# “Sound and Fury”: Nonlinear Functionals of Volatility Matrix in the Presence
of Jump and Noise

Richard Y<EMAIL_ADDRESS>[

###### Abstract

This paper resolves a pivotal open problem on nonparametric inference for
nonlinear functionals of volatility matrix. Multiple prominent statistical
tasks can be formulated as functionals of volatility matrix, yet a unified
statistical theory of general nonlinear functionals based on noisy data
remains challenging and elusive. Nonetheless, this paper shows it can be
achieved by combining the strengths of pre-averaging, jump truncation and
nonlinearity bias correction. In light of general nonlinearity, bias
correction beyond linear approximation becomes necessary. Resultant estimators
are nonparametric and robust over a wide spectrum of stochastic models.
Moreover, the estimators can be rate-optimal and stable central limit theorems
are obtained. The proposed framework lends itself conveniently to uncertainty
quantification and permits fully feasible inference. With strong theoretical
guarantees, this paper provides an inferential foundation for a wealth of
statistical methods for noisy high-frequency data, such as realized principal
component analysis, continuous-time linear regression, realized Laplace
transform, generalized method of integrated moments and specification tests,
hence extends current application scopes to noisy data which is more prevalent
in practice.

62M09,

60G44,

62G05,

62G15,

62G20,

nonlinear functionals,

nonlinearity bias,

stochastic volatility matrix,

microstructure noise,

jumps,

stable central limit theorems,

principal component analysis,

###### keywords:

[class=MSC]

###### keywords:

.
** This manuscript supersedes an earlier draft named “Inference for Volatility
Functionals of Multivariate Itô Semimartingales Observed with Jump and Noise”.
The overlap with the previous draft consists of stochastic model in Section 2,
rate-optimal estimator in Section 3, Theorem 1 in Section 4. What’re new in
this manuscript include statistical applications of the proposed methodology,
a new family of estimators with a new set of hyper-parameters that facilitate
applications, additional theorem and propositions that show consistency,
asymptotic Gaussianity, finite-sample positive definiteness. Section 6 is new
and we demonstrate a concrete financial application with large high-frequency
dataset and discuss its empirical implication on portfolio construction

##  1 Introduction

Functions of covariance matrix are the formulations in various eminent
statistical applications. The inverse function leads to precision matrix,
while a more prosaic example is the identity function (covariance matrix).
Their applications include, for example, graphical models and discriminant
analysis. Model selection and testing are oftentimes accomplished by functions
of covariance matrix, as statistical models imply linear and nonlinear
relationships among its elements and blocks. As for factor models and
principal component analysis, eigenvalues or associated spectral functions are
differentiable nonlinear functions of the corresponding covariance matrix, so
are eigenvectors.

Accordingly, functionals of stochastic volatility matrix are critical in
applications using time-dependent high-frequency data. High-frequency data
often arises from financial transactions, neurological measurements, satellite
weather recordings, gravitational wave detectors etc. To demonstrate a
concrete application, we will study principal component analysis from the
perspective of nonlinear functionals in Section 5; and we will present an
empirical analysis of transaction data from TAQ (Trade and Quote) database in
Section 6.

Extant literature on functionals of volatility matrix demands the assumption
that there is no noise in the data. This no-noise assumption implies, as the
sampling frequency becomes higher, consistent estimators converge to
parameters of interest. However, this assumption imposes a restricting
limitation on the type of data that we can use, as the empirical fact of high-
frequency data tells us an opposing story of divergence at higher frequencies.
Figure 1 presents how estimators behave as sampling interval $\Delta_{n}$
shrinks. The subscript $n$ in $\Delta_{n}$ denotes sample size. Larger sample
sizes come with smaller sampling intervals, i.e., $\Delta_{n}\to 0$ as
$n\to\infty$. Strikingly the estimator for noiseless data does not converge
for large sample sizes.

Figure 1: Estimators applied to S&P 100 transaction data during 12/16/2019 -
12/20/2019

Noise affects statistical significance and severely biases estimation. The
presence of microstructure noise is commonly acknowledged. Solving the noise
problem is vital to statistics of high-frequency data. The literature has
called for a solution to this challenging open problem for years, e.g. [1, 2,
3, 4, 5, 6].111For reference, see p. 1108 in [1]; p. 1617 in [2]; p. 38 in
[3]; Section 6 in [4]; p. 159 in [5]; Remark 1 in [6]. In this paper we
provide a viable approach to this much sought-after solution.

Formally, functionals of volatility matrix are defined as

$S(g)_{t}=\int_{0}^{t}g(c_{s})\,\mathrm{d}s,$ (1.1)

here $t$ is finite; $g:\mathbb{R}^{d\times d}\mapsto\mathbb{R}^{r}$ is a twice
continuously differentiable function on some convex cone of positive-definite
matrices; $c_{s}$ is stochastic volatility matrix, viz., instantaneous
covariation of the continuous part of Itô semimartingale (e.g. jump-diffusion
model). (1.1) is the inferential objective and it is of paramount statistical
importance. By suitable choices of $g$, the quantity $S(g)_{t}$ can aid
researchers and practitioners in uncovering valuable yet hidden information
about cross sections and temporal evolutions from datasets of interest.

Indeed, a large amount of research efforts have been devoted to this kind of
questions in the last twenty years. Early literature largely concentrated in
stochastic volatility matrix and linear functionals; recent research advanced
on general nonlinear functionals. It may come as no surprise that general
functionals of stochastic volatility matrix provide a powerful unifying
framework.

For linear functionals of stochastic volatility, the problem of microstructure
noise has been resolved. Some influential approaches are two-scales realized
volatility [7, 8], realized kernel method [9, 10], pre-averaging method [11,
12, 13, 14], quasi-maximum likelihood method [15, 16, 17], spectral local
method of moments [18, 19].

For general nonlinear functionals, existing literature has absolved the
inferential problem in the absence of microstructure noise. The pioneering
work [20] put forward a statistical framework based on contiguity and local
likelihood for nonlinear transformations of volatility. The seminal paper [21]
proposed a plug-in inferential framework for nonlinear functionals, laid out
the statistical theory, and spurred a long line of follow-up work on
statistical methodologies and empirical applications. A more recent work [22]
proposed a spectral method to circumvent asynchronous observations across
dimensions when conducting statistical inference for nonlinear functionals of
volatility matrices.

Detailed study of nonlinear functionals of statistical importance and
resultant empirical results using noise-free data blossomed in recent years,
some examples are

  * •

realized Laplace transform [1, 23, 24];

  * •

generalized method of integrated moments [2];

  * •

continuous-time linear regressions [3];

  * •

realized principal component analysis [4, 27];

  * •

estimation of asymptotic variances, e.g. the quantity called quarticity [21];

  * •

model specification tests [25, 26].

These aforementioned applications utilized high-frequency transaction data to
answer statistical questions in economics and finance, such as model
calibration [1, 23, 24], testing economic models [2, 25, 26], measuring
financial risks [3, 4, 27], and uncertainty quantification [21]. Behind this
powerful formulation and the wide range of applicability, however, there lurk
difficult challenges.

A substantial challenge is nonlinearity. General nonlinearity induces high-
order derivatives which contributes to the estimation error. Local linear
approximation of nonlinear functionals only guarantees consistency and a
suboptimal rate of convergence. In order to improve estimation accuracy and
establish central limit theorems, bias correction of the high-order
nonlinearity terms is indispensable, cf. [21, 28]. On one hand, direction
correction of these nonlinearity biases requires computing gradients and
Hessian matrices. On the other hand, nonlinearity bias correction requires
careful tuning in nonparametric estimation. Creative and groundbreaking
advancements have been made, under the no-noise assumption, to relax tuning
restriction, facilitate bias correction and variance estimation by multiscale
jackknife [5] and matrix calculus [26]. In situations where noise must be
considered, notwithstanding one more layer of complexity, we find novel
nonlinearity bias terms and a new tuning range that allows successful bias
correction.

The second significant challenge is jumps. Discontinuity is frequently
encountered in functional data. It is also a stylized fact in financial time
series. In consequence, nonparametric jump-diffusion models are widely
accepted by the literature. When estimating functionals of volatility matrix,
the technique to handle discontinuity is jump truncation, which has been used
in [5, 26, 21, 28]. However, jump and noise exhibit subtle interplay and
delicate intricacy as documented in [29, 30]. Jump truncation for the purpose
of volatility matrix functionals in the noisy setting is far from being
straightforward. Despite this complexity, we establish a theoretical base that
takes account of noise and jump simultaneously. It is one of the contributions
of this paper.

Nevertheless, the central challenge is to find an effective statistical method
and theory robust to microstructure noise. Figure 1 has demonstrated the curse
of noise, especially in the regime where the sampling interval $\Delta_{n}$ is
small. As made manifest in Figure 2, $\Delta_{n}$ being a few seconds or
smaller is typical for high-frequency financial data. Yet, because of the no-
noise assumption in the extant literature on nonlinear functionals, the
current available statistical methods require researchers to subsample high-
frequency data so that $\Delta_{n}$ is between 1 minute to 5 minutes, e.g. [4,
31, 32]. The goal of subsampling is to mitigate the degree of noise
accumulation, so that estimators designed for noise-free data are able to
yield sound results, cf. Figure 1.

Figure 2: Daily average frequencies of S&P 100 stock transactions

In contrast to mitigation, we adopt a “bottom-up” approach, i.e., to start
with a more realistic stochastic model that incorporates microstructure noise,
then design noise-robust statistical methods, and then provide relevant
statistical theory. We do not dispute the validity of subsampling as a
mitigation, but we would like to emphasize that utilizing data of higher
frequencies in a noise-robust fashion indeed gains statistical benefits,
namely better estimation accuracy and sharper confidence intervals. It is an
exemplar that data gives feedback to the model, and the attendant change
delivers a better statistical performance.

The limitation imposed by the no-noise assumption on the amount of usable data
is a pressing issue, and will be more so as we are observing explosions of
data in the data-revolution age. The TAQ database of financial time series is
no exception. The size of TAQ transaction data is depicted in Figure 3. TAQ is
timestamped at millisecond latency and is subject to noise contamination. If
equipped with a noise-robust statistical methodology, we can improve
$\Delta_{n}$ from 5 minutes to, for example, 1 second. In this case there
would immediately be 300 times more data at our disposal. Consequently this
paper enables applications of volatility functionals to harvest data at much
higher frequencies than the state-of-the-art methods allow.

Figure 3: Accumulated sizes of TAQ transaction data

To cope with noise, we conduct local time-domain smoothing by the pre-
averaging method [12, 13, 14]. To estimate general nonlinear functionals, we
embed the pre-averaging method into the plug-in framework of [21]. In this
direction, the technical advancement of this work is two-fold. One is to
generalize the pre-averaging method from volatility matrix to its general
functionals. The other is to extend the framework of functional inference to
accommodate noisy data.

