[a]Orlando Oliveira

# Looking at the analytic structure of Landau gauge propagators

Alexandre F. Falcão  Paulo J. Silva

###### Abstract

We report on a study of the analytical structure of the Landau gauge gluon,
ghost and quark propagators taken from lattice simulations using large
physical volumes, to better access the IR region, and large gauge ensembles to
reduce the statistical uncertainties. The investigation uses Padé approximants
to look at poles and branch cuts for each of the propagators. For the gluon
propagator we identify complex conjugate poles and a branch point. For the
ghost propagator the procedure identifies a pole at zero momentum and a branch
point for Minkowski-like momenta. The quark propagator appears to have a pole
for Minkowski-like momenta that is correlated with the pion mass as expected
from PCAC.

##  1 Introduction and Elements of Padé Approximants

Lattice simulations are first principles computations that, for the
propagators of the fundamental fields, deliver a table of numbers. The Landau
gauge propagators in QCD computed on the lattice are functions of the
Euclidean space momentum. In general, the interpretation of lattice results is
difficult and the computation of poles and branch cuts in the complex plane
requires going beyond the numerical simulations. The motivation to look at the
analytical structure is multiple as it is connected with confinement, the
generation of bound states, the connection between Euclidean and Minkowski
momenta, etc.

Our approach to study the analytical structure of the propagators is based on
the use of Padé approximants and on Pommerenke’s theorem [1]. The use of Padé
approximants requires describing the lattice data in Euclidean space by a
ratio of polynomials, i.e. a lattice function $\mathcal{G}(p^{2})$ computed on
a finite number of Euclidean momenta is approximated by

$\mathcal{G}(p^{2})\approx
P^{M}_{N}(p^{2})=\frac{Q_{M}(p^{2})}{P_{N}(p^{2})}=[M|N](p^{2})$ (1)

where $Q_{M}$ and $P_{N}$ are polynomials of order $M$ and $N$, respectively,
in $p^{2}$ and assume that this parameterization of the lattice data is valid
on the all complex plane. Pommerenke’s theorem ensures that for meromorphic
functions the Padé sequences $[M|M+k]$ with fixed $k$ converge in any compact
set of the complex plane to $\mathcal{G}(p^{2})$. In the analysis of the
lattice data we consider sequences of Padé approximants $[N|N+1]$ and look how
the poles and zeros evolve with $N$. Only those poles whose position does not
change with $N$ can be meaningful.

In order to build a Padé approximant to the lattice propagators we use two
global optimisation methods, namely Differential Evolution (DE) and Simulated
Annealing (SA), and minimise the function

$\sum_{j}\left(\frac{\mathcal{G}(p^{2}_{j})-\mathcal{G}_{L}(p^{2}_{j})}{\sigma(p^{2}_{j})}\right)^{2}$
(2)

where the sum runs over the lattice momenta, $\mathcal{G}_{L}(p^{2}_{j})$ is
the lattice estimation of the propagator and $\sigma(p^{2}_{j})$ the
corresponding statistical error.

The details of the analysis of the gluon and ghost propagators that uses data
from [2] and [3], respectively, can be found in [4] with the Master thesis [5]
giving further details. Herein, we report also on a similar analysis for the
quark propagator using a subset of the results published in [6].

##  2 Gluon and Ghost Propagators

Although in [4] we have analysed the lattice data for the gluon and the ghost
propagator for various lattice volumes, herein, due to the lack of space, we
only report on the results of the largest physical volume.

Figure 1: Sequence of poles as given by $[N|N+1]$ Padé approximants for the
$128^{4}$ gluon propagator data for off-axis momenta. The upper two plots are
the results computed with the DE method, while the lower two plot show the
results obtained with the SA method. The colour code refers to the absolute
value of the residua. Figure 2: Estimation of the gluon complex poles using
different methods: DE and SA estimations (see text for details); RGZ are the
estimations of the refined Gribov-Zwanziger analysis of the lattice gluon
propagator [2]; SPM-Lattice and SPM-DSE are the estimations of [7] using
either the lattice data or the propagator from solving the gluon and ghost
Dyson-Schwinger equations for pure Yang-Mills SU(3) theory, respectively; “new
result (DSE)” are the results of [8].

Figure 3: Sequence of poles (circles) and zeros (crosses) as given by
$[N|N+1]$ Padé approximants for the $128^{4}$ gluon propagator data for on-
axis momenta. The left plot are the results computed with the DE method, while
the right plot shows the results computed with the SA method. The bottom plots
are zooms of the region near the origin of the complex plane. The colour code
refers to the absolute value of the residua.

In Fig. 1 we show the sequence of poles, computed with the two optimisation
methods, as a function of $N$ for $[N|N+1]$ Padé approximants and for the
$128^{4}$ gluon propagator data for off-axis momenta. As seen, the dominant
poles, i.e. those with higher residuum (in red), are stable and appear at
relatively small $N$. Moreover, their positions in the complex plane as given
by the DE or the SA methods is essentially the same. We take this result as an
indication that the gluon has complex conjugate poles and, from a selection of
the data in Fig. 1, it comes that the poles111Note that the numbers quoted
below do not match exactly those represented in Fig. 2 that use all the
information produced by the Padé analysis [4, 5]. are at

$p^{2}=\left\\{\begin{array}[]{lcl}-(0.343-0.220)\pm i(0.301-0.546)\text{
GeV}^{2},&\mbox{ }&\text{DE method},\\\ -(0.220-0.150)\pm i(0.227-0.444)\text{
GeV}^{2},&\mbox{ }&\text{SA method}.\end{array}\right.$ (3)

In Fig. 2 we collect the results of several investigations of the Landau gauge
gluon propagator where complex conjugate poles are observed. We note the good
agreement from the different studies.

In Fig. 3 we show the zeros (crosses) and poles (circles) as given by the Padé
approximants for on-axis momenta. The two methods do not give compatible
results, with the SA method resulting in a collection of zeros for Minkowski
type of momenta that suggests the presence of a branch cut. The corresponding
branch point occurs at the Euclidean momenta $p^{2}\sim-0.5$ GeV2 or is closer
to the origin of the complex plane.

Figure 4: Sequence of poles (circles) and zeros (crosses) as given by
$[N|N+1]$ Padé approximants for the $80^{4}$ ghost propagator data for on-axis
momenta. The left plot are the results computed with the DE method, while the
right plot shows the results computed with the SA method. The bottom plots are
zooms of the region near the origin of the complex plane. The colour code
refers to the absolute value of the residua.

The same analysis of the ghost data shows no complex poles; see [4] for
details. However, for on-axis momenta, see Fig. 4, the Padé approximants
identify a pole at the origin of the complex plane that has the highest
absolute value of the residuum, meaning that the ghost propagator is of type
$Z(p^{2})/p^{2}$. Besides this pole the Padé analysis shows a sequence of
zeros that suggests also the presence of a branch cut, with the corresponding
branch point being at $p^{2}\sim-0.1$ GeV2.

##  3 Quark Propagator

Figure 5: The quark propagator.

Figure 6: Poles (full circles) and zeros (open circles) from the analysis with
Padé approximants for the quark propagator associated with the ensemble with a
pion mass of 290 MeV. In the upper plot we show the sequence poles and zeros
for the term that multiplies $p\\!\\!\\!/$ and in the bottom plot we show the
Dirac scalar term.

For the analysis of the quark propagator we consider the subset of the
ensembles generated in [6] with lattice spacing $a=0.071$ fm. Of these data we
take for analysis only the simulations with a pion mass of 422 MeV, 290 MeV
and 150 MeV. The quark propagator is diagonal in colour space and its Dirac
structure reads

$S(p^{2})=Z(p^{2})\,\frac{p\\!\\!\\!/+M(p^{2})}{p^{2}+M^{2}(p^{2})}\ .$ (4)

Fig. 5 shows the two reconstructed functions associated with the different
Dirac structures for the ensembles.

The Padé analysis of the vector and scalar Dirac functions show no complex
poles for any of these propagators. However, surprisingly, it points towards
poles on the negative real axis, i.e. for Minkowski type of momenta. To
illustrate the results of the Padé approximant analysis, in Fig. 6 we report
on the results for the quark propagator associated a pion mass of 290 MeV. A
stable pole associated with $p^{2}\sim-0.2$ GeV2 is clearly observed. The
outcome of the analysis of the three quark propagators points to poles at

$M_{\pi}$ | 150 MeV | 290 MeV | 422 MeV
---|---|---|---
Vector | 0.19(6) | 0.22(3) | 0.26(3)
Scalar | 0.16(3) | 0.19(6) | 0.21(6)

where the pole positions are given in GeV2; recall that the poles appear for
Minkowski-like momentum. From our results we make the following observations.
The pole position obtained from the Dirac vector and the Dirac scalar
functions are consistent within errors. The pole mass correlates with the pion
mass and increases when the pion mass increases. Moreover, the results also
show that the pole mass is compatible with $M^{2}_{\pi}\propto
m_{\text{pole}}$ behaviour as predicted by PCAC. This results should be read
with care due to the large statistical errors on the pole mass that allow for
different types of power law. Finally, the computed mass poles are of the same
order of magnitude as the constituent quark mass used in many quark models.

