}}|\sum_{t}[\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)E_{n_{t},j_{t}}/\sigma_{j_{t}}-\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta^{-})E_{n_{t},(j_{t}-1)}/\sigma_{j_{t}-1}]\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle\leq$
$\displaystyle\sqrt{T}^{-1}\max_{t,j_{t}}|(E_{n_{t},j_{t}}/\hat{\sigma}_{j_{t}}-E_{n_{t},(j_{t})}/\sigma_{j_{t}})\\{f_{t}-E(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle+\sqrt{T}^{-1}\max_{t,j_{t}}|(E_{n_{t},j_{t}-1}/\hat{\sigma}_{j_{t}-1}-{E_{n_{t},j_{t}-1}}/\sigma_{j_{t}-1})\\{{f_{t}}{-E}(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle+\sqrt{T}^{-1}\max_{t,j_{t}}|(E_{n_{t},j_{t}}/\sigma_{j_{t}}-E_{n_{t},(j_{t}-1)}/\sigma_{j_{t}-1})\\{f_{t}{-E}(f_{t}|\mathcal{F}_{t-1})\\}|$
$\displaystyle\lesssim_{p}$
$\displaystyle\sqrt{T}^{-1}T^{1/2q}(J_{u}^{1/2q}{J}^{-1}\vee(\delta_{1}+\delta_{2}))\max_{t}\|f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1})\|_{2q}^{1/2}.$

Thus
$r_{1}\lesssim_{p}\sqrt{T}^{-1}T^{1/2q}\max_{t}\|f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1})\|_{2q}$.
The second last inequality is due the fact that $\hat{\beta}_{(j_{t})t}$ are
not equal with probability one for different $j_{t}$ and $t$, and the jump
points occurs $\hat{\beta}_{(j),t}$. The last inequality is due to the
boundedness of the $2q$th moment of
$f_{t}{-\mathbb{E}}(f_{t}|\mathcal{F}_{t-1})$, the moment of
$(E_{n_{t},j_{t}}/\sigma_{j_{t}}-E_{n_{t},(j_{t}-1)}/\sigma_{j_{t}-1})$ is
order $J_{t}^{-1}$ (c.f. Assumption 29), and
$\max_{j_{t}}\sigma_{j_{t}}-\hat{\sigma}_{j_{t}}=\mbox{\tiny$\mathcal{O}$}_{p}(1)$.
The follow steps proves the statement (3). By the maximal inequality for
Gaussian random variables, we have