Besides nonlinearity bias, jumps, and noise discussed earlier, we need to
tackle the following technicalities on the journey to asymptotic theory
valuable to statistics.

  * •

Stochastic volatility. Our model is nonparametric and can reproduce the
salient characteristics of empirical financial data. An primary feature is
stochastic volatility which models the volatility matrix as a stochastic
process. However, to develop a robust and model-free methodology with
satisfactory statistical guarantees, it becomes crucial to balance statistical
error (dictated by the extent of local smoothing) and target error
(attributable to time-varying parameters).

  * •

Dependence. The pre-averaging method uses overlapping windows, hence the local
moving averages are highly correlated for which standard central limit theorem
is not valid. We use the “big block - small block” technique of [12]. It is
instrumental to derive central limit theorems (hereafter CLT) for highly
correlated high-frequency data, but in proof it induces technicality and
involved notation.

  * •

Stable convergence.222Readers can read section 2.2.1 and section 2.2.2 in [33]
for an exposition of stable convergence. For statistical purpose, we need to
establish stable convergence which is stronger than weak convergence
(convergence in distribution). The reason is that sampling distributions of
the estimators depend on the sample path of stochastic volatility matrix. We
need a mode of convergence in which the estimators converge jointly with other
variables, so that we can consistently estimate the asymptotic variances for
the purpose of statistical inference.

  * •

Unbounded derivatives. Some statistical applications, such as precision matrix
estimation, specification tests, and linear regression, correspond to
functionals with singularities in derivatives near the origin. The original
framework [21] is not applicable in this situation. [3] proposed a spatial
localization procedure that accommodates this type of functionals under the
no-noise assumption. In the interest of applications using noisy data, we
extend the spatial localization by establishing uniform convergence of the
pre-averaging method plus jump truncation in the presence of discontinuity.

We state the stochastic model and our assumptions in Section 2. After
presenting the appropriate range of tuning parameters, we show consistency,
convergence rates, and associated CLT. There are two types of general results.
The first satisfies the rate-optimal CLT, hence it is asymptotically superior;
the second ensures positive semi-definiteness in finite samples, and in a
sense requires less computation for statistical inference. We will give
details about them in Section 3 and Section 4.

We focus on specific functionals which are the basis of principal component
analysis (hereafter PCA) using noisy high-frequency data in Section 5. It has
generated a lot of statistical interest and has been studies in, for example,
[4, 27]. This paper significantly improves the accuracy of PCA either by being
able to harness more data at higher frequencies, or by increasing the
convergence rate. We also present recent examples among other interesting
applications. Based on PCA of more than 16 years of noisy high-frequency data,
a large-scale empirical analysis is presented in Section 6. Finally we
conclude in Section 7. Proofs and simulations are in the appendices.

##  2 Setting

We describe our data generating mechanism and set up notation in this section.

The model of high-frequency data with microstructure noise consists of three
components: signal process, noise process, sampling scheme.

Itô Semimartingale $X$Noisy Process $Y$Noisy Data $Y^{n}_{i}$$(Q_{t})$sampling

###  2.1 Signal

The underlying signal process is modeled by an Itô semimartingale $X$, i.e.,
$X$ is a solution to the stochastic differential equation

$X_{t}=X_{0}+\int_{0}^{t}b_{s}\,\mathrm{d}s+\int_{0}^{t}\sigma_{s}\,\mathrm{d}W_{s}+J_{t}$
(2.1)

where $b_{s}\in\mathbb{R}^{d}$, $\sigma_{s}\in\mathbb{R}^{d\times d^{\prime}}$
with $d\leq d^{\prime}$, $b$ and $c$ are adapted and càdlàg333“continue à
droite, limite à gauche”. It means “right continuous with left limits”, i.e.,
the sample paths lie in the Skorokhod space.; $W$ is a
$d^{\prime}$-dimensional standard Brownian motion; $J$ is a purely
discontinuous process.

The mathematical description of $J$ is relegated to (A.1), which can
accommodate compound Poisson process, Lévy process etc. The stochastic
volatility matrix is defined as $c_{s}=\sigma_{s}\sigma_{s}^{\mathrm{T}}$,
which is stochastic process specified in (A.2).

The full signal model assumption is given in Appendix A, below is a summary of
its features:

  * •

the drift $b$ has a smooth trajectory in the mean-square sense;

  * •

the volatility $c$ is a spatially constrained Itô semimartingale444However, it
is important to accommodate long-memory volatility model. The volatility
functional inference in long-memory and noisy setting is an open question
under investigation. such that $c$ is locally bounded and well-conditioned;

  * •

the celebrated leverage effect (correlation between price and volatility) is
allowed, and is defined through the tensor product of $\sigma$ and
$\sigma^{(c)}$;

  * •

$J$ may exhibit finite and infinite activities, but has finite variation
(i.e., finite-length trajectory) on compact sets;

These assumptions are necessary for the existence of CLT. The assumption that
$c$ is locally well-conditioned is for the applicability of some functionals
of statistical interest, such as some specification tests and linear
regression.

This signal model has no parametric restriction and allows time-varying
parameters, e.g. the drift $b_{s}$ and diffusion coefficient $\sigma_{s}$
could be stochastic processes. Moreover it permits unrestricted non-
stationarity and dependence among the state variables.

###  2.2 Noise

We adopt the probabilistic model in [12, 13] to describe noise. The advantage
of this formalism is that it incorporates additive white noise, rounding
error, the combination thereof as special cases. The gist is to suppose for
$t>0$ there is a probability transition kernel $Q_{t}$ linking the value of a
noisy process $Y$ with that of $X$ at time $t$. The abstract formalism is
given in Appendix A.

The noise assumption is that under the conditional probability measure
$Q_{t}$, the process $Y-X$ has zero mean, finite eighth moment, no serial
correlation (though cross-sectional correlation is allowed). An example of
this noisy process is given below.

######  Example 1.

$Y_{t}=f(X_{t},e_{t})$ (2.2)

where $f:\mathbb{R}^{d}\times\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ and $e$ is a
$\mathbb{R}^{d}$-valued perturbation process such that
$\varepsilon_{t}=f(X_{t},e_{t})-X_{t}$ is centered white noise process
conditional on $X$.

###  2.3 Sampling

This work treats regularly sampled observations and considers in-fill
asymptotics.555aka fixed-domain asymptotics, high-frequency asymptotics,
small-interval asymptotics Specifically, the samples are observed every
$\Delta_{n}$ time units on a finite time interval $[0,t]$ where $n=\lfloor
t/\Delta_{n}\rfloor$ is the sample size up to time $t$. As $\Delta_{n}\to 0$,
$n\to\infty$ while $t$ is fixed. We establish functional convergence and our
statistical theory holds for any $t>0$.

Throughout this paper, $U^{n}_{i}$ is written for $U_{i\Delta_{n}}$ where $U$
can be a process or filtration. For example, $c^{n}_{i}$ denotes the
stochastic volatility matrix at time $i\Delta_{n}$; let be $(\mathcal{F}_{t})$
a filtration, $\mathcal{F}^{n}_{i}$ is the information up to time
$i\Delta_{n}$. For any process $U$, $\Delta^{n}_{i}U$ represents the increment
$U^{n}_{i}-U^{n}_{i-1}$.

###  2.4 Notations

For $r\in\mathbb{N}^{+}$, $\mathcal{C}^{r}(\mathcal{S})$ denotes the space of
$r$-time continuously differentiable functions on the domain $\mathcal{S}$;
$\mathcal{S}^{+}_{d}$ is the convex cone of $d\times d$ positive semidefinite
matrices; $\mathbf{I}_{r}$ is the $r\times r$ identity matrix; $\|\cdot\|$
denotes a norm on vectors, matrices or tensors; given $a\in\mathbb{R}$,
$\lfloor a\rfloor$ denotes the largest integer no more than $a$; $a\vee
b=\max\\{a,b\\}$, $a\wedge b=\min\\{a,b\\}$; $a_{n}=O_{p}(b_{n})$ means
$\forall\epsilon>0$, $\exists M>0$ such that
$\sup_{n}\mathbb{P}(a_{n}/b_{n}>M)<\epsilon$; $\mathbf{A}^{\mathrm{T}}$ is the
transpose of the vector or matrix $\mathbf{A}$; for a multidimensional array,
the entry index is written in the superscript, e.g.,
$X_{t}=(X^{1}_{t},\cdots,X^{d}_{t})^{\mathrm{T}}$, $\mathbf{A}^{jk}$ denotes
the $(j,k)$-th entry in the matrix $\mathbf{A}$; $\otimes$ is the outer
product, e.g., for two vectors $\mathbf{u}$ and $\mathbf{v}$,
$\mathbf{u}\otimes\mathbf{v}=\mathbf{u}\mathbf{v}^{\mathrm{T}}$;
$\partial_{jk}g$ and $\partial^{2}_{jk,lm}g$ denote the gradient and Hessian
of $g$ with respect to the $(j,k)$-th and $(l,m)$-th entries in its matrix
argument; for a scalar function $f$ define on a subset of $\mathbb{R}$,
$f^{\prime}$ denotes its gradient.
$\overset{\mathcal{L}-s(f)}{\longrightarrow}$ (resp.
$\overset{\mathcal{L}-s}{\longrightarrow}$) denotes stable convergence in law
of processes (resp. variables); $\overset{\mathcal{L}}{\longrightarrow}$
denotes convergence in law (weak convergence);
$\overset{u.c.p.}{\longrightarrow}$ denotes uniform convergence in probability
on compact sets; $\mathcal{MN}(0,V)$ is a mixed Gaussian distribution with
zero mean and conditional covariance matrix $V$.

##  3 Methodology

The estimation methodology is comprised of five steps:

  1. i.

local moving averages of noisy data by a smoothing kernel;

  2. ii.

jump truncation operated on local moving averages;

  3. iii.

spot volatility estimator $\widehat{c}^{n}_{i}$’s for estimating
$c^{n}_{i}$’s;

  4. iv.

Riemann sum of $g(\widehat{c}^{n}_{i})$’s for approximating the integral
functional $\int g(c_{s})\,\mathrm{d}s$;

  5. v.

nonlinearity bias correction.

The idea of local moving average is due to the pre-averaging method [12, 13];
the truncation is modified from (16.4.4) in [33]; the Riemann sum and bias
correction are inspired by [21].

Though nonlinear bias does not impact on statistical consistency, we stress
that we must carefully take into account the nonlinear terms since they lead
to bias in the second order. To improve convergence rate and to establish CLT,
nonlinearity bias correction is crucial.