##  4 Summary and Conclusions

Padé approximants allow to access the analytic structure of the Landau gauge
lattice propagators. Accordingly, the gluon propagator is described by a pair
of complex conjugate poles and has a branch cut. The ghost propagator has a
pole at zero momentum and a branch cut. The quark propagator has a pole at
Minkowski momenta that grows (linearly) with the pion mass squared. A possible
branch cut at $p<1$ GeV was also identified.

We plan to improve the precise location of the branch cut for the bosonic
propagators and to investigate the presence of multiple poles.

## Acknowledgments

This work was supported by national funds from FCT – Fundação para a Ciência e
a Tecnologia, I.P., Portugal, within projects UID/FIS/04564/2019,
UID/FIS/04564/2020 and CERN/FIS-COM/0029/2017. A.F.F. acknowledges financial
support from FCT (Portugal) under the Project No. UIDB/04564/2020. P. J. S.
acknowledges financial support from FCT (Portugal) under contract
CEECIND/00488/2017.

## References

  * [1] C. Pommerenke, J. Math. Anal. Appl. 41, 775 (1973).
  * [2] D. Dudal, O. Oliveira and P. J. Silva, Annals Phys. 397, 351-364 (2018) doi:10.1016/j.aop.2018.08.019 [arXiv:1803.02281 [hep-lat]].
  * [3] A. G. Duarte, O. Oliveira and P. J. Silva, Phys. Rev. D 94, no.1, 014502 (2016) doi:10.1103/PhysRevD.94.014502 [arXiv:1605.00594 [hep-lat]].
  * [4] A. F. Falcão, O. Oliveira and P. J. Silva, Phys. Rev. D 102, no.11, 114518 (2020) doi:10.1103/PhysRevD.102.114518 [arXiv:2008.02614 [hep-lat]].
  * [5] A. F. Falcão, “Padé Approximants and the analytic structure of the gluon and ghost propagators,” [arXiv:2009.08961 [hep-lat]], Master thesis, University of Coimbra, 2020.
  * [6] A. Kızılersü, O. Oliveira, P. J. Silva, J. I. Skullerud and A. Sternbeck, Phys. Rev. D 103, no.11, 114515 (2021) doi:10.1103/PhysRevD.103.114515 [arXiv:2103.02945 [hep-lat]].
  * [7] D. Binosi and R. A. Tripolt, Phys. Lett. B 801, 135171 (2020) doi:10.1016/j.physletb.2019.135171 [arXiv:1904.08172 [hep-ph]].
  * [8] C. S. Fischer and M. Q. Huber, Phys. Rev. D 102, no.9, 094005 (2020) doi:10.1103/PhysRevD.102.094005 [arXiv:2007.11505 [hep-ph]].


to $([p],\widetilde{E})$ in the sense that $\\{k,(i,j)\\}\in[p]\times
E\rightarrow\\{\pi(k),(\pi(i),\pi(j))\\}\in[p]\times\widetilde{E}$. Hence we
complete the proof of Theorem 1. $\hfill\Box$

##  Appendix D Proof of Theorem 2

Let
$\nu_{n}=\Xi^{2}p_{\dagger}^{1-\alpha}\mathcal{M}_{y}^{1-\alpha}(n^{-1}\log
p)^{(1-\alpha)/{2}}$, where $\Xi=s_{1}s_{2}(2p_{\dagger}+1)$ with
$p_{\dagger}=\max_{l\in[q]}p_{l}$. Since $\rho^{-1}\nu_{n}\to 0$, the result
in (S.3) holds. This, together with (S.4) and the remark for Lemma 1 of Chang
et al. (2018), yields that

$\displaystyle\max_{l\in[q]}D\\{{\mathcal{C}}(\widehat{\boldsymbol{\Pi}}_{l}),{\mathcal{C}}({\mathbf{A}}_{l})\\}\lesssim\rho^{-1}\|\widehat{{\mathbf{W}}}_{y}-{\mathbf{W}}_{y}\|_{2}=O_{\rm
p}(\rho^{-1}\nu_{n})\,.$ (S.10)

Recall that
$\widehat{\boldsymbol{\Pi}}_{l}=(\widehat{\boldsymbol{\gamma}}_{i})_{i\in
G_{l}}$ and
$\widehat{\mathbf{A}}_{l}=(\widehat{\boldsymbol{\gamma}}_{i})_{i\in\widehat{G}_{l}}$.
Theorem 1 implies that there exists a permutation $\tilde{\pi}:[q]\to[q]$ such
that
$\mathbb{P}[\bigcap_{l=1}^{q}\\{\widehat{G}_{\tilde{\pi}(l)}=G_{l}\\}\,,\,\hat{q}=q]\rightarrow
1$. Let $\Omega_{l}=\\{\widehat{G}_{\tilde{\pi}(l)}=G_{l}\,,\,\hat{q}=q\\}$
and $d_{n}=\rho^{-1}\nu_{n}$. For any $\epsilon>0$, by (S.10), there exists a
constant $C>0$ such that

$\mathbb{P}\bigg{[}\max_{l\in[q]}d_{n}^{-1}D\\{{\mathcal{C}}(\widehat{\boldsymbol{\Pi}}_{l}),{\mathcal{C}}({\mathbf{A}}_{l})\\}>C\bigg{]}<\epsilon\,,$

which implies

$\begin{split}&\mathbb{P}\bigg{[}\max_{l\in[q]}\min_{j\in[\hat{q}]}d_{n}^{-1}D\\{{\mathcal{C}}(\widehat{\mathbf{A}}_{j}),{\mathcal{C}}({\mathbf{A}}_{l})\\}>C\bigg{]}\\\
&~{}~{}~{}~{}~{}~{}\leq\mathbb{P}\bigg{[}\max_{l\in[q]}\min_{j\in[\hat{q}]}d_{n}^{-1}D\\{{\mathcal{C}}(\widehat{\mathbf{A}}_{j}),{\mathcal{C}}({\mathbf{A}}_{l})\\}>C,\bigcap_{l=1}^{q}\Omega_{l}\bigg{]}+\mathbb{P}\bigg{(}\bigcup_{l=1}^{q}\Omega_{l}^{c}\bigg{)}\\\
&~{}~{}~{}~{}~{}~{}\leq\mathbb{P}\bigg{[}\max_{l\in[q]}d_{n}^{-1}D\\{{\mathcal{C}}(\widehat{\boldsymbol{\Pi}}_{l}),{\mathcal{C}}({\mathbf{A}}_{l})\\}>C\bigg{]}+o(1)<\epsilon+o(1)\,.\end{split}$

Hence,
$\max_{l\in[q]}\min_{j\in[\hat{q}]}D\\{{\mathcal{C}}(\widehat{\mathbf{A}}_{j}),{\mathcal{C}}({\mathbf{A}}_{l})\\}=O_{\rm
p}(d_{n})$. We complete the proof of Theorem 2. $\hfill\Box$

##  Appendix E Proof of Theorem 3

Recall that
$\boldsymbol{\Pi}_{l}={\mathbf{A}}_{l}\boldsymbol{\Gamma}_{z,l}{\mathbf{H}}_{l}=(\boldsymbol{\gamma}_{i})_{i\in
G_{l}}$ and
$\widehat{\mathbf{A}}_{l}=(\widehat{\boldsymbol{\gamma}}_{i})_{i\in G_{l}}$.
Write
${\mathbf{M}}_{k}^{(l)}(u,v)=\boldsymbol{\Pi}_{l}^{\mathrm{\scriptscriptstyle\top}}\boldsymbol{\Sigma}_{y,k}(u,v)\boldsymbol{\Pi}_{l}\equiv\\{M_{k,ij}^{(l)}(u,v)\\}_{i,j\in[p_{l}]}$
and
$\widehat{\mathbf{M}}_{k}^{(l)}(u,v)=\widehat{\mathbf{A}}_{l}^{{\mathrm{\scriptscriptstyle\top}}}{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)\widehat{\mathbf{A}}_{l}\equiv\\{\widehat{M}_{k,ij}^{(l)}(u,v)\\}_{i,j\in[p_{l}]}.$
Let
$\nu_{n}=\Xi^{2}p_{\dagger}^{1-\alpha}\mathcal{M}_{y}^{1-\alpha}(n^{-1}\log
p)^{(1-\alpha)/{2}}$, where $\Xi=s_{1}s_{2}(2p_{\dagger}+1)$ with
$p_{\dagger}=\max_{l\in[q]}p_{l}$. By a similar decomposition to (S.6) and
$\rho^{-1}\Xi p_{\dagger}^{1-\alpha}\nu_{n}\to 0$, we obtain that
$\max_{i,j\in[p_{l}],l\in[q]}\|\widehat{M}_{k,ij}^{(l)}-M_{k,ij}^{(l)}\|_{\mathcal{S}}=O_{\rm
p}(\rho^{-1}\Xi p_{\dagger}^{1-\alpha}\nu_{n})$. Hence,