$r_{3}\lesssim_{p}\sqrt{\log(J_{a})}\sqrt{J}^{-1}.$

The follow steps prove the statement (2) by verifying conditions of Theorem
C.3(i)-iv)). We mention the correspondence between subjects in the Theorem and
our case. Recall that for the grid
$\overline{\beta}_{v}=[\beta_{1},\beta_{2},\cdots,\beta_{c/\tilde{\delta}}]$.
This corresponds to ${J}_{a,\delta}$ distinct value of $\beta$. From
Assumption 29 for avoiding singularity of the variance covariance matrix. We
have
$\sigma(\beta_{j_{t}},\beta_{j_{t}^{\prime}})\neq\sigma(\beta_{j_{t}},\beta_{j_{t}})\quad\mbox{or}\quad\sigma(\beta_{j_{t}^{\prime}},\beta_{j_{t}^{\prime}})$.
In the theorem $n=T$,
$X_{t}=\sqrt{T}^{-1}\sum_{t=1}^{T}\sum_{j_{t}}\hat{p}_{t,j_{t}}(\overline{\beta}_{v})E_{n_{t},j_{t}}[f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1})]/\hat{\sigma}(\beta)^{1/2}$.
Also $p=J_{a,\tilde{\delta}}$. Thus the $\Sigma_{z}$ in the theorem therein
corresponds to
$[\hat{\sigma}(\beta_{i},\beta_{i^{\prime}})/{\hat{\sigma}^{1/2}\hat{\sigma}(\beta_{i^{\prime}})^{1/2}}]_{i,i^{\prime}}$,
which is a $\Sigma_{J_{a,\delta},J_{a,\delta}}$ dimension matrix. We shall
assume that
$c<\lambda_{\min}(\Sigma_{J_{a},\delta})<\lambda_{\max}(\Sigma_{J_{a},\delta})<C^{\prime}$,
with $C^{\prime},c>0$. This proves iii). Moreover, to prove ii), we shall
assume that $m=\textit{O}(T^{1/6})$, $M=\textit{O}(T^{1/3})$,
$J_{a,\delta}\lesssim\exp(T^{\varepsilon^{\prime}})$, with
$\varepsilon^{\prime}=1/9$. To derive the strong approximation for
$\sup_{\beta\in
B_{\tilde{\delta}}}T^{-1/2}\sum_{t}\sum_{j_{t}}\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$
uniformly over $\beta$. We note that it is equivalent to look at strong
approximation for
$\max_{\mathbb{J}}T^{-1/2}\sum_{t}\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$.
$\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$
is involved, and it is still martingale difference sequence with respect to
$\mathcal{F}_{t-1}$, as $\hat{p}_{j_{t},t}(\beta)$ is measurable with respect
to $\mathcal{F}_{t-1}$. It remains to check the dependence adjusted norm for
$\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$.
We see that since $\hat{p}_{j_{t},t}(\beta)$ is uniformly bounded by $1$. For
martingale differences, let
$e_{t,j_{t}}=(\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}$,
and $\delta_{q,m}(e_{.,j_{t},l})=0$ for $l>0$ (recall that $e_{.,j_{t},l}$ is
defined by $e_{.,j_{t}}$ replaced with an iid copy of the $l$th lag.) Thus due
to the property of martingale differences the dependence adjusted norm is just
the norm itself
$\Theta_{m^{0},q}(e_{.},j_{t})=\|\hat{p}_{j_{t},t}(\beta)(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))E_{n_{t},j_{t}}\|_{q}\leq\|(f_{t}-\mathbb{E}(f_{t}|\mathcal{F}_{t-1}))\|_{2q}\|E_{n_{t},j_{t}}\|_{2q}$.
Then by Assumption 29, ii) is verified regarding the bounded dependence
adjusted norm.

∎

###  C.16 Proof of Theorem B.23

###### Proof.

Define the standardized process as the following,

$\displaystyle\tilde{U}_{T}(\beta_{1},\beta_{2},\beta_{3})=$
$\displaystyle|T^{-1}\sum_{t}\sqrt{Tn_{t}/{J_{t}}}\\{\hat{\mu}_{t}(\beta_{1})+\hat{\mu}_{t}(\beta_{3})-2\hat{\mu}_{t}(\beta_{2})-(\mu_{t}(\beta_{1})+\mu_{t}(\beta_{3})-2\mu_{t}(\beta_{2}))\\}/\hat{\sigma}_{D}(\beta_{1,2,3})^{1/2}|.$
$\displaystyle{U}_{T}(\beta_{1},\beta_{2},\beta_{3})=$
$\displaystyle|T^{-1}\sum_{t}\sqrt{Tn_{t}/{J_{t}}}\\{\hat{\mu}_{t}(\beta_{1})+\hat{\mu}_{t}(\beta_{3})-2\hat{\mu}_{t}(\beta_{2})-(\mu_{t}(\beta_{1})+\mu_{t}(\beta_{3})-2\mu_{t}(\beta_{2}))\\}/{\sigma}_{D}(\beta_{1,2,3})^{1/2}|.$