The specific recipe is given next.

###  3.1 Estimator of stochastic volatility matrix

For the local moving averages, we choose a smoothing kernel $\varphi$ such
that

$\begin{array}[]{l}\text{supp}(\varphi)\subset(0,1),\,\int_{0}^{1}\varphi^{2}(s)\,\mathrm{d}s>0\\\
\varphi\in\mathcal{C}\text{ is piecewise
}\mathcal{C}^{1};\,\varphi^{\prime}\text{ is piecewise Lipschitz}\end{array}$
(3.1)

We choose an integer $l_{n}$ as the number of observations in each smoothing
window, define $\varphi^{n}_{h}=\varphi(h/l_{n})$ and
$\psi_{n}=\sum_{h=1}^{l_{n}-1}(\varphi^{n}_{h})^{2}$. We associate the
following quantities with the noisy process $Y$,

$\begin{array}[]{lcl}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}&=&(\psi_{n})^{-1/2}\sum_{h=1}^{l_{n}-1}\varphi^{n}_{h}\,\Delta^{n}_{i+h-1}Y\\\
\widehat{Y}^{n}_{i}&=&(2\psi_{n})^{-1}\sum_{h=0}^{l_{n}-1}(\varphi^{n}_{h+1}-\varphi^{n}_{h})^{2}\,\Delta^{n}_{i+h}Y\otimes\Delta^{n}_{i+h}Y\end{array}$
(3.2)

  * •

$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{Y}^{n}_{i}$ is a local
moving average of the noisy data $Y^{n}_{i}$’s, and it serves as a proxy for
$\Delta^{n}_{i}X$;

  * •

$\widehat{Y}^{n}_{i}$ is an offset to the remaining noise in the outer product
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\otimes\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}$.

If there is jump(s) in the local window $(i\Delta_{n},(i+l_{n}-1)\Delta_{n}]$,
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{Y}^{n}_{i}$ deviates from
the target. We deem the moving average to be unreliable and veto it if
$\|\accentset{{\cc@style\underline{\mskip 10.0mu}}}{Y}^{n}_{i}\|>\nu_{n}$. The
threshold of jump truncation is $\nu_{n}=\alpha\,\Delta_{n}^{\rho}$. We let
$\alpha$ mirror the dimensionality and volatility level, and set the value of
$\rho$ according to the scaling property of Brownian motion (the choice of
$\rho$ is stated in (3.9)). There are some variants to this jump truncation.
An alternative is elementwise truncation
$|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{r,n}_{i}|>\alpha^{r}_{i}\Delta_{n}^{\rho}$ where
$\alpha^{r}_{i}$ is related to the local volatility of the $r$-th component.
We recommend jump truncation tuned to optimize finite-sample performance, and
it is used in our simulation and data analysis. We should emphasize that these
variants share the same asymptotic behavior and satisfy the same asymptotic
theory.

Based on these ingredients, we choose $k_{n}$ such that $k_{n}>l_{n}$, and
design the following estimator of stochastic volatility matrix:

$\widehat{c}^{n}_{i}\equiv\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}\Big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\otimes\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\|\leq\nu_{n}\\}}-\widehat{Y}^{n}_{i+h}\Big{)}$ (3.3)

This offset term $\widehat{Y}^{n}_{i}$ is needed to achieve the optimal
convergence rate in CLT. Yet a disadvantage is that $\widehat{c}^{n}_{i}$ can
not be guaranteed to be positive semidefinite in finite samples. A fix is to
project $\widehat{c}^{n}_{i}$ onto the convex cone $\mathcal{S}^{+}_{d}$ in
Frobenius norm. Suppose $\widehat{c}^{n}_{i}=Q\Lambda Q^{\mathrm{T}}$ is the
eigenvalue factorization, the positive semidefinite projection is
$\widehat{c}^{\prime n}_{i}=Q\Lambda_{+}Q^{\mathrm{T}}$ where
$\Lambda_{+}^{ij}=(\Lambda^{jj}\vee 0)\mathds{1}_{\\{i=j\\}}$.

Stimulated by [14], we design another estimator of stochastic volatility
matrix to fulfill the positive semidefinite requirement:

$\widetilde{c}^{n}_{i}\equiv\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\otimes\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i+h}\|\leq\nu_{n}\\}}$ (3.4)

In addition to being positive semidefinite which some readers may prioritize,
(3.4) leads to easier and more practical bias correction and uncertainty
quantification for general functionals of volatility matrix (our end goal).
This kind of advantage of (3.4) is pronounced in PCA. Detailed comparison
between these two estimators is given in Remark 2.

The estimators of stochastic volatility matrix use local moving averages
computed on overlapping windows. Local moving averages on overlapping windows
fully utilize data and can lead to the optimal convergence rate. The
computation can be done via convolution of the noisy data with a sequence
induced by the smoothing kernel $\varphi$. Hence (3.3) and (3.4) can be
implemented by a fast Fourier transform (FFT) algorithm and they are scalable
for very large datasets.

###  3.2 Correction of nonlinearity bias

As an exposition of the necessity of nonlinear bias correction, let first
consider the case of $d=1$ and constant volatility. By Taylor expansion, the
estimation error of the plug-in estimator $g(\widehat{c})$ can be decomposed
as

$g(\widehat{c})-g(c)=\underbrace{\partial
g(c)(\widehat{c}-c)}_{\text{limit}}+\underbrace{\frac{1}{2}\partial^{2}g(c)(\widehat{c}-c)^{2}}_{\text{nonlinearity
bias}}+\underbrace{O_{p}(|\widehat{c}-c|^{3})}_{\text{asymptotically
negligible}}$

As it turns out (shown in Appendix C, D), the first-order term converges to
the limit in CLT. The bias arises inevitably, as long as $g$ is nonlinear
(with non-zero Hessian), from the second-order derivative and the quadratic
form of estimation error.

In the multivariate case, the bias term is more involved. Let
$\widehat{B}(g)^{n}_{i}$ be the nonlinear bias correction term to
$g(\widehat{c}^{n}_{i})$, we find that $\widehat{B}(g)^{n}_{i}$ need to be

$\widehat{B}(g)^{n}_{i}=\frac{1}{2k_{n}\Delta_{n}^{1/2}}\sum^{d}_{j,k,l,m=1}\partial^{2}_{jk,lm}g(\widehat{c}^{n}_{i})\times\Xi\big{(}\widehat{c}^{n}_{i},\widehat{\gamma}^{n}_{i}\big{)}^{jk,lm}$
(3.5)

where $\Xi$ is a tensor-valued function defined in (4.3), and
$\widehat{\gamma}^{n}_{i}$ is an estimator of the instantaneous covariance
matrix of noise at time $i\Delta_{n}$:

$\widehat{\gamma}^{n}_{i}\equiv\frac{1}{2m_{n}}\sum_{h=1}^{m_{n}}\Delta^{n}_{i+h}Y\otimes\Delta^{n}_{i+h}Y$
(3.6)

with $m_{n}=\lfloor\theta^{\prime}\Delta_{n}^{-1/2}\rfloor$ and
$\theta^{\prime}$ being positive finite.

The nonlinearity bias correction term $\widetilde{B}(g)^{n}_{i}$ to
$g(\widetilde{c}^{n}_{i})$ is

$\widetilde{B}(g)^{n}_{i}=\frac{1}{2k_{n}\Delta_{n}^{1/2+\delta}}\sum^{d}_{j,k,l,m=1}\partial^{2}_{jk,lm}g(\widetilde{c}^{n}_{i})\times\Sigma\big{(}\widetilde{c}^{n}_{i}\big{)}^{jk,lm}$
(3.7)

where $\delta$ is introduced in (3.12), $\Sigma$ is defined in (4.1).

###  3.3 Estimators of functionals

######  Definition 1.

Let $N^{n}_{t}=\lfloor t/(k_{n}\Delta_{n})\rfloor$, a rate-optimal estimator
of $(\ref{def.S(g)})$ is defined as

$\widehat{S}(g)^{n}_{t}\equiv
k_{n}\Delta_{n}\sum_{i=0}^{N^{n}_{t}-1}\left[g(\widehat{c}^{n}_{ik_{n}})-\widehat{B}(g)^{n}_{ik_{n}}\right]\times
a^{n}_{t}$

$\widehat{c}^{n}_{h}$ is defined in (3.3), $\widehat{B}(g)^{n}_{h}$ is defined
in (3.5), $a^{n}_{t}=t/(N^{n}_{t}k_{n}\Delta_{n})$ is a finite-sample
adjustment.

We choose to compute $\widehat{c}^{n}_{h}$’s over disjoint windows for
estimating $S(g)_{t}$. One may choose to compute $\widehat{c}^{n}_{h}$’s on
overlapping windows instead and define an alternative estimator as
$\widehat{S}(g)^{\prime n}_{t}\equiv\Delta_{n}\sum_{i=0}^{\lfloor
t/\Delta_{n}\rfloor-1}[g(\widehat{c}^{n}_{i})-B(g)^{n}_{i}]$. But we advice
against this alternative for two reasons. The first one is that
$\widehat{S}(g)^{n}_{t}$ and $\widehat{S}(g)^{\prime n}_{t}$ has the same
asymptotic properties such as convergence rate and limiting distributions, cf.
[4, 21]. The second one, more essential for some applications, is that
computing $\widehat{S}(g)^{\prime n}_{t}$ is unfortunately impractical in some
applications, such as the data analysis using PCA in Section 6. In fact,
should $\widehat{S}(g)^{\prime n}_{t}$ be employed, we need to compute more
than 100,000 eigenvalue factorizations of $90\times 90$ matrices for data of
every week, with more than 850 weeks our data analysis would take too long to
finish.

######  Definition 2.

Let $N^{n}_{t}=\lfloor t/(k_{n}\Delta_{n})\rfloor$, an estimator of
$(\ref{def.S(g)})$ with positive semidefinite plug-ins is defined as

$\widetilde{S}(g)^{n}_{t}\equiv
k_{n}\Delta_{n}\sum_{i=0}^{N^{n}_{t}-1}\left[g(\widetilde{c}^{n}_{ik_{n}})-\widetilde{B}(g)^{n}_{ik_{n}}\right]\times
a^{n}_{t}$

where $\widetilde{c}^{n}_{h}$ is defined in (3.4), $\widetilde{B}(g)^{n}_{h}$
is defined in (3.7), and $a^{n}_{t}=t/(N^{n}_{t}k_{n}\Delta_{n})$ is an
adjustment for the right-edge effect.

Our estimators, given the smoothing kernel $\varphi$, entail three tuning
parameters.