$\max_{l\in[q]}\|\widehat{\mathbf{M}}_{k}^{(l)}-{\mathbf{M}}_{k}^{(l)}\|_{{\mathcal{S}},\text{F}}=\max_{l\in[q]}\bigg{(}\sum_{i,j\in[p_{l}]}\|\widehat{M}_{k,ij}^{(l)}-M_{k,ij}^{(l)}\|_{\mathcal{S}}^{2}\bigg{)}^{1/2}=O_{\rm
p}(\rho^{-1}\Xi p_{\dagger}^{2-\alpha}\nu_{n})\,.$

Write
${\mathbf{Z}}_{t}^{(l)}(\cdot)=\\{Z_{t,1}^{(l)}(\cdot),\dots,Z_{t,p_{l}}^{(l)}(\cdot)\\}^{{\mathrm{\scriptscriptstyle\top}}}.$
It follows from Cauchy–Schwartz inequality that

$\begin{split}\max_{l\in[q]}\|{\mathbf{M}}_{k}^{(l)}\|_{{\mathcal{S}},\text{F}}^{2}&=\max_{l\in[q]}\sum_{i,j=1}^{p_{l}}\int\int\\{M_{k,ij}^{(l)}(u,v)\\}^{2}\,{\rm
d}u{\rm d}v\\\
&\leq\max_{l\in[q]}\sum_{i=1}^{p_{l}}\int\mathbb{E}[\\{Z_{t,i}^{(l)}(u)\\}^{2}]\,{\rm
d}u\cdot\max_{l\in[q]}\sum_{j=1}^{p_{l}}\int\mathbb{E}[\\{Z_{t+k,j}^{(l)}(u)\\}^{2}]\,{\rm
d}u=O(p_{\dagger}^{2})\,.\end{split}$

Observe that
$\widehat{\mathbf{K}}^{(l)}(u,v)-{\mathbf{K}}^{(l)}(u,v)=\sum_{k=1}^{k_{0}}\int_{\mathcal{U}}{\mathbf{M}}_{k}^{(l)}(u,w)\\{\widehat{\mathbf{M}}_{k}^{(l)}(v,w)-{\mathbf{M}}_{k}^{(l)}(v,w)\\}^{{\mathrm{\scriptscriptstyle\top}}}\,{\rm
d}w+\sum_{k=1}^{k_{0}}\int_{\mathcal{U}}\\{\widehat{\mathbf{M}}_{k}^{(l)}(u,w)-{\mathbf{M}}_{k}^{(l)}(u,w)\\}{\mathbf{M}}_{k}^{(l)}(v,w)^{{\mathrm{\scriptscriptstyle\top}}}\,{\rm
d}w+\sum_{k=1}^{k_{0}}\int_{\mathcal{U}}\\{\widehat{\mathbf{M}}_{k}^{(l)}(u,w)-{\mathbf{M}}_{k}^{(l)}(u,w)\\}\\{\widehat{\mathbf{M}}_{k}^{(l)}(v,w)-{\mathbf{M}}_{k}^{(l)}(v,w)\\}^{{\mathrm{\scriptscriptstyle\top}}}\,{\rm
d}w$. Together with Inequality 1, $\rho^{-1}\Xi
p_{\dagger}^{1-\alpha}\nu_{n}\to 0$ and fixed $k_{0}$, it holds that

$\displaystyle\begin{split}\max_{l\in[q]}\|\widehat{\mathbf{K}}^{(l)}-{\mathbf{K}}^{(l)}\|_{{\mathcal{S}},\text{F}}&\leq\max_{l\in[q]}\sum_{k=1}^{k_{0}}\|\widehat{\mathbf{M}}_{k}^{(l)}-{\mathbf{M}}_{k}^{(l)}\|_{{\mathcal{S}},\text{F}}^{2}+2\max_{l\in[q]}\sum_{k=1}^{k_{0}}\|{\mathbf{M}}_{k}^{(l)}\|_{{\mathcal{S}},\text{F}}\|\widehat{\mathbf{M}}_{k}^{(l)}-{\mathbf{M}}_{k}^{(l)}\|_{{\mathcal{S}},\text{F}}\\\
&=O_{\rm p}(\rho^{-2}\Xi^{2}p_{\dagger}^{4-2\alpha}\nu_{n}^{2})+O_{\rm
p}(\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n})=O_{\rm p}(\rho^{-1}\Xi
p_{\dagger}^{3-\alpha}\nu_{n})\,.\end{split}$

This, together with Lemma A5, implies that

$\max_{l\in[q],j\in[r_{l}]}|\hat{\theta}_{j}^{(l)}-\theta_{j}^{(l)}|=O_{\rm
p}(\rho^{-1}\Xi
p_{\dagger}^{3-\alpha}\nu_{n})\,,~{}~{}\max_{l\in[q],j\in[r_{l}]}\|\widehat{\boldsymbol{\psi}}_{j}^{(l)}-\boldsymbol{\psi}_{j}^{(l)}\|=O_{\rm
p}(\Delta^{-1}\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n})\,,$ (S.11)

where
$\Delta=\min_{l\in[q],j\in[r_{l}]}\\{\theta_{j}^{(l)}-\theta_{j+1}^{(l)}\\}$.

Recall that

$\hat{r}_{l}=\arg\max_{j\in[n-k_{0}]}\frac{\hat{\theta}_{j}^{(l)}+\tilde{\delta}_{n}}{\hat{\theta}_{j+1}^{(l)}+\tilde{\delta}_{n}}\,.$

Note that the condition
$\tilde{\delta}_{n}\max_{l\in[q]}\theta_{1}^{(l)}/\min_{l\in[q]}\\{\theta_{r_{l}}^{(l)}\\}^{2}\to
0$ implies that ${\tilde{\delta}_{n}}=o(\min_{l\in[q]}\theta_{r_{l}}^{(l)}).$
By $\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n}/\tilde{\delta}_{n}\to 0$ and
$\tilde{\delta}_{n}/\min_{l\in[q]}\theta_{r_{l}}^{(l)}\to 0$, we can find some
$\tilde{h}_{n}$ such that $\rho^{-1}\Xi
p_{\dagger}^{3-\alpha}\nu_{n}\ll\tilde{h}_{n}\ll\tilde{\delta}_{n}\ll\min_{l\in[q]}\theta_{r_{l}}^{(l)}$.
Let
$\widecheck{\Omega}=\\{\max_{l\in[q],j\in[r_{l}]}|\hat{\theta}_{j}^{(l)}-\theta_{j}^{(l)}|\leq\tilde{h}_{n}\leq\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2\\}$.
Under the event $\widecheck{\Omega}$, we thus have
$\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2\leq\hat{\theta}_{j}^{(l)}\leq\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2+\max_{l\in[q]}\theta_{1}^{(l)}$
if $j\in[r_{l}]$ and $0\leq\hat{\theta}_{j}^{(l)}\leq\tilde{h}_{n}$ if
$j>r_{l}$, for each $l\in[q]$. Due to the definition of $\hat{r}_{l}$, for
each $l\in[q]$, we have that

$\displaystyle\frac{\hat{\theta}_{\hat{r}_{l}}^{(l)}+\tilde{\delta}_{n}}{\hat{\theta}_{\hat{r}_{l}+1}^{(l)}+\tilde{\delta}_{n}}\geq\frac{\hat{\theta}_{r_{l}}^{(l)}+\tilde{\delta}_{n}}{\hat{\theta}_{r_{l}+1}^{(l)}+\tilde{\delta}_{n}}\geq\frac{\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2+\tilde{\delta}_{n}}{\tilde{h}_{n}+\tilde{\delta}_{n}}\asymp\frac{\min_{l\in[q]}\theta_{r_{l}}^{(l)}}{\tilde{\delta}_{n}}$
(S.12)

under $\widecheck{\Omega}$. For each $l\in[q]$, if $\hat{r}_{l}<r_{l}$, under
$\widecheck{\Omega}$,

$\displaystyle\frac{\hat{\theta}_{\hat{r}_{l}}^{(l)}+\tilde{\delta}_{n}}{\hat{\theta}_{\hat{r}_{l}+1}^{(l)}+\tilde{\delta}_{n}}\leq\frac{\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2+\max_{l\in[q]}\theta_{1}^{(l)}+\tilde{\delta}_{n}}{\min_{l\in[q]}\theta_{r_{l}}^{(l)}/2+\tilde{\delta}_{n}}\asymp\frac{\max_{l\in[q]}\theta_{1}^{(l)}}{\min_{l\in[q]}\theta_{r_{l}}^{(l)}}\,.$
(S.13)

Since
$\tilde{\delta}_{n}\max_{l\in[q]}\theta_{1}^{(l)}/\min_{l\in[q]}\\{\theta_{r_{l}}^{(l)}\\}^{2}\to
0$, (S.12) and (S.13) imply
$\mathbb{P}[\bigcup_{i=1}^{q}\\{\hat{r}_{l}<r_{l}\\}\,|\,\widecheck{\Omega}]\to
0$. Similarly, for each $l\in[q]$, if $\hat{r}_{l}>r_{l}$, under
$\widecheck{\Omega}$,

$\frac{\hat{\theta}_{\hat{r}_{l}}^{(l)}+\tilde{\delta}_{n}}{\hat{\theta}_{\hat{r}_{l}+1}^{(l)}+\tilde{\delta}_{n}}\leq\frac{\tilde{h}_{n}+\tilde{\delta}_{n}}{0+\tilde{\delta}_{n}}\to
1\,.$