Define the centered and standardized process as the leading term as follows,

$\displaystyle
Z_{T}(\beta_{1},\beta_{2},\beta_{3})=\sqrt{T}T^{-1}\sum_{t}\sqrt{n_{t}/J_{t}}n_{t}^{-1}\sum_{i}\sum_{j_{t}}(\hat{p}_{j_{t},t}(\beta_{1})+\hat{p}_{j_{t},t}(\beta_{3})-2\hat{p}_{j_{t},t}(\beta_{2}))$
$\displaystyle\tilde{q}_{j_{t}}^{-1}\Phi_{i,j_{t},t}^{*}\varepsilon_{it}/{\sigma}_{D}(\beta_{1,2,3})^{1/2}.$

We shall analyze the leading term of the above statistics object,

$\displaystyle\sup_{\beta_{1,}\beta_{2},\beta_{3}}|\tilde{U}_{T}(\beta_{1},\beta_{2},\beta_{3})|$
$\displaystyle=$
$\displaystyle\sup_{\beta_{1,}\beta_{2},\beta_{3}}|{U}_{T}(\beta_{1},\beta_{2},\beta_{3})|+\textit{O}_{p}(r_{1,2,3}),$
$\displaystyle=$
$\displaystyle\sup_{\beta_{1,}\beta_{2},\beta_{3}}|Z_{T}(\beta_{1},\beta_{2},\beta_{3})|+\textit{O}_{p}(r_{1,2,3}\vee\sqrt{Tn_{u}/J}J^{-1}\vee\sqrt{Tn_{u}/J}h),$

where the first equality is due to Assumption 27, and the second equality is
due to Theorem B.14. We shall then prove the following steps:

  * •

(Coupling with a Normal Vector)let the limit variance covariance matrix be
$\Sigma(\beta_{\mathbb{J}_{1}},\beta_{\mathbb{J}_{2}},\beta_{\mathbb{J}_{3}})=\lim_{T\to\infty}\mathbb{C}\mathrm{ov}({{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta_{\mathbb{J}_{1}},\beta_{\mathbb{J}_{2}},\beta_{\mathbb{J}_{3}})$,
denote $\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}$ as the number of
elements within $B_{\tilde{\delta}}$, there exists
$G_{T}\circ\pi_{\delta}(\beta_{\mathbb{J}})\sim
N\left(0,\Sigma(\beta_{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}}))\right]$
such that

$r_{2}^{\prime}=|\tilde{\mathbb{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta_{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}})-Z_{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}}|_{\infty}=\textit{O}_{p}(T^{-\varepsilon^{\prime}}\vee\sqrt{T}(J^{-1}\vee
h))=\textit{o}\left((nT)^{-\varepsilon^{\prime}}\right);$

Let $Z_{T}(\beta)$ be a mean zero Gaussian process, with variance
$\sigma_{d}(\beta)=\mbox{lim}_{T\to\infty}T^{-1}\sum_{t}\sigma_{t}^{2}\tilde{q}_{j_{t}}^{-1}\hat{p}_{j_{t},t}(\beta)$.
Since we have

$\displaystyle|\tilde{\mathbb{U}}_{T}\circ\pi_{\tilde{\delta}}(\beta_{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}})-Z_{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}}|_{\infty}$
$\displaystyle\leq$ $\displaystyle
C\mbox{sup}_{\beta\in\beta_{v}}|\sum_{t}\sqrt{T}^{-1/2}\sqrt{n_{t}J_{t}}^{-1}\sum_{j_{t}}\Phi_{i,j_{t},t}^{*}\tilde{q}_{j_{t}}^{-1}\varepsilon_{it}\hat{p}_{j_{t},t}(\beta)-Z_{T}(\beta)|\max_{\beta}(\sigma^{1/2}(\beta))/\min_{\beta}\sigma_{D}^{1/2}(\beta_{1,2,3}).$