Table 1: Tuning parameters | description
---|---
$l_{n}$ | length of overlapping windows for local moving averages
$k_{n}$ | length of disjoint windows for stochastic volatility matrix estimation
$\nu_{n}$ | threshold of jump truncation

A judicially tuning choice is an indispensable if not the most important part
of sound estimation and inference. As for $k_{n}$, there are three levels of
considerations:

  1. level 1:

$k_{n}$ should be chosen to result in consistency and a satisfactory
convergence rate;

  2. level 2:

$k_{n}$ should be chosen so that we can establish CLT for inference;

  3. level 3:

$k_{n}$ should be chosen in a way that allows convenient correction of
nonlinear bias.

A severe pitfall looms when level 3 is dismissed. If falling prey to this
pitfall, researchers have to estimate the volatility of volatility and
volatility jumps (both are notoriously hard to estimate) in order to establish
CLT [28]. Under the no-noise assumption, this pitfall is documented and
avoided by [21]. Recently [5] proposed the multiscale jackknife method and
[26] used matrix calculus for practical nonlinear bias correction, and they
investigated how to choose $k_{n}$ more robustly under the no-noise
assumption. Their results greatly facilitate statistical applications using
sparsely subsampled data.

When we must confront the noise problem to improve statistical quality, there
is one more tuning parameter, namely $l_{n}$, to consider. However, when noise
is taken into account, the three levels of considerations for $k_{n}$ are
missing so far, and given a legitimate $l_{n}$ how to adjust $v_{n}$ is
unknown.

We provide an answer to this tuning problem. We finding the range of tuning
parameters when noise and jump are both present. The benefits include more
usable data and better estimation accuracy. Importantly, practical bias
correction and uncertainty quantification are possible under our tuning
configuration. We will show, with suitable choices of the combination
$(l_{n},\,k_{n},\,\nu_{n})$, our estimators are widely applicable to any
$g:\mathcal{S}^{+}_{d}\mapsto\mathbb{R}^{r}$ that satisfies

$g\in\mathcal{C}^{3}(\mathcal{S})$ (3.8)

where $\mathcal{S}\subset\mathcal{S}^{+}_{d}$ is the subspace containing an
enlargement of the sample path of $c$. Precisely,
$\mathcal{S}\supset\cup_{m}\mathcal{S}^{\epsilon}_{m}$ for some $\epsilon>0$,
$\mathcal{S}^{\epsilon}_{m}=\big{\\{}A\in\mathcal{S}^{+}_{d}:\inf_{M\in\mathcal{S}_{m}}\|A-M\|\leq\epsilon\big{\\}}$
is an enlargement of $\mathcal{S}_{m}$ which is identified in Assumption
A-$\nu$.

###  3.4 Tuning parameters for the optimal convergence rate

A proper combination of the tuning parameters is the requisite for
consistency, the existence of CLT, and the optimal convergence rate. To
fulfill these objectives for $\widehat{S}(g)^{n}_{t}$, we need

$\left\\{\begin{array}[]{rcll}l_{n}&=&\lfloor\theta\Delta_{n}^{-1/2}\rfloor&\\\
k_{n}&=&\lfloor\varrho\Delta_{n}^{-\kappa}\rfloor&\hskip
14.22636pt\kappa\in\left(\frac{2}{3}\vee\frac{2+\nu}{4},\frac{3}{4}\right)\\\
\nu_{n}&=&\alpha\Delta_{n}^{\rho}&\hskip
14.22636pt\rho\in\big{[}\frac{1}{4}+\frac{1-\kappa}{2-\nu},\frac{1}{2}\big{)}\end{array}\right.$
(3.9)

where $\theta,\varrho,\alpha>0$ are positive finite, and $\nu\in[0,1)$ is
introduced in Assumption A-$\nu$ which dictates the jump intensity.

Here we offer an intuition for (3.9). The reader can skip this part without
affecting understanding of the main results in Section 4.

  * •

$l_{n}$ influences the convergence rate
Under the noise model (2.2), according to the definition (3.2),

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}=\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{i}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\varepsilon}^{n}_{i}$

where $\accentset{{\cc@style\underline{\mskip 10.0mu}}}{X}^{n}_{i}$ and
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{\varepsilon}^{n}_{i}$ are
defined analogously to $\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}$ (see (3.2)). We can write
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\varepsilon}^{n}_{i}=-\psi_{n}^{-1/2}\sum_{h=0}^{l_{n}-1}(\varphi^{n}_{h+1}-\varphi^{n}_{h})\,\varepsilon^{n}_{i+h}$.
By the conditional independence of $\varepsilon^{n}_{i}$’s,
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\varepsilon}^{n}_{i}=O_{p}(l_{n}^{-1})$;
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{i}=O_{p}(\Delta_{n}^{-1/2})$ by (C.7) in Appendix C.2. By
taking $l_{n}\asymp\Delta_{n}^{-1/2}$ the orders of
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{X}^{n}_{i}$ and
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{\varepsilon}^{n}_{i}$ are
equal. It turns out that this choice of local smoothing window will deliver
the optimal rate of convergence.

  * •

$k_{n}$ dictates bias-correction and the CLT form
Here for exposition let’s suppose $d=1$ and $X$ is an continuous Itô
semimartingale, then

$\widehat{c}^{n}_{i}-c^{n}_{i}=d^{n}_{i}+s^{n}_{i}$

where

$\displaystyle d^{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\int_{i\Delta_{n}}^{(i+k_{n}-l_{n}+1)\Delta_{n}}(c_{s}-c^{n}_{i})\,\mathrm{d}s$
$\displaystyle s^{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\bigg{[}\widehat{c}^{n}_{i}(k_{n}-l_{n})\Delta_{n}-\int_{i\Delta_{n}}^{(i+k_{n}-l_{n}+1)\Delta_{n}}c_{s}\,\mathrm{d}s\bigg{]}$

    * –

We call $d^{n}_{i}$ “target error”; $d^{n}_{i}=O_{p}((k_{n}\Delta_{n})^{1/2})$
by (B.2) in Appendix B; its value depends on the temporal evolution of the
underlying stochastic volatility.

    * –

We call $s^{n}_{i}$ “statistical error”;
$s^{n}_{i}\approx\frac{1}{(k_{n}-l_{n})\Delta_{n}}\Delta_{n}^{1/4}(\chi^{n}_{i+k_{n}-l_{n}+1}-\chi^{n}_{i})$,
where $\chi$ is another continuous Itô semimartingale; this result is due to
(3.8) in [12]; so $s^{n}_{i}=O_{p}((k_{n}\Delta_{n}^{1/2})^{-1/2})$.

Balancing the orders of $d^{n}_{i}$ and $s^{n}_{i}$ by setting $\kappa=3/4$
brings the error $\widehat{c}^{n}_{i}-c^{n}_{i}$ down to the minimum order.
However, for the purpose of general nonlinear functionals, choosing
$\kappa\geq 3/4$ causes the bias to be substantially influenced by $d^{n}_{i}$
which depends on volatility of volatility and volatility jump (difficult to
estimate and de-bias in applications). Therefore, it is advisable to choose
$\kappa<3/4$, in which case $s^{n}_{i}$ dominates and $d^{n}_{i}$ vanishes
asymptotically, thereby the thorny terms are circumvented. Besides, to achieve
successful de-biasing of statistical error and negligibility of higher-order
Taylor-expansion terms, we need $\kappa>2/3$.

$\kappa$ for $k_{n}$$1/2$$2/3$$3/4$$1$order of
$\log(|\widehat{c}^{n}_{i}-c^{n}_{i}|)$$d^{n}_{i}$dominated bytarget
error$s^{n}_{i}$dominated bystatistical errorerrorminimizing $\kappa$ Figure
4: A schematic plot of tuning $k_{n}$

When jump is present and its activity index $\nu>2/3$, the lower-bound
requirement on $\kappa$ increase from $2/3$ to $(2+\nu)/4$. This increase is
required for uniform convergence in the spatial localization procedure, which
in turn is necessary for functionals with a singularity at the origin.

Section 3.1, 3.2 of [21] give an analogous discussion and their conclusion is
$1/3<\kappa<1/2$ under the no-noise assumption. With noise and local moving
averages, our conclusion is markedly different.

  * •

$\nu_{n}$ disentangles volatility from jump variation
According to (C.1) in Appendix C.1, $\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\|=O_{p}(\Delta_{n}^{1/2})$ if there is no jump over
$[i\Delta_{n},(i+l_{n})\Delta_{n}]$. By choosing $\rho<1/2$, the truncation
level $\nu_{n}>\Delta_{n}^{1/2}$, hence jump truncation keeps the diffusion
movements largely intact and discards jumps in a certain sense. To effectively
filter out the jumps, the truncation level should be bounded from above and
the upper bounds depends on the jump activity index $\nu$.

######  Remark 1.

If the reader is interested to estimate $(\ref{def.S(g)})$ with $g$ satisfying

$\|\partial_{h}g(x)\|\leq K_{h}(1+\|x\|^{r-h}),\,\,h=0,1,2,3,\,r\geq 3$ (3.10)

the requirements on $k_{n}$ and $\nu_{n}$ can be loosened and become

$\begin{array}[]{rcll}k_{n}&=&\lfloor\varrho\Delta_{n}^{-\kappa}\rfloor&\hskip
14.22636pt\kappa\in\left(\frac{2}{3},\frac{3}{4}\right)\\\
\nu_{n}&=&\alpha\Delta_{n}^{\rho}&\hskip
14.22636pt\rho\in\big{[}\frac{1}{4}+\frac{1}{4(2-\nu)},\frac{1}{2}\big{)}\end{array}$
(3.11)

For generality and wider applicability, we choose to accommodate the
functional space (3.8) and retain the requirement (3.9).

###  3.5 Tuning parameters for p.s.d. plug-ins

If readers choose to use $\widetilde{S}(g)^{n}_{i}$, the tuning parameters
should satisfy

$\left\\{\begin{array}[]{rcll}l_{n}&=&\lfloor\theta\Delta_{n}^{-1/2-\delta}\rfloor&\hskip
14.22636pt\delta\in\big{(}\frac{1}{10},\frac{1}{2}\big{)}\\\
k_{n}&=&\lfloor\varrho\Delta_{n}^{-\kappa}\rfloor&\hskip
14.22636pt\kappa\in\big{(}\big{(}\frac{2}{3}+\frac{2\delta}{3}\big{)}\vee\big{(}\frac{2+\nu}{4}+\frac{(2-\nu)\delta}{2}\big{)},\frac{3}{4}+\frac{\delta}{2}\big{)}\\\
\nu_{n}&=&\alpha\Delta_{n}^{\rho}&\hskip
14.22636pt\rho\in\big{[}\frac{1}{4}+\frac{\delta}{2}+\frac{1-\kappa}{2-\nu},\frac{1}{2}\big{)}\end{array}\right.$
(3.12)

The key insight is that we can increase $l_{n}$ to attenuate noise in
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{Y}^{n}_{i}$ and dispense
with the noise offset term $\widehat{Y}^{n}_{i}$. However, the positive
semidefinite estimator does not possess the optimal convergence rate (see the
next section).