This together with (S.12) and
$\tilde{\delta}_{n}/\min_{l\in[q]}\theta_{r_{l}}^{(l)}\to 0$ yields that
$\mathbb{P}[\bigcup_{i=1}^{q}\\{\hat{r}_{l}>r_{l}\\}\,|\,\widecheck{\Omega}]\to
0$. Thus
$\mathbb{P}[\bigcap_{i=1}^{q}\\{\hat{r}_{l}=r_{l}\\}\,|\,\widecheck{\Omega}]\to
1$. By (S.11), $\mathbb{P}(\widecheck{\Omega})\to 1$. We complete the proof of
Theorem 3. $\hfill\Box$

##  Appendix F Proof of Theorem 4

Let
$\boldsymbol{\psi}_{j}^{(l)}(\cdot)=\\{\psi_{j1}^{(l)}(\cdot),\dots,\psi_{jp_{l}}^{(l)}(\cdot)\\}^{{\mathrm{\scriptscriptstyle\top}}}$
and
$\widehat{\boldsymbol{\psi}}_{j}^{(l)}(\cdot)=\\{\hat{\psi}_{j1}^{(l)}(\cdot),\dots,\hat{\psi}_{jp_{l}}^{(l)}(\cdot)\\}^{{\mathrm{\scriptscriptstyle\top}}}$.
Due to the orthonormality of $\boldsymbol{\psi}_{j}^{(l)}(\cdot)$ and
$\widehat{\boldsymbol{\psi}}_{j}^{(l)}(\cdot),$ we obtain that

$\displaystyle\bigg{\|}\sum_{j=1}^{r_{l}}\big{[}\widehat{\boldsymbol{\psi}}_{j}^{(l)}\otimes\\{\widehat{\boldsymbol{\psi}}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}-\boldsymbol{\psi}_{j}^{(l)}\otimes\\{\boldsymbol{\psi}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}\big{]}\bigg{\|}_{{\mathcal{S}},\text{F}}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\int\int\sum_{k=1}^{p_{l}}\sum_{m=1}^{p_{l}}\bigg{[}\sum_{j=1}^{r_{l}}\big{\\{}\hat{\psi}_{jk}^{(l)}(u)\hat{\psi}_{jm}^{(l)}(v)-\psi_{jk}^{(l)}(u)\psi_{jm}^{(l)}(v)\big{\\}}\bigg{]}^{2}{\rm
d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}=\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\int\int\sum_{k=1}^{p_{l}}\sum_{m=1}^{p_{l}}\hat{\psi}_{ik}^{(l)}(u)\hat{\psi}_{im}^{(l)}(v)\hat{\psi}_{jk}^{(l)}(u)\hat{\psi}_{jm}^{(l)}(v)\,{\rm
d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\int\int\sum_{k=1}^{p_{l}}\sum_{m=1}^{p_{l}}\psi_{ik}^{(l)}(u)\psi_{im}^{(l)}(v)\psi_{jk}^{(l)}(u)\psi_{jm}^{(l)}(v)\,{\rm
d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-2\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\int\int\sum_{k=1}^{p_{l}}\sum_{m=1}^{p_{l}}\psi_{ik}^{(l)}(u)\psi_{im}^{(l)}(v)\hat{\psi}_{jk}^{(l)}(u)\hat{\psi}_{jm}^{(l)}(v)\,{\rm
d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}=\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\langle\widehat{\boldsymbol{\psi}}_{i}^{(l)},\widehat{\boldsymbol{\psi}}_{j}^{(l)}\rangle^{2}+\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\langle\boldsymbol{\psi}_{i}^{(l)},\boldsymbol{\psi}_{j}^{(l)}\rangle^{2}-2\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\langle\widehat{\boldsymbol{\psi}}_{i}^{(l)},\boldsymbol{\psi}_{j}^{(l)}\rangle^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=2r_{l}-2\sum_{i=1}^{r_{l}}\sum_{j=1}^{r_{l}}\langle\widehat{\boldsymbol{\psi}}_{i}^{(l)},\boldsymbol{\psi}_{j}^{(l)}\rangle^{2}\,.$

Denote by
$\widetilde{\mathcal{C}}_{l}=\text{span}\\{\widehat{\boldsymbol{\psi}}_{1}^{(l)}(\cdot),\dots,\widehat{\boldsymbol{\psi}}_{r_{l}}^{(l)}(\cdot)\\}$
the dynamic space spanned by $r_{l}$ estimated eigenfunctions. By the
definition of $\widetilde{D}(\widetilde{\mathcal{C}}_{l},{\mathcal{C}}_{l})$,
we thus have
$\sqrt{2r_{l}}\widetilde{D}(\widetilde{\mathcal{C}}_{l},{\mathcal{C}}_{l})=\|\sum_{j=1}^{r_{l}}[\widehat{\boldsymbol{\psi}}_{j}^{(l)}\otimes\\{\widehat{\boldsymbol{\psi}}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}-\boldsymbol{\psi}_{j}^{(l)}\otimes\\{\boldsymbol{\psi}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}]\|_{{\mathcal{S}},\text{F}}.$
Let
$\nu_{n}=\Xi^{2}p_{\dagger}^{1-\alpha}\mathcal{M}_{y}^{1-\alpha}(n^{-1}\log
p)^{(1-\alpha)/{2}}$, where $\Xi=s_{1}s_{2}(2p_{\dagger}+1)$ with
$p_{\dagger}=\max_{l\in[q]}p_{l}$. Since $\Delta^{-1}\rho^{-1}\Xi
p_{\dagger}^{3-\alpha}\nu_{n}\to 0$, the result in (S.11) holds. This,
together with Equality 1, fixed $r_{l}$ and the orthonormality of
$\boldsymbol{\psi}_{j}^{(l)}(\cdot)$, leads to

$\begin{split}\max_{l\in[q]}\sqrt{2r_{l}}\widetilde{D}(\widetilde{\mathcal{C}}_{l},{\mathcal{C}}_{l})=\,&\max_{l\in[q]}\bigg{\|}\sum_{j=1}^{r_{l}}\left[\widehat{\boldsymbol{\psi}}_{j}^{(l)}\otimes\\{\widehat{\boldsymbol{\psi}}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}-\boldsymbol{\psi}_{j}^{(l)}\otimes\\{\boldsymbol{\psi}_{j}^{(l)}\\}^{\mathrm{\scriptscriptstyle\top}}\right]\bigg{\|}_{{\mathcal{S}},\text{F}}\\\
\leq\,&\max_{l\in[q]}\sum_{j=1}^{r_{l}}\|\widehat{\boldsymbol{\psi}}_{j}^{(l)}-\boldsymbol{\psi}_{j}^{(l)}\|^{2}+2\max_{l\in[q]}\sum_{j=1}^{r_{l}}\|\boldsymbol{\psi}_{j}^{(l)}\|\|\widehat{\boldsymbol{\psi}}_{j}^{(l)}-\boldsymbol{\psi}_{j}^{(l)}\|\\\
=\,&O_{\rm
p}(\Delta^{-2}\rho^{-2}\Xi^{2}p_{\dagger}^{6-2\alpha}\nu_{n}^{2})+O_{\rm
p}(\Delta^{-1}\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n})\\\ =\,&O_{\rm
p}(\Delta^{-1}\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n})\,.\\\ \end{split}$

Let $e_{n}=\Delta^{-1}\rho^{-1}\Xi p_{\dagger}^{3-\alpha}\nu_{n}$. Theorem 3
implies that $\mathbb{P}[\bigcap_{l=1}^{q}\\{\hat{r}_{l}=r_{l}\\}]\to 1$.
Thus, for any $\epsilon>0$, there exists a constant $C>0$ such that

$\mathbb{P}\bigg{\\{}\max_{l\in[q]}e_{n}^{-1}\widetilde{D}(\widetilde{\mathcal{C}}_{l},{\mathcal{C}}_{l})>C\bigg{\\}}<\epsilon\,,$

which implies

$\begin{split}&\mathbb{P}\bigg{\\{}\max_{l\in[q]}e_{n}^{-1}\widetilde{D}(\widehat{\mathcal{C}}_{l},{\mathcal{C}}_{l})>C\bigg{\\}}\\\
&~{}~{}~{}~{}~{}~{}\leq\mathbb{P}\bigg{[}\max_{l\in[q]}e_{n}^{-1}\widetilde{D}(\widehat{\mathcal{C}}_{l},{\mathcal{C}}_{l})>C,\,\bigcap_{l=1}^{q}\\{\hat{r}_{l}=r_{l}\\}\bigg{]}+\mathbb{P}\bigg{\\{}\bigcup_{l=1}^{q}[\hat{r}_{l}\neq
r_{l}\\}\bigg{]}\\\
&~{}~{}~{}~{}~{}~{}\leq\mathbb{P}\bigg{\\{}\max_{l\in[q]}e_{n}^{-1}\widetilde{D}(\widetilde{\mathcal{C}}_{l},{\mathcal{C}}_{l})>C\bigg{\\}}+o(1)<\epsilon+o(1)\,.\end{split}$