Apply Theorem C.3 again, with $n$ correspond to $\sum_{t}n_{t}\asymp n_{u}T$.
$X_{t}$ corresponds to
$\Phi_{i,j_{t},t}^{*}\varepsilon_{it}\hat{p}_{j_{t},t}(\beta_{v})\tilde{q}_{j_{t}}$.
$\Sigma_{z}$ corresponds to $\operatorname*{diag}(\sigma_{d}(\beta_{v}))$.
Since by Assumption 27, $\max_{\beta\in\beta_{v}}\sigma_{d}(\beta)$ is bounded
from the above and $\min_{\beta\in\beta_{v}}\sigma_{d}(\beta)$ is bounded from
the below. In addition, we have by Assumption 27, the $q$th moment of
$\max_{j_{t},t}\|\Phi_{i,j_{t},t}^{*}\tilde{q}_{j_{t}}^{-1}\varepsilon_{it}\|_{q}\lesssim
J^{1-1/q}$. Thus we can prove that

$\displaystyle\mbox{sup}_{\beta\in\beta_{v}}|\sum_{t}\sqrt{T}^{-1/2}\sqrt{n_{t}J_{t}}^{-1}\sum_{j_{t}}\Phi_{i,j_{t},t}^{*}\tilde{q}_{j_{t}}^{-1}\varepsilon_{it}\hat{p}_{j_{t},t}(\beta)-Z_{T}(\beta)|=\textit{O}_{p}((n_{u}T)^{-\varepsilon^{\prime}}),$
(C.31)

for a constant $0<\varepsilon^{\prime}<1/6$.

∎

###  C.17 Strong approximation for weakly dependent processes

Let us first derive results for nonstationary high dimensional time series. We
denote $X_{t}$ ($t\in 1,\cdots,n$) as a $p-$ dimension zero mean time series,
we let $X_{t,j}=H_{t,j}(\varepsilon_{t},\varepsilon_{t-1},\cdots)$. We let
$X_{t,j}=H_{t,j}(\varepsilon_{t},\varepsilon_{t-1},\cdots,\varepsilon_{t-k}^{*},\cdots)$,
where $\varepsilon_{t-k}^{*}$ is an independent i.i.d. copy of
$\varepsilon_{t-k}$ . We can denote the dependence adjusted norm as

$\delta_{j,k,q}=\sup_{t}\|X_{t,j}-X_{t,j}^{k*}\|.\\\ $ (C.32)

We denote $\Gamma_{\alpha,q}=\sup_{l\geq 0}l^{-\alpha}(\sum_{j}(\sum_{k\geq
l}\delta_{j,k,q})^{q})^{1/q}$. And
$\Theta_{q,\alpha}=\||X_{.}|_{\infty}\|_{q,\alpha}(\log
p)^{3/2}\wedge\Gamma_{\alpha,q}$. $\Psi_{2,\alpha}=\max_{j}\sup_{l\geq
0}l^{-\alpha}(\sum_{k\geq l}\delta_{j,k,q})$.

######  Theorem C.3.

Suppose that $X_{t}$ is a $p-$ dimensional mean zero nonstationary time series
then on a rich probability space, there exists a Gaussian random variable $Z$
such that i)
$\Sigma_{z}=\lim_{n\to\infty}n^{-1}\sum_{t}\sum_{l\geq
0}\mathbb{E}(X_{t}X_{t-l}^{\top})$, and $Z\sim\operatorname{N}(0,\Sigma_{z})$.
Assume that $X_{t}$ has elementwise bounded $q$th moment ($q>4$). The element
of $X_{t}$ is $X_{t,j}$. ii) $\log(p)^{3}\leq L$ and $\log p\ll(M/m)^{1/3}$.
The dependence adjusted norm for $X_{t}$, i.e. $\Theta_{q,\alpha}$ and
$\Psi_{2,\alpha}$ are bounded. We define the dependence adjusted norm for
$Z_{t}$ as
$\Phi_{\psi_{1},\alpha^{\prime}}=\max_{q}q^{-\alpha^{\prime}}\max_{j}\Theta^{z}_{\alpha^{\prime},q,j}.$
Let $\beta=2/(1+2\alpha^{\prime})$. Let $\alpha^{\prime}=1,$ $\beta=2/3$. iii)
We assume that for the long run variance $\Sigma_{z}$,
$\lambda_{\max}(\Sigma_{z})\leq C$ and $\lambda_{\min}(\Sigma_{z})\geq c>0$,
then we have,