##  4 Asymptotics and Inference

###  4.1 Elements

Before stating the asymptotic result, we need to introduce some elements in
our asymptotic theory. We associate the following quantities with the
smoothing kernel $\varphi$ for $l,m=0,1$:

$\begin{array}[]{ll}\phi_{0}(s)=\int_{s}^{1}\varphi(u)\varphi(u-s)\,\mathrm{d}u,&\phi_{1}(s)=\int_{s}^{1}\varphi^{\prime}(u)\varphi^{\prime}(u-s)\,\mathrm{d}u\\\
\Phi_{lm}=\int_{0}^{1}\phi_{l}(s)\phi_{m}(s)\,\mathrm{d}s,&\Psi_{lm}=\int_{0}^{1}s\,\phi_{l}(s)\phi_{m}(s)\,\mathrm{d}s\end{array}$
(4.1)

We define $\Sigma$, $\Theta$, $\Upsilon$ as $\mathbb{R}^{d\times d\times
d\times d}$-valued functions, such that for $x,z\in\mathbb{R}^{d\times d}$,
$j,k,l,m=1,\cdots,d$,

$\displaystyle\Sigma(x)^{jk,lm}$ $\displaystyle=$
$\displaystyle\frac{2\theta\,\Phi_{00}}{\phi_{0}(0)^{2}}\,\big{(}x^{jl}x^{km}+x^{jm}x^{kl}\big{)}$
$\displaystyle\Theta(x,z)^{jk,lm}$ $\displaystyle=$
$\displaystyle\frac{2\Phi_{01}}{\theta\,\phi_{0}(0)^{2}}\,\big{(}x^{jl}z^{km}+x^{jm}z^{kl}+x^{km}z^{jl}+x^{kl}z^{jm}\big{)}$
(4.2) $\displaystyle\Upsilon(z)^{jk,lm}$ $\displaystyle=$
$\displaystyle\frac{2\Phi_{11}}{\theta^{3}\,\phi_{0}(0)^{2}}\,\big{(}z^{jl}z^{km}+z^{jm}z^{kl}\big{)}$

where $\theta$ is introduced in (3.9). We define $\Xi$ as a tensor-valued
function

$\Xi(x,z)=\Sigma(x)+\Theta(x,z)+\Upsilon(z)$ (4.3)

Now we are ready to describe limiting processes.

######  Definition 3.

Suppose $g$ satisfies (3.8) or (3.10), $Z_{\Xi}(g)$ and $Z_{\Sigma}(g)$ are
two stochastic processes defined on an extension of the probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ specified in (A.4), such
that conditioning on $\mathcal{F}$, $Z_{\Xi}(g)$ and $Z_{\Sigma}(g)$ are
centered Gaussian processes with conditional covariance matrices being

$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{E}[Z_{\Xi}(g)Z_{\Xi}(g)^{\mathrm{T}}|\mathcal{F}]$ $\displaystyle=$
$\displaystyle V_{\Xi}(g)$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{E}[Z_{\Sigma}(g)Z_{\Sigma}(g)^{\mathrm{T}}|\mathcal{F}]$
$\displaystyle=$ $\displaystyle V_{\Sigma}(g)$

where $\accentset{{\cc@style\underline{\mskip 10.0mu}}}{E}$ is the conditional
expectation operator on the extended probability space and

$\displaystyle V_{\Xi}(g)_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{j,k,l,m=1}^{d}\partial_{jk}g(c_{s})\,\partial_{lm}g(c_{s})^{\mathrm{T}}\,\Xi(c_{s},\gamma_{s})^{jk,lm}\,\mathrm{d}s$
(4.4) $\displaystyle V_{\Sigma}(g)_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{j,k,l,m=1}^{d}\partial_{jk}g(c_{s})\,\partial_{lm}g(c_{s})^{\mathrm{T}}\,\Sigma(c_{s})^{jk,lm}\,\mathrm{d}s$
(4.5)

where the process $\gamma$ is the conditional instantaneous covariance matrix
of noise defined in (A.5).

###  4.2 Central limit theorems

######  Theorem 1.

Assume assumptions A-$\nu$, A-$\gamma$, and suppose $g$ satisfies (3.8). We
select the tuning parameters according to (3.9), then we have the following
stable convergence in law of discretized process to a conditional continuous
Itô semimartingale on compact subsets of $\mathbb{R}^{+}$:

$\Delta_{n}^{-1/4}\Big{[}\widehat{S}(g)^{n}-S(g)\Big{]}\overset{\mathcal{L}-s(f)}{\longrightarrow}Z_{\Xi}(g)$
(4.6)

where $S(g)$ is defined in (1.1), $\widehat{S}(g)^{n}$ is from Definition 1,
$Z(g)$ is identified in Definition 3.

Theorem 1 is valid over the functional space (3.8), which is as general as the
state-of-the-art literature can achieve. If applications require functionals
whose derivatives satisfy a stronger condition (3.10) of polynomial growth, we
can put less restrictions on the tuning parameters.

######  Proposition 1.

Assume Assumptions A-$\nu$, A-$\gamma$. Replace the functional space (3.8)
with (3.10), replace the tuning conditions (3.9) on $k_{n}$, $\nu_{n}$ with
(3.11), then the functional stable convergence (4.6) still holds true.

However, Proposition 1 rules out functionals that involve matrix inversion, it
is not applicable to, for instance, estimation of precision matrices and
inference of linear regression models. In the rest of this paper, we focus on
the results over the general functional space (3.8).

We also establish a limit theorem for $\widetilde{S}(g)^{n}$ using positive
semidefinite plug-ins:

######  Theorem 2.

Assume assumptions A-$\nu$, A-$\gamma$, and suppose $g$ satisfies (3.8). We
select the tuning parameters according to (3.12), then we have the following
stable convergence in law of discretized process to a conditional continuous
Itô semimartingale on compact subsets of $\mathbb{R}^{+}$:

$\Delta_{n}^{-(1/4-\delta/2)}\Big{[}\widetilde{S}(g)^{n}-S(g)\Big{]}\overset{\mathcal{L}-s(f)}{\longrightarrow}Z_{\Sigma}(g)$
(4.7)

where $\delta$ is specified in (3.12), $Z_{\Sigma}(g)$ is identified in
Definition 3.

The foregoing asymptotic results are stated in terms of converging stochastic
processes, which is necessary to express functional stable convergence in law
(the strongest mode of convergence one can obtain in our setting). Below is a
point-wise formulation which is more relevant for statistical applications.
For finite $t$, and under the conditions of Theorem 1,

$n^{1/4}\left[\widehat{S}(g)^{n}_{t}-S(g)_{t}\right]\overset{\mathcal{L}-s}{\longrightarrow}\mathcal{MN}\big{(}0,\sqrt{t}\,V_{\Xi}(g)_{t}\big{)}$
(4.8)

Moreover, under the conditions of Theorem 2,

$n^{1/4-\delta/2}\left[\widetilde{S}(g)^{n}_{t}-S(g)_{t}\right]\overset{\mathcal{L}-s}{\longrightarrow}\mathcal{MN}\big{(}0,t^{1/2-\delta}\,V_{\Sigma}(g)_{t}\big{)}$
(4.9)

###  4.3 Confidence intervals

The asymptotic variances (4.4) and (4.5) are also functionals of volatility
matrix, so we can also estimate the asymptotic variances by the plug-in
framework, i.e. plugging in estimates of stochastic volatility matrices (and
estimates of instantaneous noise covariance matrices) to the corresponding
functional forms.

######  Definition 4.

For $g$ that satisfies (3.8), we define

$\widehat{V}_{\Xi}(g)^{n}_{t}\equiv
k_{n}\Delta_{n}\sum_{i=0}^{N^{n}_{t}-1}\sum_{j,k,l,m=1}^{d}\partial_{jk}g(\widehat{c}^{n}_{ik_{n}})\,\partial_{lm}g(\widehat{c}^{n}_{ik_{n}})^{\mathrm{T}}\;\Xi(\widehat{c}^{n}_{ik_{n}},\widehat{\gamma}^{n}_{ik_{n}})^{jk,lm}$
(4.10)

as the estimator for $V_{\Xi}(g)_{t}$ for which we abide the tuning (3.9).

Furthermore, we define

$\widetilde{V}_{\Sigma}(g)^{n}_{t}\equiv
k_{n}\Delta_{n}\sum_{i=0}^{N^{n}_{t}-1}\sum_{j,k,l,m=1}^{d}\partial_{jk}g(\widetilde{c}^{n}_{ik_{n}})\,\partial_{lm}g(\widetilde{c}^{n}_{ik_{n}})^{\mathrm{T}}\;\Sigma(\widetilde{c}^{n}_{ik_{n}})^{jk,lm}$
(4.11)

as the estimator for $V_{\Sigma}(g)_{t}$ for which we abide the tuning (3.12).

We have that both $\widehat{V}_{\Xi}(g)^{n}$ and
$\widetilde{V}_{\Sigma}(g)^{n}$ are consistent:

######  Proposition 2.

Suppose Assumptions A-$\nu$, A-$\gamma$, $g$ satisfies (3.8) and $t$ is
finite.

With the tuning (3.9), we have

$\big{\|}\widehat{V}_{\Xi}(g)^{n}_{t}-V_{\Xi}(g)_{t}\big{\|}=O_{p}(\Delta_{n}^{\kappa-1/2})$

With the tuning (3.12), we have

$\big{\|}\widetilde{V}_{\Sigma}(g)^{n}_{t}-V_{\Sigma}(g)_{t}\big{\|}=O_{p}(\Delta_{n}^{\kappa-1/2-\delta})$

###### Proof.

The asymptotic variance $V_{\Xi}(g)_{t}$ is a smooth functional of volatility
matrix and instantaneous noise covariance matrix. Its consistency follows from
the consistence of the volatility matrix estimator (3.3) and the instantaneous
noise covariance matrix estimator (3.6). According to Lemma 5 in Appendix C.2
and (C.28) in Appendix C.4, the error rate of $\widehat{V}_{\Xi}(g)^{n}_{t}$
is determined by the estimation error of (3.3). Therefore, the error rate of
$\widehat{V}_{\Xi}(g)^{n}_{t}$ is the same as the error rate of the volatility
functional estimator without bias correction, which is
$(k_{n}\Delta_{n}^{1/2})^{-1}$, then the proposition follows from (3.9).