Hence,
$\max_{l\in[q]}\widetilde{D}(\widehat{\mathcal{C}}_{l},{\mathcal{C}}_{l})=O_{\rm
p}(e_{n})$. We complete the proof of Theorem 4. $\hfill\Box$

##  Appendix G Proofs of auxiliary lemmas

###  G.1 Proof of Inequality 1

By Cauchy–Schwartz inequality, we notice that

$\displaystyle\bigg{\|}\int\int{\mathbf{B}}_{1}(u,v){\mathbf{B}}_{2}(u,v)^{{\mathrm{\scriptscriptstyle\top}}}\,{\rm
d}u{\rm d}v\bigg{\|}_{1}$
$\displaystyle=\max\limits_{j\in[p]}\sum_{i=1}^{p}\bigg{|}\int\int\sum_{k=1}^{p}B_{1,ik}(u,v)B_{2,jk}(u,v)\,{\rm
d}u{\rm d}v\bigg{|}$
$\displaystyle\leq\max\limits_{j\in[p]}\sum_{i=1}^{p}\sum_{k=1}^{p}\|B_{1,ik}\|_{{\mathcal{S}}}\|B_{2,jk}\|_{{\mathcal{S}}}$
(S.14)
$\displaystyle\leq\max\limits_{k\in[p]}\sum_{i=1}^{p}\|B_{1,ik}\|_{{\mathcal{S}}}\cdot\max\limits_{j\in[p]}\sum_{k=1}^{p}\|B_{2,jk}\|_{{\mathcal{S}}}=\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},1}\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},\infty}\,.$

By similar argument, we obtain that

$\bigg{\|}\int\int{\mathbf{B}}_{1}(u,v){\mathbf{B}}_{2}(u,v)^{{\mathrm{\scriptscriptstyle\top}}}{\rm
d}u{\rm
d}v\bigg{\|}_{\infty}\leq\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},\infty}\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},1}\,.$
(S.15)

Combining (G.1) and (S.15) and applying the inequality
$\|{\mathbf{E}}\|^{2}\leq\|{\mathbf{E}}\|_{\infty}\|{\mathbf{E}}\|_{1}$ for
any matrix ${\mathbf{E}}\in\mathbb{R}^{p\times p},$ we complete the proof of
part (i).

Let $\EuScriptBF
C(u,v)=\int{\mathbf{B}}_{1}(u,w){\mathbf{B}}_{2}(v,w)^{{\mathrm{\scriptscriptstyle\top}}}{\rm
d}w=\\{\mathcal{C}_{ij}(u,v)\\}_{i,j\in[p]}$. It then follows from
Cauchy–Schwartz inequality that

$\displaystyle\|\EuScriptBF C\|_{{\mathcal{S}},\text{F}}^{2}=$
$\displaystyle\,\sum_{i,j=1}^{p}\int\int\mathcal{C}_{ij}^{2}(u,v)\,{\rm
d}u{\rm d}v=\sum_{i,j=1}^{p}\int\int\bigg{\\{}\sum_{k=1}^{p}\int
B_{1,ik}(u,w)B_{2,jk}(v,w)\,{\rm d}w\bigg{\\}}^{2}{\rm d}u{\rm d}v$
$\displaystyle\leq$
$\displaystyle\,\sum_{i,j=1}^{p}\int\int\bigg{\\{}\sum_{k=1}^{p}\int
B_{1,ik}^{2}(u,w)\,{\rm d}w\cdot\sum_{k=1}^{p}\int B_{2,jk}^{2}(u,w)\,{\rm
d}w\bigg{\\}}\,{\rm d}u{\rm
d}v=\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},{\rm\text{F}}}^{2}\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm\text{F}}}^{2}\,.$

Hence, we complete the proof of part (ii).

By Cauchy–Schwartz inequality, we further obtain that

$\displaystyle\|{\mathbf{B}}_{1}+{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm
F}}^{2}=\,$
$\displaystyle\sum_{i,j=1}^{p}\int\int\\{B_{1,ij}(u,v)+B_{2,ij}(u,v)\\}^{2}\,{\rm
d}u{\rm d}v$ $\displaystyle=\,$ $\displaystyle 2\sum_{i,j=1}^{p}\int\int
B_{1,ij}^{2}(u,v)B_{2,ij}^{2}(u,v)\,{\rm d}u{\rm
d}v+\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},{\rm
F}}^{2}+\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm F}}^{2}$
$\displaystyle\leq\,$ $\displaystyle 2\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},{\rm
F}}\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm
F}}+\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},{\rm
F}}^{2}+\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm
F}}^{2}=(\|{\mathbf{B}}_{1}\|_{{\mathcal{S}},{\rm
F}}+\|{\mathbf{B}}_{2}\|_{{\mathcal{S}},{\rm F}})^{2}\,.$

Hence, we complete the proof of part (iii). $\hfill\Box$

###  G.2 Proof of Inequality 2

By elementary calculations and Inequality 1, we obtain that

$\displaystyle\|{\mathbf{b}}_{1}^{\mathrm{\scriptscriptstyle\top}}{\mathbf{B}}{\mathbf{b}}_{2}\|_{\mathcal{S}}^{2}=$
$\displaystyle\,\int\int{\mathbf{b}}_{1}^{\mathrm{\scriptscriptstyle\top}}{\mathbf{B}}(u,v){\mathbf{b}}_{2}{\mathbf{b}}_{2}^{\mathrm{\scriptscriptstyle\top}}{\mathbf{B}}(u,v)^{\mathrm{\scriptscriptstyle\top}}{\mathbf{b}}_{1}\,{\rm
d}u{\rm d}v$ $\displaystyle\leq$
$\displaystyle\,\int\int\|{\mathbf{b}}_{2}{\mathbf{b}}_{2}^{{\mathrm{\scriptscriptstyle\top}}}\|_{2}\|{\mathbf{B}}(u,v)^{\mathrm{\scriptscriptstyle\top}}{\mathbf{b}}_{1}\|_{2}^{2}\,\,{\rm
d}u{\rm d}v$ $\displaystyle\leq$
$\displaystyle\,\|{\mathbf{b}}_{2}\|_{2}^{2}\int\int{\mathbf{b}}_{1}^{\mathrm{\scriptscriptstyle\top}}{\mathbf{B}}(u,v){\mathbf{B}}(u,v)^{\mathrm{\scriptscriptstyle\top}}{\mathbf{b}}_{1}\,{\rm
d}u{\rm
d}v\leq\|{\mathbf{b}}_{1}\|_{2}^{2}\|{\mathbf{b}}_{2}\|_{2}^{2}\|{\mathbf{B}}\|_{{\mathcal{S}},\infty}\|{\mathbf{B}}\|_{{\mathcal{S}},1}\,,$

which completes our proof. $\hfill\Box$

###  G.3 Proof of Inequality 3

Let
${\boldsymbol{f}}(u)=\\{f_{1}(u),\dots,f_{p}(u)\\}^{{\mathrm{\scriptscriptstyle\top}}}$
and ${\boldsymbol{g}}(u)=\int{\mathbf{B}}(u,v){\boldsymbol{f}}(v)\,{\rm
d}v=\\{g_{1}(u),\dots,g_{p}(u)\\}^{{\mathrm{\scriptscriptstyle\top}}}$. By
Cauchy–Schwartz inequality, we obtain that

$\begin{split}\|{\boldsymbol{g}}\|^{2}=\,&\sum_{i=1}^{p}\int
g_{i}(u)^{2}\,{\rm d}u=\sum_{i=1}^{p}\int\bigg{\\{}\sum_{k=1}^{p}\int
B_{ik}(u,v)f_{k}(v)\,{\rm d}v\bigg{\\}}^{2}\,{\rm d}u\\\
\leq\,&\sum_{i=1}^{p}\int\bigg{\\{}\sum_{k=1}^{p}\int B_{ik}^{2}(u,v)\,{\rm
d}v\sum_{k=1}^{p}\int f_{k}^{2}(v)\,{\rm d}v\bigg{\\}}\,{\rm
d}u=\|{\mathbf{B}}\|_{{\mathcal{S}},{\rm
F}}^{2}\|{\boldsymbol{f}}\|^{2}\,,\end{split}$

This further leads to
$\|{\mathbf{B}}\|_{{\mathcal{L}}}={\text{sup}}_{\|{\boldsymbol{f}}\|\leq
1,{\boldsymbol{f}}\in\mathbb{H}}\|{\mathbf{B}}({\boldsymbol{f}})\|\leq{\text{sup}}_{\|{\boldsymbol{f}}\|\leq
1,{\boldsymbol{f}}\in\mathbb{H}}\|{\mathbf{B}}\|_{{\mathcal{S}},{\rm
F}}\|{\boldsymbol{f}}\|\leq\|{\mathbf{B}}\|_{{\mathcal{S}},{\rm F}}$, which
completes our proof. $\hfill\Box$

###  G.4 Proof of Equality 1

For ${\boldsymbol{f}}=(f_{1},\dots,f_{p})^{{\mathrm{\scriptscriptstyle\top}}}$
and
${\boldsymbol{g}}=(g_{1},\dots,g_{p})^{{\mathrm{\scriptscriptstyle\top}}}\in\mathbb{H},$
it holds that