$\mathbb{P}(|\sqrt{n}^{-1/2}\sum_{t}X_{t}-Z|_{\infty}\geq\delta(p,L,m,q,\alpha))\to
0.$ (C.33)

Note that we shall let the small blocks $m\ll M$ and $L=\lfloor
n/(m+M)\rfloor$. iv) $\delta(p,L,m,q,\alpha)\to 0$. We let $n\gg mL(\log
p)^{2}$, and $\delta(p,L,m,q,\alpha)\lesssim L^{-1/6}(\log p)^{3/2}\vee
L^{-1/2+1/q}(\log p)^{1-1/q}p^{1/q}\vee m^{-\alpha}\sqrt{\log p}\vee
m^{1/2-1/q-\alpha}n^{1/q-1/2}\ll 1$ for $\alpha>1/2-1/q$. And
${L}^{-1/2}\sqrt{\log p^{3}}\log(p\vee L)\ll\delta(p,L,m,q,\alpha))$. $(\log
p)^{1/{\beta}}m^{-\alpha}\vee\sqrt{mL}(\log p)^{1/\beta}\ll\sqrt{n}$.
$\\{(\log p)^{1/{\beta}}m^{-\alpha}\vee\sqrt{mL}(\log
p)^{1/{\beta}}\\}/{\sqrt{n}}\lesssim\delta(p,L,m,q,\alpha)\ll 1$.

######  Remark C.4.

(discussion of rates) We see that $q,m,M,L,p$ interplays with each other.
Compared to the iid case, we have observation loss, with respect to blocks.
The term $L^{-1/6}(\log p)^{3/2}\vee L^{-1/2+1/q}(\log p)^{1-1/q}p^{1/q}$
corresponds to the rate in the iid case with $L$ replaced by $n$ in Theorem
2.1 Chernozhukov, Chetverikov, and Kato (2016). We show an example, when we
set $q$ to be large enough and $p$ to be small enough such that
$L^{-1/2+1/q}(\log p)^{1-1/q}p^{1/q}\vee m^{1/2-1/q-\alpha}n^{1/q-1/2}$ are of
small order. We analyze $L^{-1/6}(\log p)^{3/2}\vee m^{-\alpha}\sqrt{\log p}$.
For example let $L=n^{2/3}$, then $n^{1/(9\alpha)}\ll m\ll M\lesssim n^{1/3}$.
Then $L^{-1/6}(\log p)^{3/2}\vee m^{-\alpha}\sqrt{\log p}\lesssim
n^{-1/9}(\log p)^{3/2}$. It would also be noted that if the dependence is
rather weak $\alpha$ can be very large and therefore the rate of $m$ can be
small and the block size can diverge less slowly. Thus we can see that for $q$
to be large enough, the rate $\delta(p,L,m,q,\alpha)$ can be of the following
order ${L}^{-1/2}\sqrt{\log p^{3}}\log(p\vee L)\ll\delta(p,L,m,q,\alpha)\asymp
L^{-1/6}(\log p)^{3/2}\vee(\log p)^{3/2}\sqrt{m/M}$.

###### Proof.

  * Step 1 divide observations into block, approximate by m dependent blocks.
We denote
$X_{t,m}=\mathbb{E}(X_{t}|\varepsilon_{t-m},\cdots,\varepsilon_{t})$. We shall
divide the observations into big blocks of size $M$ and small blocks of size
$m$.