The consistency of $\widetilde{V}_{\Sigma}(g)^{n}_{t}$ is established by a
similar argument. ∎

Based on Theorem 1, Theorem 2, Proposition 2 and the property of stable
convergence, we have the following feasible central limit theorems:

######  Corollary 1.

Suppose $g$ satisfies (3.8) and Assumptions A-$\nu$, A-$\gamma$ hold. With the
tuning (3.9), we have

$\big{[}\Delta_{n}^{1/2}\,\widehat{V}_{\Xi}(g)^{n}_{t}\big{]}^{-1/2}\left[\widehat{S}(g)^{n}_{t}-S(g)_{t}\right]\overset{\mathcal{L}}{\longrightarrow}\mathcal{N}\big{(}0,\mathbf{I}_{r}\big{)}$
(4.12)

in restriction to the event
$\\{\omega\in\Omega,\widehat{V}_{\Xi}(g)^{n}_{t}\text{ is positive
definite}\\}$, where $\Omega$ is defined in (A.4).

With the tuning (3.12), we have

$\big{[}\Delta_{n}^{1/2-\delta}\,\widetilde{V}_{\Sigma}(g)^{n}_{t}\big{]}^{-1/2}\left[\widetilde{S}(g)^{n}_{t}-S(g)_{t}\right]\overset{\mathcal{L}}{\longrightarrow}\mathcal{N}\big{(}0,\mathbf{I}_{r}\big{)}$
(4.13)

in restriction to the event
$\\{\omega\in\Omega,\widetilde{V}_{\Sigma}(g)^{n}_{t}\text{ is positive
definite}\\}$.

##  5 Applications

###  5.1 PCA of high-frequency data

PCA is one of powerful statistical methods that help us obtain information and
gain insights from large and complex datasets. It reduces dimensionality,
facilitates data visualization, reveals common trends et cetera. For high-
frequency data, PCA can be done by inference of spectral structure of
stochastic volatility matrix

$c_{t}=\left[q_{t}^{1},\cdots,q_{t}^{d}\right]\left[\begin{array}[]{ccc}\lambda_{t}^{1}&&\\\
&\ddots&\\\
&&\lambda_{t}^{d}\end{array}\right]\left[\begin{array}[]{c}q_{t}^{1,\mathrm{T}}\\\
\vdots\\\ q_{t}^{d,\mathrm{T}}\end{array}\right]$ (5.1)

The eigenvalues and eigenvectors can be considered as functionals of the
associated stochastic volatility matrix once directions and lengths of the
eigenvectors are specified and some regularity conditions hold. As per this
functional idea we write
$\lambda^{k}(c_{t})=\lambda^{k}_{t},\;q^{k}(c_{t})=q^{k}_{t}$. Based on this
insight, [4] applied PCA to nonstationary financial data by conducting
inference on the realized eigenvalue
$\int_{0}^{t}\lambda_{s}^{k}\,\mathrm{d}s$, realized eigenvector
$\int_{0}^{t}q_{s}^{k}\,\mathrm{d}s$, realized principal component
$\int_{0}^{t}q_{s-}^{k}\,\mathrm{d}X_{s}$. More recently, [27] extends the
realized PCA to asynchronously observed high-frequency noisy data. Here we
extend [4] to noisy data, and improve the convergence rate of [27] by a
considerable margin.

Suppose for $t\in[0,T]$,

$\lambda_{t}^{r_{0}+1}=\cdots=\lambda_{t}^{r_{1}}>\lambda_{t}^{r_{1}+1}=\cdots=\lambda_{t}^{r_{2}}>\cdots\cdots>\lambda_{t}^{r_{K-1}+1}=\cdots=\lambda_{t}^{r_{K}}$
(5.2)

where $r_{0}=0,\,r_{K}=d$. Let $\mathcal{K}_{k}=\\{r_{k-1}+1,\cdots,r_{k}\\}$
for $k=1,\cdots,K$. Define the following functions of volatility matrix

$\lambda(c_{t})=\Big{(}\frac{1}{r_{1}-r_{0}}\sum_{r\in\mathcal{K}_{1}}\lambda_{t}^{r},\cdots,\frac{1}{r_{K}-r_{K-1}}\sum_{r\in\mathcal{K}_{K}}\lambda_{t}^{r}\Big{)}^{\mathrm{T}}$

If $r_{k}=r_{k-1}+1$, $\lambda^{k}_{t}$ is called a simple eigenvalue,
otherwise a repeated eigenvalue. If $\lambda^{k}_{t}$ is a repeated
eigenvalue, it is not differentiable with respect to $c_{t}$, and establishing
its asymptotic distribution would be problematic. Nevertheless, $\lambda$
defined above is a differentiable function regardless of singleton or
multicity (see Remark 3 in Appendix E), so we can do nonlinear bias correction
and derive its CLT.

To do PCA, we need to estimate

$\displaystyle S(\lambda)_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\lambda(c_{s})\,\mathrm{d}s$ $\displaystyle
S(q^{k})_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}q^{k}(c_{s})\,\mathrm{d}s,\hskip
11.38109ptk=1\cdots,d$

Given $\widetilde{c}^{n}_{i}$ defined in (3.4), we compute its spectral
factorization:

$\widetilde{c}^{n}_{i}=\left[\widetilde{q}^{1,n}_{i},\cdots,\widetilde{q}^{d,n}_{i}\right]\left[\begin{array}[]{ccc}\widetilde{\lambda}^{1,n}_{i}&&\\\
&\ddots&\\\
&&\widetilde{\lambda}^{d,n}_{i}\end{array}\right]\left[\widetilde{q}^{1,n}_{i},\cdots,\widetilde{q}^{d,n}_{i}\right]^{\mathrm{T}}$

Define $\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\lambda}^{k,n}_{i}=(r_{k}-r_{k-1})^{-1}\sum_{r\in\mathcal{K}_{k}}\widetilde{\lambda}^{r,n}_{i}$,
our estimator of $S(\lambda)_{t}$ is

$\displaystyle\widetilde{S}(\lambda)^{n}_{t}$ $\displaystyle=$
$\displaystyle\big{[}\widetilde{S}(\lambda)^{1,n}_{t},\cdots,\widetilde{S}(\lambda)^{K,n}_{t}\big{]}^{\mathrm{T}}$
$\displaystyle\widetilde{S}(\lambda)^{k,n}_{t}$ $\displaystyle\equiv$
$\displaystyle
k_{n}\Delta_{n}\sum_{h=0}^{N^{n}_{t}-1}\left[1-\frac{2\theta\Phi_{00}}{\phi_{0}(0)^{2}\,k_{n}\,\Delta_{n}^{1/2+\delta}}\sum_{v\notin\mathcal{K}_{k}}\frac{\widetilde{\lambda}^{v,n}_{hk_{n}}}{\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\lambda}^{k,n}_{hk_{n}}-\widetilde{\lambda}^{v,n}_{hk_{n}}}\right]\accentset{{\cc@style\underline{\mskip
10.0mu}}}{\lambda}^{k,n}_{hk_{n}}$ (5.3)

where $\theta$, $\Phi_{00}$, $\phi_{0}(0)$ are defined in (3.9), (4.1). Note
that $\widetilde{S}(\lambda)^{k,n}_{t}$ is a Riemann sum of simple eigenvalues
or averages of repeated eigenvalues, with a multiplicative bias-correction
factor. We have an expression for this multiplicative factor in terms of
eigenvalue estimates. Since we do not have to evaluate Hessian of the
functional, (5.1) is very advantageous for applications.

This advantage is also carried over to eigenvectors. We derive a closed-form
expression for the nonlinear bias correction to the eigenvectors and define
the estimator of $S(q^{k})_{t}$ to be

$\widetilde{S}(q^{k})^{n}_{t}\equiv
k_{n}\Delta_{n}\sum_{h=0}^{N^{n}_{t}-1}\left[1+\frac{\theta\Phi_{00}}{\phi_{0}(0)^{2}\,k_{n}\,\Delta_{n}^{1/2+\delta}}\sum_{v\neq
k}\frac{\widetilde{\lambda}^{k,n}_{hk_{n}}\widetilde{\lambda}^{v,n}_{hk_{n}}}{\big{(}\widetilde{\lambda}^{k,n}_{hk_{n}}-\widetilde{\lambda}^{v,n}_{hk_{n}}\big{)}^{2}}\right]\widetilde{q}^{k,n}_{hk_{n}}$
(5.4)

######  Proposition 3.

Assume Assumption A-$\nu$, A-$\gamma$, and suppose (5.2). We select the
bandwidths $l_{n}$, $k_{n}$ and the truncation threshold $\nu_{n}$ according
to (3.12). We have the following functional stable convergence in law:

$\Delta_{n}^{-1/4+\delta/2}\left[\widetilde{S}(\lambda)^{n}-S(\lambda)\right]\overset{\mathcal{L}-s(f)}{\longrightarrow}Z(\lambda)$

where $Z(\lambda)$ is a process defined on an extension of the space
$\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P}\right)$, such
that conditioning on $\mathcal{F}$ it is a continuous centered Gaussian
martingale with variance

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{E}[Z(\lambda)Z(\lambda)^{\mathrm{T}}|\mathcal{F}]=V(\lambda)$

where $\accentset{{\cc@style\underline{\mskip 10.0mu}}}{E}$ is the conditional
expectation operator on the extended probability space and

$V(\lambda)_{t}=\frac{4\theta\Phi_{00}}{\phi_{0}(0)^{2}}\left[\begin{array}[]{ccc}(r_{1}-r_{0})^{-1}\int_{0}^{t}(\lambda^{r_{1}}_{s})^{2}\,\mathrm{d}s&&\\\
&\ddots&\\\
&&(r_{K}-r_{K-1})^{-1}\int_{0}^{t}(\lambda^{r_{K}})^{2}\,\mathrm{d}s\end{array}\right]$
(5.5)

If $\lambda^{k}_{t}$ is a simple eigenvalue, the corresponding eigenvector
$q^{k}_{t}$ is differentiable with respect to $c_{t}$ (see Lemma 10 in
Appendix E). In this case we can derive an inferential theory for the
integrated eigenvector estimator (5.4).