$\|{\boldsymbol{f}}\otimes{\boldsymbol{g}}^{\mathrm{\scriptscriptstyle\top}}\|_{{\mathcal{S}},{\rm\text{F}}}^{2}=\sum_{i,j=1}^{p}\int\int
f_{i}^{2}(u)g_{j}^{2}(v)\,{\rm d}u{\rm d}v=\bigg{\\{}\sum_{i=1}^{p}\int
f_{i}^{2}(u)\,{\rm d}u\bigg{\\}}\bigg{\\{}\sum_{j=1}^{p}\int
g_{j}^{2}(v)\,{\rm
d}v\bigg{\\}}=\|{\boldsymbol{f}}\|^{2}\|{\boldsymbol{g}}\|^{2}\,.$

Hence, we complete our proof. $\hfill\Box$

###  G.5 Proof of Lemma A1

This lemma follows directly from Theorem 1 of Fang et al. (2022) and Theorem 2
of Guo and Qiao (2023)and hence the proof is omitted here. $\hfill\Box$

###  G.6 Proof of Lemma A2

Recall that $\max_{i\in[p]}\int\mathbb{E}\\{Z_{ti}^{2}(u)\\}\,{\rm d}u=O(1).$
Hence,

$\displaystyle{\max_{i,j\in[p],|k|\leq k_{0}\vee
m}\|\Sigma_{z,k,ij}\|_{\mathcal{S}}^{2}}$
$\displaystyle=\max_{i,j\in[p],|k|\leq k_{0}\vee
m}\int\int[\mathbb{E}\\{Z_{ti}(u)Z_{(t+k)j}(v)\\}]^{2}\,{\rm d}u{\rm d}v$
$\displaystyle\leq\max_{i\in[p]}\int\mathbb{E}\\{Z_{ti}^{2}(u)\\}\,{\rm
d}u\cdot\max_{j\in[p]}\int\mathbb{E}\\{Z_{(t+k)j}^{2}(v)\\}\,{\rm d}v=O(1)\,.$
(S.16)

Let $p_{\dagger}=\max_{l\in[q]}p_{l}$. Since
$\boldsymbol{\Sigma}_{y,k}(u,v)={\mathbf{A}}\boldsymbol{\Sigma}_{z,k}(u,v){\mathbf{A}}^{{\mathrm{\scriptscriptstyle\top}}}$
and $\Sigma_{z,k,lm}=0$ for $|l-m|>p_{\dagger}$, then
$\Sigma_{y,k,ij}(u,v)=\sum_{l,m=1}^{p}A_{il}\Sigma_{z,k,lm}(u,v)A_{jm}=\sum_{|l-m|\leq
p_{\dagger}}A_{il}\Sigma_{z,k,lm}(u,v)A_{jm}$. By the inequality
$(a+b)^{\alpha}\leq a^{\alpha}+b^{\alpha}$ for $a,b\geq 0$ and
$\alpha\in[0,1),$ we obtain that

$\displaystyle\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}^{\alpha}$
$\displaystyle=\sum_{i=1}^{p}\bigg{\|}\sum_{|l-m|\leq
p_{\dagger}}A_{il}\Sigma_{z,k,lm}A_{jm}\bigg{\|}_{\mathcal{S}}^{\alpha}\leq\sum_{i=1}^{p}\bigg{(}\sum_{|l-m|\leq
p_{\dagger}}\big{\|}A_{il}\Sigma_{z,k,lm}A_{jm}\big{\|}_{\mathcal{S}}\bigg{)}^{\alpha}$
$\displaystyle\leq\sum_{i=1}^{p}\sum_{|l-m|\leq
p_{\dagger}}|A_{il}|^{\alpha}|A_{jm}|^{\alpha}\|\Sigma_{z,k,lm}\|_{\mathcal{S}}^{\alpha}$
$\displaystyle\leq\max_{l,m\in[p],|k|\leq k_{0}\vee
m}{\|\Sigma_{z,k,lm}\|_{\mathcal{S}}^{\alpha}}\cdot\max_{l\in[p]}\sum_{i=1}^{p}|A_{il}|^{\alpha}\cdot\sum_{|l-m|\leq
p_{\dagger}}|A_{jm}|^{\alpha}$ $\displaystyle\leq\max_{l,m\in[p],|k|\leq
k_{0}\vee m}{\|\Sigma_{z,k,lm}\|_{\mathcal{S}}^{\alpha}}\cdot
s_{2}\cdot(2p_{\dagger}+1)\sum_{m=1}^{p}|A_{jm}|^{\alpha}$
$\displaystyle=O(\Xi)\,,$

where $\Xi=s_{1}s_{2}(2p_{\dagger}+1)$. By (G.6) and the block structure of
$\boldsymbol{\Sigma}_{z,k}(u,v)$, we further obtain that
$\|\boldsymbol{\Sigma}_{z,k}\|_{{\mathcal{S}},1}=\max_{j\in[p]}\sum_{i=1}^{p}\|\Sigma_{z,k,ij}\|_{\mathcal{S}}=O(p_{\dagger})$.
Similarly, we also have
$\|\boldsymbol{\Sigma}_{z,k}\|_{{\mathcal{S}},\infty}=O(p_{\dagger}).$
Together with Inequality 2 and the orthonormality of the rows in
${\mathbf{A}}$, it holds that

$\|\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},1}=\max_{j\in[p]}\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\leq\max_{i,j\in[p]}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}^{1-\alpha}\cdot\max_{j\in[p]}\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}^{\alpha}=O(\Xi
p_{\dagger}^{1-\alpha})\,.$

Recall $k_{0}$ and $m$ are fixed integers. Similarly, we can prove the rest of
this lemma. $\hfill\Box$

###  G.7 Proof of Lemma A3

Denote by
${\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})$
the $(i,j)$-th component of
${\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k}).$
Due to the fact that
${\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})=\widehat{\Sigma}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}(u,v)I\\{\|\widehat{\Sigma}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{{\mathcal{S}}}\geq\omega_{k}\\}$,
we have
$\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\widehat{\Sigma}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\leq\omega_{k}$.
Under the event
$\Omega=\\{\max_{i,j\in[p]}\|{\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}}\|_{\mathcal{S}}\leq\tilde{\theta}\omega_{k}\\}$
for $\tilde{\theta}\in(0,1)$ and $\omega_{k}=c_{k}\mathcal{M}_{y}(n^{-1}\log
p)^{1/2}$, we have

$\displaystyle\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\Sigma_{y,k,ij}\|_{\mathcal{S}}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}<\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\widehat{\Sigma}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\max_{j\in[p]}\sum_{i=1}^{p}\|\widehat{\Sigma}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},\|\Sigma_{y,k,ij}\|_{\mathcal{S}}<\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\max_{j\in[p]}\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}<\omega_{k}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\underbrace{\omega_{k}\sum_{i=1}^{p}I\\{\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k}\\}}_{Q_{1}}+\underbrace{\max_{j\in[p]}\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\Sigma_{y,k,ij}\|_{\mathcal{S}}<2\omega_{k}\\}}_{Q_{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\underbrace{\max_{j\in[p]}\sum_{i=1}^{p}\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},\|\Sigma_{y,k,ij}\|_{\mathcal{S}}<\omega_{k}\\}}_{Q_{3}}\,.$

By Lemma A2, we have
$Q_{1}+Q_{2}\lesssim\omega_{k}^{1-\alpha}\sum_{i=1}^{p}\|\Sigma_{y,k,ij}\|_{\mathcal{S}}^{\alpha}\lesssim\omega_{k}^{1-\alpha}\Xi$
under the event $\Omega$. Also,

$\begin{split}Q_{3}\leq\,&\max_{j\in[p]}\sum_{i=1}^{p}\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},\|\Sigma_{y,k,ij}\|_{\mathcal{S}}<(1-\tilde{\theta})\omega_{k}\\}\\\
&+\max_{j\in[p]}\sum_{i=1}^{p}\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}\|_{\mathcal{S}}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}\|_{\mathcal{S}}\geq\omega_{k},(1-\tilde{\theta})\omega_{k}\leq\|\Sigma_{y,k,ij}\|_{\mathcal{S}}<\omega_{k}\\}\\\
\leq\,&\omega_{k}\max_{j\in[p]}\sum_{i=1}^{p}I\\{\|\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij}-\Sigma_{y,k,ij}\|_{\mathcal{S}}>\tilde{\theta}\omega_{k}\\}+\omega_{k}\max_{j\in[p]}\sum_{i=1}^{p}I\\{\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\geq(1-\tilde{\theta})\omega_{k}\\}\\\
=\,&\omega_{k}\max_{j\in[p]}\sum_{i=1}^{p}I\\{\|\Sigma_{y,k,ij}\|_{\mathcal{S}}\geq(1-\tilde{\theta})\omega_{k}\\}\lesssim\omega_{k}^{1-\alpha}\Xi\end{split}$

under the event $\Omega$. By Lemma A1, if $n\gtrsim\log p$ and
$\tilde{c}{\tilde{\theta}}^{2}c_{k}^{2}>2$, then $\mathbb{P}(\Omega^{c})\leq
8p^{2-\tilde{c}{\tilde{\theta}}^{2}c_{k}^{2}}\to 0$. Combining the above
results, we thus have