$\displaystyle L_{b}=[(b-1)(M+m)+1,bM+(b-1)m]$ (C.34) $\displaystyle
S_{b}=[bM+(b-1)m+1,b(M+m)].$ (C.35)

This in total leads to $L=\lfloor T/(m+M)\rfloor$. We then throw away the
observations in small blocks to construct approximation of the partial sums by
partial sum independent blocks. Namely we denote

$\displaystyle Y_{b,m}=\sum_{t\in L_{b}}X_{t}$ $\displaystyle
T_{b}=\sum_{b=1}^{L_{b}}Y_{b,m}.$

The m- dependent counterpart is denoted as

$\displaystyle\tilde{Y}_{b,m}=\sum_{t\in L_{b}}X_{t,m}$
$\displaystyle\tilde{T}_{b}=\sum_{b=1}^{L_{b}}\tilde{Y}_{b,m}.$

According to similar steps as in Lemma 7.1 Zhang and Wu (2017), we have that
if $\Theta_{q,\alpha}<\infty$. We let $\alpha>1/2-1/q$. Then we have

$\mathbb{P}(|\sum_{t}X_{t}-\sum_{b=1}^{L_{b}}\tilde{Y}_{b,m}|_{\infty}\geq
y)\leq f(y,m),$ (C.36)

where $f(y,m)$ is denoted as

$y^{-q}nm^{q/2-1-\alpha
q}\Theta^{q}_{q,\alpha}+p\exp(-C_{q,\alpha}y^{2}m^{2\alpha}/n\Psi^{2}_{2,\alpha}).$
(C.37)

By choose sufficient large $m$ we can arrange $\sum_{b}\tilde{Y}_{b,m}$ to be
sufficiently close to $\sum_{t}X_{t}$. If $\Theta_{q,\alpha}$ and
$\Psi_{2,\alpha}$ are bounded, we need to assume
$n^{1/q}m^{1/2-1/q-\alpha}n^{-1/2}\ll 1$ and $n^{1/2}m^{-\alpha}\sqrt{\log
p}n^{-1/2}\ll 1$. $y$ shall be set $n^{1/q}m^{1/2-1/q-\alpha}n^{-1/2}\vee
n^{1/2}m^{-\alpha}\sqrt{\log p}n^{-1/2}\ll yn^{-1/2}\ll 1$.

  * Step 2

Approximate independent blocks by Gaussian random variables.

As $\sum_{b}\tilde{Y}_{b,m}$ are partial sum of independent blocks by
construction, we shall apply Theorem 3.1 as in Chernozhukov, Chetverikov, and
Kato (2016). We denote $Y_{b,m,j}$ as the $j$th element of $Y_{b,m}$.
$g(\delta)=\frac{\log^{2}p}{\delta^{3}\sqrt{L}}\\{L_{n}+M_{L,1}(\delta)+M_{L,2}(\delta)\\}$.
We denote $Z_{b}$ as a $p$ dimension mean zero Gaussian random variable with
variance covariance structure as $\mathbb{E}(Y_{b}Y_{b}^{\top})$, then

$\displaystyle\mathbb{P}(\max_{j}|\sum_{b}\sqrt{L}^{-1/2}\tilde{Y}_{b,m,j}|\in
A)-\mathbb{P}(\max_{j}|\sum_{b}\sqrt{L}^{-1/2}Z_{b,j}|\in A^{\delta})$
$\displaystyle\leq g(\delta),$ (C.38)

where the term in $g(\delta)$ involves
$L_{n}=\mbox{max}_{b,j}\mathbb{E}(|Y_{b,j}|^{3})$,

$M_{L,1}(\delta)=L^{-1}\sum_{b}\mathbb{E}[\max_{j}|Y_{b,j}|^{3}\bm{1}\\{\max_{j}|Y_{b,j}|\geq\delta\sqrt{L}/\log
p\\}],$

and

$M_{L,2}(\delta)=L^{-1}\sum_{b}\mathbb{E}[\max_{j}|Z_{b,j}|^{3}\bm{1}\\{\max_{j}|Z_{b,j}|\geq\delta\sqrt{L}/\log
p\\}].$