######  Proposition 4.

Assume Assumption A-$\nu$, A-$\gamma$, and suppose $\lambda^{k}_{t}$ is a
simple eigenvalue of $c_{t},\,\forall t\in[0,T]$. We control the bandwidths
$l_{n}$, $k_{n}$ and the truncation threshold $\nu_{n}$ according to (3.12).
We have the following functional stable convergence in law:

$\Delta_{n}^{-1/4+\delta/2}\left[\widetilde{S}(q^{k})^{n}-S(q^{k})\right]\overset{\mathcal{L}-s(f)}{\longrightarrow}Z(q^{k})$

where $Z(q^{k})$ is a process defined on an extension of the space
$\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P}\right)$, such
that conditioning on $\mathcal{F}$ it is a continuous centered Gaussian
martingale with variance

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{E}[Z(q^{k})Z(q^{k})^{\mathrm{T}}|\mathcal{F}]=V(q^{k})$

where $\accentset{{\cc@style\underline{\mskip 10.0mu}}}{E}$ is the conditional
expectation operator on the extended probability space and

$V(q^{k})_{t}=\frac{2\theta\Phi_{00}}{\phi_{0}(0)^{2}}\int_{0}^{t}\sum_{v\neq
k}\frac{\lambda^{k}_{s}\lambda^{v}_{s}}{(\lambda^{k}_{s}-\lambda^{v}_{s})^{2}}\,q^{v}_{s}\,q^{v,\mathrm{T}}_{s}\,\mathrm{d}s$
(5.6)

######  Remark 2.

Our asymptotic theory for PCA is based on the general result of
$\widetilde{S}(g)^{n}_{t}$. We compare $\widehat{S}(g)^{n}_{t}$ and
$\widetilde{S}(g)^{n}_{t}$ below in view of the asymptotics and practicality
of PCA.

  * •

Convergence rates: $\widehat{S}(g)^{n}_{t}$ attains the optimal convergence
rate $n^{1/4}$; the convergence rate of $\widetilde{S}(g)^{n}_{t}$ is
$n^{1/4-\delta/2}$ which is less than $n^{1/5}$ according to (3.12).

  * •

Dealing with negative eigenvalue estimate(s): If $\widehat{c}^{n}_{i}$ is not
positive semidefinite, we can project it to $\mathcal{S}^{d}_{+}$. This is
equivalent to setting negative eigenvalue estimate(s) to zero.
$\widetilde{S}(g)^{n}_{t}$ is positive semidefinite by definition and does not
suffer this problem.

  * •

Bias correction and uncertainty quantification: For $\widehat{S}(g)^{n}_{t}$
we need to evaluate $\partial_{jk}g$ and $\partial^{2}_{jk,lm}g$ for
$j,k,l,m=1,\cdots,d$ in order to compute bias correction
$\widehat{B}(g)^{n}_{i}$ and asymptotic variance
$\widehat{V}_{\Xi}(g)^{n}_{t}$; this can be quite difficult in applications
and motivated [5] to propose the multiscale jackknife method. For
$\widetilde{S}(g)^{n}_{t}$, we can use estimates
$\widetilde{\lambda}^{k,n}_{i}$ and $\widetilde{q}^{k,n}_{i}$ to compute
$\widetilde{B}(g)^{n}_{i}$ and $\widetilde{V}_{\Sigma}(g)^{n}_{t}$. Evident
from (5.1), (5.4), (5.5), (5.6), they are straightforward to compute without
much computational or memory overhead.

We apply Proposition 3 and Proposition 4 to conduct PCA on TAQ data in Section
6. We choose PCA based on $\widetilde{S}(g)^{n}_{t}$ because it scales better
for the large TAQ data.

###  5.2 More examples

######  Example 2 (Uncertainty quantification).

Asymptotic variances are often functionals of volatility. In the univariate
setting, the so-called “quarticity” defined as
$\int_{0}^{t}c_{s}^{2}\,\mathrm{d}s$ appears in the asymptotic variances of
many estimators of integrated volatility. The multivariate counterpart
involves $\int_{0}^{t}c_{s}^{jl}c_{s}^{km}+c_{s}^{jm}c_{s}^{kl}\,\mathrm{d}s$
in [21] and $\Xi(c_{s},\gamma_{s})$ in (4.4). For this reason, the volatility
functional estimator effectively facilitates uncertainty quantification for
various volatility-related estimators.

######  Example 3 (Laplace transform).

[1] put forward an estimator of the realized Laplace transform of volatility
defined as

$\int_{0}^{t}e^{iwc_{s}}\,\mathrm{d}s$

This transform can be viewed as the characteristic function of volatility
under the occupation measure. By matching the the moments of realized Laplace
transform with those induced by a model, we can estimate model parameter(s) or
test the model. An open question noted by [1] is the estimation of realized
Laplace transform using noisy data. By the nonparametric estimation of
volatility path in the first stage and the Riemann summation of bias-corrected
functional plug-ins in the second stage, this paper presents a rate-optimal
solution to the open question.

######  Example 4 (Generalized method of moments (GMM)).

[2] proposed the generalized method of integrated moments for financial high-
frequency data. In estimating an option pricing model, one observes the
process $Z_{t}=(t,X_{t},r_{t},d_{t})$ where $X_{t}$ is the price of the
underlying observed without any noise, $r_{t}$ is the short-term interest
rate, $d_{t}$ is the dividend yield. One model of the arbitrage-free option
price under the risk-neutral probability measure is

$\beta_{t}=f(Z_{t},c_{t};\theta^{*})$

where $f$ is deterministic, $\theta^{*}$ is the true model parameter. The
observed option price is often modeled as

$O_{i\Delta_{n}}=\beta_{i\Delta_{n}}+\epsilon_{i}$

where $\epsilon_{i}$ is pricing error and $\mathbb{E}(\epsilon_{i})=0$. Let
$g(Z_{t},c_{t};\theta)=\mathbb{E}[O_{t}-f(Z_{t},c_{t};\theta)]$, then we have
the following integrated moment condition:

$\int_{0}^{t}g(Z_{s},c_{s};\theta^{*})\,\,\mathrm{d}s=0$

Utilizing noisy observations at higher frequencies, this paper provides a
means to compute a bias-corrected sample moment function of GMM.

######  Example 5 (Linear regression).

In the practice of linear factor models and financial hedging, one faces the
tasks of computing the factor loadings and the hedge ratios. These tasks can
be formulated as the estimation of the coefficient $\beta$ in the time-series
linear regression model

$Z_{t}^{c}=\beta^{\mathrm{T}}S_{t}^{c}+R_{t}$

where

$\left\\{\begin{array}[]{cl}S_{t}=&S_{0}+\int_{0}^{t}b^{S}_{u}\,\mathrm{d}u+\int_{0}^{t}\sigma^{S}_{u}\,\mathrm{d}W^{S}_{u}+J^{S}_{t}\\\
Z_{t}=&Z_{0}+\int_{0}^{t}b^{Z}_{u}\,\mathrm{d}u+\beta^{\mathrm{T}}\int_{0}^{t}\sigma^{S}_{u}\,\mathrm{d}W^{S}_{u}+\int_{0}^{t}\sigma^{R}_{u}\,\mathrm{d}W^{R}_{u}+J^{Z}_{t}\end{array}\right.$

$\langle W^{S},W^{R}\rangle=0$, $S_{t}\in\mathbb{R}^{d-1}$,
$Z_{t}\in\mathbb{R}$, and $S^{c}$, $Z^{c}$ are the continuous parts of the Itô
semimartingales.

Let $X=(S^{\mathrm{T}},Z)^{\mathrm{T}}$, we can write
$X_{t}=X_{0}+\int_{0}^{t}b_{u}\,\mathrm{d}u+\int_{0}^{t}\sigma_{u}\,\mathrm{d}W_{u}+J_{t}$
where $b=(b^{S,T},b^{Z})^{\mathrm{T}}$,
$W=(W^{S,\mathrm{T}},W^{R})^{\mathrm{T}}$,
$J=(J^{S,\mathrm{T}},J^{Z})^{\mathrm{T}}$ and

$\sigma=\left[\begin{array}[]{cc}\sigma^{S}&0\\\
\beta^{\mathrm{T}}\sigma^{S}&\sigma^{R}\end{array}\right]$

so

$c=\sigma\sigma^{\mathrm{T}}=\left[\begin{array}[]{cc}\sigma^{S}\sigma^{S,\mathrm{T}}&\sigma^{S}\sigma^{S,\mathrm{T}}\beta\\\
\beta^{\mathrm{T}}\sigma^{S}\sigma^{S,\mathrm{T}}&\beta^{\mathrm{T}}\sigma^{S}\sigma^{S,\mathrm{T}}\beta+(\sigma^{R})^{2}\end{array}\right]\coloneqq\left[\begin{array}[]{cc}c^{SS}&c^{SZ}\\\
c^{ZS}&c^{ZZ}\end{array}\right]$

hence by letting $g(c)=c^{SS,-1}c^{SZ}$, we have $\beta=t^{-1}S(g)_{t}$. [3]
proposed a method under the no-noise assumption. When the observations contain
noise, the methodology of this paper can extend the method of [3] to wider
applicability.

##  6 Empirical analysis of TAQ data

We analyze the transaction data of S&P 100 constituents. Each stock is sampled
every second during 09:35 - 15:55 EST, so we have 22,800 observations for each
stock on each business day. We focus on data from 09/10/2003 to 12/31/2019.
During this period there were 4106 business days in 852 weeks. By focusing on
the time window 09:35 - 15:55 EST, we remove overnight returns which contain
big jumps and returns incurred in the first and last five minutes of trading
hours which are relatively volatile. 10 illiquid assets are excluded from our
statistical analysis.

Instantaneous volatility matrix and its eigenvalues and eigenvectors were
estimated for every business day. The realized eigenvalues and realized
eigenvectors were computed weekly by aggregating the instantaneous eigenvalues
and instantaneous eigenvectors.

The temporal evolution of four leading eigenvalue estimates along with
confidence intervals are plotted in Figure 5. It shows
$\lambda_{q}/\sum_{j}\lambda_{j}$’s which are the proportions of total
variation that can be explained by the corresponding principal components. The
first eigenvalues are conspicuously separated from the others and indicate the
first principal component explains more than 60% of the cross-section
variation for the majority of time. Moreover, it is evident from this dataset
that the first four leading eigenvalues were simple rather than repeated. The
estimates of eigenvalues tell us that a low-dimensional model can largely
explain the systematic comovement of these stocks.

Figure 5: Weekly integrated eigenvalues and the confidence intervals

It is reasonable to interpret the first principal component as a market
indicator on how the economy performs in general. As all the major
corporations and their stock values are moving with the overall economy, the
eigenvector corresponding to the first principal component is expected to
comprise only positive entries.