$\max_{j\in[p]}\sum_{i=1}^{p}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\Sigma}{}^{{\scriptscriptstyle\textup{S}}}_{y,k,ij})-\Sigma_{y,k,ij}\|_{\mathcal{S}}=O_{\rm
p}\bigg{\\{}\Xi\mathcal{M}_{y}^{1-\alpha}\bigg{(}\frac{\log
p}{n}\bigg{)}^{(1-\alpha)/{2}}\bigg{\\}}\,.$

Recall $k_{0}$ and $m$ are fixed integers. We have the first result. The
second result can be proved in the similar manner. Due to
${\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)^{\otimes
2}-\boldsymbol{\Sigma}_{y,k}(u,v)^{\otimes
2}=\\{{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)-\boldsymbol{\Sigma}_{y,k}(u,v)\\}^{\otimes
2}+\boldsymbol{\Sigma}_{y,k}(u,v)\\{{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)-\boldsymbol{\Sigma}_{y,k}(u,v)\\}^{{\mathrm{\scriptscriptstyle\top}}}+\\{{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)-\boldsymbol{\Sigma}_{y,k}(u,v)\\}\boldsymbol{\Sigma}_{y,k}(u,v)^{{\mathrm{\scriptscriptstyle\top}}}$,
it follows from Inequality 1 and Lemma A2 that

$\begin{split}&\bigg{\|}\int\int\big{\\{}{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})(u,v)^{\otimes
2}-\boldsymbol{\Sigma}_{y,k}(u,v)^{\otimes 2}\big{\\}}\,{\rm d}u{\rm
d}v\bigg{\|}_{2}\\\ &~{}~{}~{}~{}~{}~{}\leq
2\|\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},1}^{1/2}\|\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},\infty}^{1/2}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})-\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},1}^{1/2}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})-\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},\infty}^{1/2}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}+\|{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})-\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},1}\|{\mathcal{T}}_{\omega_{k}}(\widehat{\boldsymbol{\Sigma}}{}^{{\scriptscriptstyle\textup{S}}}_{y,k})-\boldsymbol{\Sigma}_{y,k}\|_{{\mathcal{S}},\infty}\\\
&~{}~{}~{}~{}~{}~{}=O_{\rm
p}\bigg{\\{}\Xi^{2}p_{\dagger}^{1-\alpha}\mathcal{M}_{y}^{1-\alpha}\bigg{(}\frac{\log
p}{n}\bigg{)}^{(1-\alpha)/{2}}\bigg{\\}}+O_{\rm
p}\bigg{\\{}\Xi^{2}\mathcal{M}_{y}^{2-2\alpha}\bigg{(}\frac{\log
p}{n}\bigg{)}^{1-\alpha}\bigg{\\}}\,.\end{split}$

Since $p_{\dagger}^{-2}\mathcal{M}_{y}^{2}\log p=o(n)$, we have the third
result. $\hfill\Box$

###  G.8 Proof of Lemma A5

By the definition of spectral decomposition, we have that
$\theta_{j}=\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}}$
and
$\hat{\theta}_{j}=\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|\widehat{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}},$
where
${\mathcal{L}}_{j-1}=\\{{\mathbf{B}}:{\mathbf{B}}\in{\mathcal{L}},\,\text{dim}(\text{Im}({\mathbf{B}}))\leq
j-1\\}$. Thus,
$\theta_{j}=\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}}\leq\|{\mathbf{Q}}-\widehat{\mathbf{Q}}\|_{{\mathcal{L}}}+\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|\widehat{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}}=\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{L}}}+\hat{\theta}_{j}$.
Similarly, we have
$\hat{\theta}_{j}=\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|\widehat{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}}\leq\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{L}}}+\min_{{\mathbf{B}}\in{\mathcal{L}}_{j-1}}\|{\mathbf{Q}}-{\mathbf{B}}\|_{{\mathcal{L}}}=\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{L}}}+\theta_{j}$.
Combining the two above results with Inequality 3, we obtain that

$|\hat{\theta}_{j}-\theta_{j}|\leq\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{L}}}\leq\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{S}},\text{F}}$
(S.17)

holds for all $j\geq 1,$ which completes our proof of part (i).

Without loss of generality, we assume that
$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle\geq 0$.
Since
$\sum_{l=1}^{\infty}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}=\|\widehat{\boldsymbol{\phi}}_{j}\|^{2}=1$
and
$0\leq\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle\leq
1$, it holds that

$\displaystyle\|\widehat{\boldsymbol{\phi}}_{j}-\boldsymbol{\phi}_{j}\|^{2}=\,$
$\displaystyle\sum_{l=1}^{\infty}\big{(}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}-\mbox{$\langle\boldsymbol{\phi}_{j},\boldsymbol{\phi}_{l}\rangle$}\big{)}^{2}=\big{(}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle$}-1\big{)}^{2}+\sum_{l\neq
j}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}$
$\displaystyle=\,$
$\displaystyle\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle$}^{2}-2\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle$}+\sum_{l=1}^{\infty}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}+\sum_{l\neq
j}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}$
$\displaystyle=\,$ $\displaystyle 2\sum_{l\neq
j}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}+2(\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle$}^{2}-\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{j}\rangle$})\leq
2\sum_{l\neq
j}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}\,.$
(S.18)

Observe that
${\mathbf{Q}}(\widehat{\boldsymbol{\phi}}_{j})(u)-\theta_{j}\widehat{\boldsymbol{\phi}}_{j}(u)=({\mathbf{Q}}-\widehat{\mathbf{Q}})(\widehat{\boldsymbol{\phi}}_{j})(u)+(\hat{\theta}_{j}-\theta_{j})\widehat{\boldsymbol{\phi}}_{j}(u)$.
This together with Inequality 3, (S.17) and the orthonormality of
$\widehat{\boldsymbol{\phi}}_{j}$ implies that

$\|{\mathbf{Q}}(\widehat{\boldsymbol{\phi}}_{j})-\theta_{j}\widehat{\boldsymbol{\phi}}_{j}\|\leq
2\|\widehat{\mathbf{Q}}-{\mathbf{Q}}\|_{{\mathcal{S}},\text{F}}\,.$ (S.19)

We further write

$\displaystyle\|{\mathbf{Q}}(\widehat{\boldsymbol{\phi}}_{j})-\theta_{j}\widehat{\boldsymbol{\phi}}_{j}\|^{2}=\,$
$\displaystyle\sum_{l=1}^{\infty}\big{(}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},{\mathbf{Q}}(\boldsymbol{\phi}_{l})\rangle$}-\mbox{$\langle\theta_{j}\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}\big{)}^{2}$
$\displaystyle=\,$ $\displaystyle\sum_{l\neq
j}(\theta_{l}-\theta_{j})^{2}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}\geq\Delta_{j}^{2}\sum_{l\neq
j}\mbox{$\langle\widehat{\boldsymbol{\phi}}_{j},\boldsymbol{\phi}_{l}\rangle$}^{2}\,.$
(S.20)

Combining (S.18)–(S.20), we complete the proof of part (ii). $\hfill\Box$

##  Appendix H Additional empirical results

###  H.1 Simulated data with VFAR model

In Section 5.3, we illustrate the usefulness of decorrelation transformation
with an example of a vector functional autoregressive (VFAR) model. In this
section, we present the corresponding details of the data generating process,
the prediction procedure of the VFAR method and some additional simulation
results.

In each simulated scenario, we generate functional variables by
$Y_{tj}(u)={\mathbf{s}}(u)^{{\mathrm{\scriptscriptstyle\top}}}\boldsymbol{\theta}_{tj}$
for $t\in[n],j\in[p]$ and $u\in{\mathcal{U}}=[0,1],$ where ${\mathbf{s}}(u)$
is a $5$-dimensional Fourier basis function and
$\boldsymbol{\theta}_{t}=(\boldsymbol{\theta}_{t1}^{{\mathrm{\scriptscriptstyle\top}}},\dots,\boldsymbol{\theta}_{tp}^{{\mathrm{\scriptscriptstyle\top}}})^{{\mathrm{\scriptscriptstyle\top}}}\in\mathbb{R}^{5p}$
are generated from a stationary vector autoregressive (VAR) model,
$\boldsymbol{\theta}_{t}={\mathbf{B}}\boldsymbol{\theta}_{t-1}+\boldsymbol{e}_{t}$,
with transition matrix ${\mathbf{B}}\in\mathbb{R}^{5p\times 5p}$, whose
entries are randomly sampled from ${\cal N}(0,1)$ and innovations
$\boldsymbol{e}_{t}=(\boldsymbol{e}_{t1}^{{\mathrm{\scriptscriptstyle\top}}},\dots,\boldsymbol{e}_{tp}^{{\mathrm{\scriptscriptstyle\top}}})^{{\mathrm{\scriptscriptstyle\top}}}\in\mathbb{R}^{5p}$
being independently sampled from ${\cal N}(\boldsymbol{0},{\mathbf{I}}_{5p}).$
To guarantee the stationarity of ${\mathbf{Y}}_{t}(\cdot)$, we rescale
${\mathbf{B}}$ by $\iota{\mathbf{B}}/\rho({\mathbf{B}})$ with
$\iota\sim\text{Uniform}[0.5,1]$. Write
${\mathbf{Q}}(u,v)=\\{Q_{ij}(u,v)\\}_{i,j\in[p]}$ and
${\mathbf{B}}=({\mathbf{B}}_{ij})_{i,j\in[p]}$ with
${\mathbf{B}}_{ij}\in\mathbb{R}^{5\times 5}$. According to Section F.3 of the
supplementary material in Guo and Qiao (2023), ${\mathbf{Y}}_{t}(\cdot)$
follows from a VFAR model of order 1 as in (33), where
$\epsilon_{tj}(u)={\mathbf{s}}(u)^{{\mathrm{\scriptscriptstyle\top}}}\boldsymbol{e}_{tj}$
and the $(i,j)$-th entry of ${\mathbf{Q}}(u,v)$ is given by
$Q_{ij}(u,v)={\mathbf{s}}(u)^{{\mathrm{\scriptscriptstyle\top}}}{\mathbf{B}}_{ij}{\mathbf{s}}(u)$.