As the statement in (Step 2) would imply

$\mathbb{P}(\max_{j}|\sqrt{L}^{-1/2}\sum_{b}(\tilde{Y}_{b,j}-Z_{b,j})|\geq\delta)\leq
g(\delta).$

Now we can see that with properly chosen $m,\delta,y$,

$\displaystyle\mathbb{P}(\sqrt{L}^{-1/2}|\sum_{t}X_{t}-\sum_{b}{Z}_{b}|_{\infty}\geq\delta+y)\leq
f(y,m)+g(\delta),$ (C.39)

we shall pick $m$ $M$, $\delta$ such that $\delta/\sqrt{M+m}\ll 1$ and
$g(\delta)\to 0$.

To analyze the rate of $g(\delta)$ we make use of the following lemma,

######  Lemma C.5 (Burkholder (1988), Rio (2009)).

Let $q>1$, $q^{\prime}=\min(q,2)$. Let $M_{n}=\sum^{n}_{t=1}\xi_{t}$; where
$\xi_{t}\in\mathcal{L}^{q}$ (i.e., $\|\xi_{t}\|_{q}<\infty$) are martingale
differences. Then

$\|M_{n}\|_{q}^{q^{\prime}}\leq
K^{q^{\prime}}_{q}\sum_{t=1}^{n}\|\xi_{t}\|^{q^{\prime}}_{q}\quad\text{where}\quad
K_{q}=\max((q-1)^{-1},\sqrt{q-1}).$

Denote $\theta_{q,j,\alpha}=\sup_{l\geq 0}l^{-\alpha}\sum_{k\geq
l}\delta_{j,k,q}$, by Lemma C.5,

$\displaystyle L_{n}$ $\displaystyle=$
$\displaystyle\mbox{max}_{b,j}\mathbb{E}(|Y_{b,j}|^{3})\lesssim
M^{3/2}\theta_{q,j,\alpha}^{3}.$

${v}_{L,q}=\max_{b}(\mathbb{E}(\max_{j}|Y_{b,j}|^{q}))^{1/q}\lesssim(M^{1/2}\theta_{q,j,\alpha})$.

$\displaystyle
M_{L,1}(\delta)=(L)^{-1}\sum_{b}\mathbb{E}\max_{j}|Y_{b,j}|^{3}\bm{1}\\{\max_{j}|Y_{b,j}|\geq\delta\sqrt{L}/\log
p\\}$ $\displaystyle\lesssim$ $\displaystyle\frac{(\log
p)^{q-3}}{(\delta\sqrt{L})^{q-3}}v^{q}_{L,q}p\lesssim\frac{(\log
p)^{q-3}}{(\delta\sqrt{L})^{q-3}}(M^{q/2}\theta^{q}_{q,j,\alpha})p,$

following the steps as the proof of Theorem 2.1 of Chernozhukov, Chetverikov,
and Kato (2016), however we have a better bound for the $M_{nT,2}(\delta)$.
For a Gaussian random variable we have, $Z_{b,j}$ follows a
$N(0,\Sigma_{b,jj})$, let $\xi=\max_{1\leq j\leq
p}|Z_{b,j}|/\Sigma_{b,jj}^{1/2}$,