We next implement the standard three-step estimation procedure in the VFAR
method to estimate ${\mathbf{Q}}(u,v)$.

  1. Step 1.

Perform FPCA on $\\{Y_{tj}(\cdot)\\}_{t\in[200]}$ thus obtaining the estimated
eigenfunctions
$\widehat{\mathbf{s}}_{j}(\cdot)=\\{\hat{s}_{j1}(\cdot),\dots,\hat{s}_{j5}(\cdot)\\}^{{\mathrm{\scriptscriptstyle\top}}}$
and the corresponding estimated principal component scores
$\widehat{\boldsymbol{\theta}}_{tj}=(\hat{\theta}_{t1},\dots,\hat{\theta}_{t5})^{{\mathrm{\scriptscriptstyle\top}}}$
with $\hat{\theta}_{tl}=\langle Y_{tj},\hat{s}_{jl}\rangle$ for each $j$.

  2. Step 2.

Write
$\boldsymbol{\Theta}_{1}=(\widehat{\boldsymbol{\theta}}_{tj})_{t\in[199],j\in[p]}\in\mathbb{R}^{199\times
5p}$ and
$\boldsymbol{\Theta}_{2}=(\widehat{\boldsymbol{\theta}}_{tj})_{t\in[200]\setminus[1],j\in[p]}\in\mathbb{R}^{199\times
5p}$. Obtain the least-squares estimator of ${\mathbf{B}}$ as
$\widehat{\mathbf{B}}=\\{(\boldsymbol{\Theta}_{1}^{{\mathrm{\scriptscriptstyle\top}}}\boldsymbol{\Theta}_{1})^{-1}\boldsymbol{\Theta}_{1}^{{\mathrm{\scriptscriptstyle\top}}}\boldsymbol{\Theta}_{2}\\}^{{\mathrm{\scriptscriptstyle\top}}}\equiv(\widehat{\mathbf{B}}_{ij})_{i,j\in[p]}.$

  3. Step 3.

Recover the functional coefficient by
$\widehat{Q}_{ij}(u,v)=\widehat{\mathbf{s}}_{i}(u)^{{\mathrm{\scriptscriptstyle\top}}}\widehat{\mathbf{B}}_{ij}\widehat{\mathbf{s}}_{j}(u)$.

Let $\widehat{Q}_{ij}(\cdot,\cdot)$ be the $(i,j)$-th entry of
$\widehat{\mathbf{Q}}(\cdot,\cdot)$. We then compute
$\int_{\mathcal{U}}\widehat{\mathbf{Q}}(\cdot,v){\mathbf{Y}}_{200}(v){\rm d}v$
as the one-step ahead prediction for the original curves
${\mathbf{Y}}_{201}(\cdot)$.

Finally, we summarize in Table S1 the effect of decorrelation transformation
on the identified group numbers $\mathring{q}_{y}$ of Seg+Y and
$\mathring{q}_{z}$ of Seg+Z. Notably, the transformation step results in the
identification of more distinct groups.

Table S1:  The means of identified group numbers for Seg+Y and Seg+Z over 500
simulation runs.

| $p=10$ | $p=15$ | $p=20$ | $p=25$ | $p=30$ | $p=35$
---|---|---|---|---|---|---
$\mathring{q}_{y}$ | 6.038 | 5.642 | 4.195 | 3.094 | 2.557 | 2.158
$\mathring{q}_{z}$ | 6.126 | 6.394 | 6.737 | 6.612 | 5.687 | 6.712

###  H.2 UK annual temperature data

The dataset, which is available at
https://www.metoffice.gov.uk/research/climate/maps-and-data/historic-station-
data, consists of monthly mean temperature collected at $p=22$ measuring
stations across Britain from 1959 to 2020 ($n=62$). Let
$\check{Y}_{tj}(v_{i})$ ($t\in[62],$ $j\in[22],$ $i\in[12]$) be the mean
temperature during month $v_{i}=i$ of year $1958+t$ measured at the $j$-th
station. The observed temperature curves are smoothed using a $10$-dimensional
Fourier basis that characterize the periodic pattern over the annual cycle.
The post-sample prediction are carried out in an identical way to Section 6.1,
we choose $k_{0}=m=3$ in our estimation procedure and treat the smoothed
curves in the first $n_{1}=41$ years and the last $n_{2}=21$ years as the
training sample and the test sample, respectively. The values of MAPE and MSPE
for $h\in\\{1,2,3\\}$ defined in (34) are summarized in Table S2. Again it is
obvious that SegV and FTSegV perform similarly well and both provide the
highest predictive accuracies among all comparison methods for all $h.$

Table S2:  MAFEs and MSFEs for eight competing methods on the UK temperature
curves for $h\in\\{1,2,3\\}.$

Method | MAFE | MSFE
---|---|---
$h=1$ | $h=2$ | $h=3$ | $h=1$ | $h=2$ | $h=3$
SegV | 0.786 | 0.806 | 0.827 | 1.073 | 1.075 | 1.155
Under.SegV | 0.805 | 0.826 | 0.883 | 1.152 | 1.135 | 1.266
Uni.SegV | 0.797 | 0.821 | 0.845 | 1.101 | 1.126 | 1.174
FTSegV | 0.789 | 0.806 | 0.828 | 1.077 | 1.073 | 1.158
Under.FTSegV | 0.791 | 0.820 | 0.872 | 1.105 | 1.112 | 1.250
Uni.FTSegV | 0.797 | 0.821 | 0.845 | 1.101 | 1.126 | 1.174
UniV | 0.936 | 0.951 | 0.976 | 1.450 | 1.450 | 1.458
GSY | 0.894 | 0.884 | 0.854 | 1.346 | 1.338 | 1.219

###  H.3 Age-specific mortality data

Table S3 gives a list of $p=29$ inclusive countries with corresponding ISO
Alpha-3 codes under our study.

Table S3:  List of inclusive countries with corresponding ISO Alpha-3 codes.

Country | Code | Country | Code | Country | Code | Country | Code
---|---|---|---|---|---|---|---
Australia | AUS | Finland | FIN | Norway | NOR | Sweden | SWE
Austria | AUT | France | FRA | Portugal | PRT | Switzerland | CHE
Belgium | BEL | Hungary | HUN | Poland | POL | Great Britain | GBR
Belarus | BLR | Ireland | IRE | Netherlands | NLD | United States | USA
Bulgaria | BGR | Italy | ITA | New Zealand | NZL | Ukraine | UKR
Canada | CAN | Japan | JPN | Russia | RUS |  |
Denmark | DEN | Lithuania | LTU | Slovakia | SVK |  |
Czech Republic | CZE | Latvia | LVA | Spain | ESP |  |

Figure S1: Functional cross-autocorrelation of the original female mortality
data

(a) Functional cross-autocorrelation of the 1st component series of the
transformed curves

(b) Functional cross-autocorrelation of the 2nd component series of the
transformed curves

(c) Functional cross-autocorrelation of the 3rd component series of the
transformed curves

Figure S2: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 4th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 5th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 6th component series of the
transformed curves

Figure S3: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 7th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 8th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 9th component series of the
transformed curves

Figure S4: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 10th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 11th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 12th component series of the
transformed curves

Figure S5: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 13th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 14th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 15th component series of the
transformed curves

Figure S6: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 16th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 17th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 18th component series of the
transformed curves

Figure S7: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 19th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 20th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 21st component series of the
transformed curves

Figure S8: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 22nd component series of the
transformed curves

(b) Functional cross-autocorrelation of the 23rd component series of the
transformed curves

(c) Functional cross-autocorrelation of the 24th component series of the
transformed curves

Figure S9: Selected functional cross-autocorrelation of the transformed female
mortality data

(a) Functional cross-autocorrelation of the 25th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 26th component series of the
transformed curves

(c) Functional cross-autocorrelation of the 27th component series of the
transformed curves

Figure S10: Selected functional cross-autocorrelation of the transformed
female mortality data

(a) Functional cross-autocorrelation of the 28th component series of the
transformed curves

(b) Functional cross-autocorrelation of the 29th component series of the
transformed curves

Figure S11: Selected functional cross-autocorrelation of the transformed
female mortality data

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