$\displaystyle\mathbb{E}|\xi|^{3}\bm{1}(\xi\geq t)$ $\displaystyle=$
$\displaystyle\mathbb{P}(\xi>t)t^{3}+3\int^{\infty}_{t}\mathbb{P}(\xi>x)x^{2}dx$
$\displaystyle\leq$ $\displaystyle
p/(t\sqrt{2\pi})\exp(-t^{2}/2)+6p\int^{\infty}_{t}\frac{1}{\sqrt{2\pi}s}\exp(-s^{2}/2)s^{2}ds$
$\displaystyle\lesssim$ $\displaystyle
pt^{2}\exp(-t^{2}/2)+(\sqrt{2\pi})^{-1}6p\int^{\infty}_{t}-d\exp(-s^{2}/2)$
$\displaystyle\lesssim$ $\displaystyle
t^{2}\exp(-t^{2}/2)p/\sqrt{2\pi}+6p\exp(-t^{2}/2)\sqrt{2\pi}^{-1},$

by the inequality of a Gaussian distribution

$\mathbb{P}(|Z|>t)\leq\frac{2}{t\sqrt{2\pi}}\exp(-t^{2}/2).$ (C.40)

It thus follows that, if we assume that $\theta_{q,j,\alpha}$ is bounded,

$M_{L,2}(\delta)\lesssim M^{3/2}\\{\delta\sqrt{L}/(\sqrt{M}\log
p)\\}^{2}[\exp(-\\{\delta\sqrt{L}/(\sqrt{M}{2\log p}^{3/2})\\}^{2})].$ (C.41)

Thus to ensure that $g(\delta)\to 0$ if the involved dependence adjusted norm
is bounded, we need to have $T_{yn2}=M^{1/2}L^{-1/6}(\log
p)^{2/3}\vee{p}^{1/(q)}M^{1/2}{(\log p)}^{1-1/q}/{\sqrt{L}^{(q-2)/q}}$ and
$\delta\lesssim T_{yn2}$. ${L}^{-1/2}\sqrt{\log p^{3}}\sqrt{M}\log(p\vee
L)\ll\delta$, and $\delta/{\sqrt{M+m}}\to 0$.

  * Step 3

Approximate Gaussian partial sums
We define $T_{yn1}=n^{1/q}m^{1/2-1/q-\alpha}n^{-1/2}\vee
n^{1/2}m^{-\alpha}\sqrt{\log p}n^{-1/2}$ as obtained in Step 1. Let
$T_{yn1}\to 0$. As both $Z$ and $\sum_{b}Z_{b}$ are Gaussian random variables,
we can derive the rate following an concentration inequality using a sub
Gaussian norm. We define
$\Phi_{\psi_{1},\alpha^{\prime}}=\max_{q}q^{-\alpha^{\prime}}\max_{j}\Theta^{z}_{\alpha^{\prime},q,j}.$
Let $\beta=2/(1+2\alpha^{\prime})$. Let $\alpha^{\prime}=1,$ $\beta=2/3$.
We define $Z_{n}=n^{-1/2}\sum_{i}^{n}Z_{i}$ such that
$\mathbb{E}(Z_{t}Z_{t-l}^{\top})=\mathbb{E}(X_{t}X_{t-l}^{\top})$. In the next
step we bound

$\mathbb{P}(|(\sqrt{n}Z_{n}-\sum_{b}Z_{b})|_{\infty}\geq y^{\prime})\leq
f^{\prime}(y^{\prime},\delta),$ (C.42)

where $f^{\prime}(y^{\prime},\delta)$ is taking form of
$p\exp(-C_{\beta}(y^{\prime}m^{\alpha}/(\sqrt{n})\Phi_{\psi_{1},\alpha})^{{}^{\beta}})+p\exp(-C_{\beta}(y^{\prime}/(\sqrt{mL}\Phi_{\psi_{1},\alpha}))^{\beta})$
by Zhang and Wu (2017)(Lemma 7.1 (ii)). Therefore, we need to ensure $(\log
p)^{1/{\beta}}m^{-\alpha}(\sqrt{n})\vee\sqrt{mL}(\log p)^{1/\beta}\ll
y^{\prime}$ to make the right hand side function
$f^{\prime}(y^{\prime},\delta)$ tends to zero. The rate is fine tuned by
combining the above three steps, $y^{\prime}/\sqrt{n}\ll 1$.

∎