An interesting question is how the first principal component is compared with
S&P 100 index. [27] calculated the accumulated returns from 2007 to 2017 of
the portfolio based on the eigenvector $q^{1}$ associated with the first
principal component. [27] normalized the eigenvector so that
$\sum_{j}^{d}q^{1,j}=1$ and used $q^{1,j}$’s as portfolio weights so that the
resultant portfolio is self-financing. A curious finding of [27] is that this
portfolio outperformed the market index hence beat the passive investment.
Here the author would like to test this portfolio using pre-averaging-based
PCA, over a longer time span.

The principal components are
$\int_{0}^{t}q^{r}(c_{s-})\,\mathrm{d}X_{s},\,r=1,2,\cdots$, where
$X_{t}^{j}=\log(S_{t}^{j})$ and $S_{t}$ is the vector of asset prices at time
$t$. However, one obviously can not transact in logarithmic currency, so a
pending question is how to find a portfolio that is analogous to the first
principal component and requires trading on the currency’s original scale.

[27] proposed the following construction. Let

$R_{t,\Delta}=\big{[}(S^{1}_{t+\Delta}-S^{1}_{t})/S^{1}_{t},\cdots,(S^{d}_{t+\Delta}-S^{d}_{t})/S^{d}_{t}\big{]}^{\mathrm{T}}$

be the vector of returns in percentage. When $\|R_{t,\Delta}\|$ is small,

$\log\big{(}1+q^{1,\mathrm{T}}R_{t,\Delta}\big{)}\approx
q^{1,\mathrm{T}}R_{t,\Delta},$

and

$q^{1,j}\frac{S^{j}_{t+\Delta}-S^{j}_{t}}{S^{j}_{t}}\approx
q^{1,j}\log\Big{(}1+\frac{S^{j}_{t+\Delta}-S^{j}_{t}}{S^{j}_{t}}\Big{)}=q^{1,j}\log\Big{(}\frac{S^{j}_{t+\Delta}}{S^{j}_{t}}\Big{)},$

so

$q^{1,\mathrm{T}}(X_{t+\Delta}-X_{t})\approx\log\big{(}1+q^{1,\mathrm{T}}R_{t,\Delta}\big{)}.$

Therefore, we consider $\sum_{t}\log(1+q^{1,\mathrm{T}}R_{t,\Delta})$ as an
approximation to the first realized principal component. It is also the
accumulated log return if there is no transaction cost and
$\sum_{j}^{d}q_{1,j}=1$. In this case,
$\exp\big{\\{}\sum_{t}\log(1+q^{1,\mathrm{T}}R_{t,\Delta})\big{\\}}-1=\prod_{t}(1+q^{1,\mathrm{T}}R_{t,\Delta})-1$
is the accumulated return of this self-financing portfolio in the absence of
transaction cost.

Since the volatility matrix and the eigenvector $q^{1}$ are updated on every
business day, the portfolio can be re-weighted daily. One caveat is that the
numerical values of eigenvector estimates are relative less stable. Roughly
speaking, the eigenvector viewed as a function of the corresponding positive
semidefinite matrix is less smooth than the eigenvalue. Due to this reason,
heuristically, an error in the covariance matrix estimation tends to result in
a larger error in the eigenvector estimate, as compared to the eigenvalue
counterpart. To stabilize eigenvector estimates, it is advisable to compute
moving averages of the initial eigenvector estimates over a window of six
business days. As a result we re-weight the portfolio on all the business days
from 09/17/2019 to 12/31/2019.

Before the portfolio re-allocation according to the new weights on the next
business day, one can either clear positions at the end of trading hours or
let positions stand overnight. In the second scenario, the portfolio bears the
overnight risk premium and absorbs overnight returns.

The accumulated log returns of the portfolio that mimics the first principal
component are plotted in Figure 6. The portfolio returns were computed using
daily prices of S&P 100 constituents which can be obtained from the Compustat
database. As we can see, return morphologies of the portfolio that mimics the
first principal component resemble that of S&P 100 index. If the portfolio was
cleared at the end of trading hours, it slightly underperformed the index.
Interestingly enough, if the portfolio that stood overnight, it manifestly
outperformed the index. It is evident that this portfolio and the factor
corresponding to the first principal component earned a significant risk
premium during the overnight periods, and implies a difference in the intraday
and overnight risk-reward patterns. This finding is consistent with the
empirical analysis of [27], though we use a method that is more accurate and
analyze data over a longer time span.

Figure 6: Accumulated log returns of the portfolio based on the first
principal component

##  7 Conclusion

In this paper, we proposed statistical methods for general nonlinear
functionals of volatility matrix, and developed statistical theory to guide
bias correction, quantify statistical uncertainty and perform statistical
inference. Because the methods can utilize data that contains microstructure
noise and jumps, and because volatility matrix functionals form a unifying
framework that accommodates many applications, this paper provides a viable
answer to the open problem of noise and extends numerous statistical methods
to much wider applicability. This brings significant statistical gains, such
as principled approaches to take advantage of much bigger datasets and
improved statistical accuracy.

The tuning choice is an essential component in the inference of volatility
matrix functionals, and it has been the main motivation of several recent
works under the no-noise assumption. The importance of tuning must not be
neglected neither in the noisy setting. We derived tuning ranges for pre-
averaging and jump truncation. We would like to stress that the tuning
parameters need to be quite different from those under the no-noise
assumption.

Asymptotic analysis shows there are non-negligible higher order biases beyond
linear terms. We can effectively debias our estimation by quadratic
approximation. Given the tuning choice and bias correction, we established
consistency, convergence rates, and asymptotic normality by functional stable
convergence of processes. We derived stable CLT for both the rate-optimal
method and the method that uses positive semidefinite plug-ins. The asymptotic
covariance matrices can be consistently estimated, which paves the way for
feasible statistical inference.

Based on the general results, we then put forward the method and theory of PCA
to analyze noisy high-frequency financial data. Particularly we gave a
demonstration on the TAQ dataset and analyzed the temporal structures of
eigenvalues and the first principal component of S&P 100 stocks. Our PCA
method is not only statistically sound in the presence of noise and jump, but
also computationally practical for large datasets. PCA is one example among
many possible applications of this paper. We demonstrate some success in
pushing the inferential framework for general functionals into a new frontier
of potentials and possibilities, and lending the effort to enable applications
to employ noisy high-frequency data where exciting new discoveries await.

## Acknowledgements

The author is indebted to Jean Jacod and Per A. Mykland for encouragement,
detailed discussion and comments. This work also benefits from discussions
with Yoann Potiron, Viktor Todorov, Dacheng Xiu.
The author was supported by NSF grant DMS 17-13129 and 2019 Stevanovich
Fellowship from the University of Chicago.

##  Appendix A Assumptions

We present formally the model specification and assumptions in this appendix.

The pure jump process $J$ in $X$ is modeled by

$J_{t}=\int_{(0,t]\times
E}\delta(s,x)\,\mathfrak{p}(\mathrm{d}s,\,\mathrm{d}x)$ (A.1)

where $\delta$ is a $\mathbb{R}^{d}$-valued predictable function on
$\mathbb{R}^{+}\times E$, $E$ is a Polish space, $\mathfrak{p}$ is a Poisson
random measure with compensator
$\mathfrak{q}(\,\mathrm{d}u,\,\mathrm{d}x)=\,\mathrm{d}u\otimes\lambda(\,\mathrm{d}x)$,
$\lambda$ is a $\sigma$-finite measure on $E$ and has no atom.

The stochastic volatility matrix is positive semidefinite almost surely and of
the form

$c_{t}=c_{0}+\int_{0}^{t}b^{(c)}_{s}\,\mathrm{d}s+\int_{0}^{t}\sigma^{(c)}_{s}\,\mathrm{d}W_{s}+J^{(c)}_{t}$
(A.2)

where $b^{(c)}\in\mathbb{R}^{d\times d}$, $\sigma^{(c)}\in\mathbb{R}^{d\times
d\times d^{\prime}}$, $b^{(c)}$ and $\sigma^{(c)}$ are adapted and càdlàg;
$J^{(c)}$ is the volatility jump, which is modeled by

$J^{(c)}_{t}=\delta^{(c)}(s,x)\mathds{1}_{\\{\|\delta^{(c)}(s,x)\|\leq
1\\}}\,(\mathfrak{p}-\mathfrak{q})(\mathrm{d}s,\,\mathrm{d}x)\\\
+\int_{(0,t]\times
E}\delta^{(c)}(s,x)\mathds{1}_{\\{\|\delta^{(c)}(s,x)\|>1\\}}\,\mathfrak{p}(\mathrm{d}s,\,\mathrm{d}x)$
(A.3)

where $\delta^{(c)}$ is a $\mathbb{R}^{d\times d}$-valued predictable function
on $\mathbb{R}^{+}\times E$.

To guarantee that $c_{t}$ is positive semidefinite, one needs to impose
additional parametric or semiparametric restrictions in concrete applications.
The results in this paper hold true for any process $c$ that is positive
semidefinite and satisfies (A.2).

The regularization on drift, diffusion coefficient and jump activity is given
below.

######  Assumption A-$\nu$ (regularity).

The drift process $b$ has $\frac{1}{2}$-Hölder sample path in the mean-square
sense, i.e., $\forall t,s\geq 0$,

$E\Big{(}\sup_{u\in[0,s]}\|b_{t+u}-b_{t}\|\big{|}\mathcal{F}^{(0)}_{t}\Big{)}\leq
Ks^{1/2},\,a.s.;$

$c$ is of the form (A.2), there is a sequence of triples
$(\tau_{m},\mathcal{S}_{m},\Gamma_{m})$, where $\tau_{m}$ is a stopping time
and $\tau_{m}\nearrow\infty$; $\mathcal{S}_{m}\subset\mathcal{S}^{+}_{d}$ is
convex, compact such that

$t\in[0,\tau_{m}]\Rightarrow c_{t}\in\mathcal{S}_{m};$

$\Gamma_{m}$ is a sequence of bounded $\lambda$-integrable functions on $E$,
such that

$t\in[0,\tau_{m}]\Longrightarrow\left\\{\begin{array}[]{l}\|b_{t}\|+\|\sigma_{t}\|+\|b^{(c)}_{t}\|+\|\sigma^{(c)}_{t}\|\leq
m\\\ \|\delta(t,x)\|^{\nu}\wedge 1\leq\Gamma_{m}(x),\,\,\nu\in[0,1)\\\
\|\delta^{(c)}(t,x)\|^{2}\wedge 1\leq\Gamma_{m}(x)\end{array}\right.$

Let
$\big{(}\Omega^{(0)},\mathcal{F}^{(0)},(\mathcal{F}^{(0)}_{t}),\mathbb{P}^{(0)}\big{)}$
be a filtered probability space with respect to which $X$ and $c$ are
$(\mathcal{F}^{(0)}_{t})$-adapted; let
$\big{(}\Omega^{(1)},\mathcal{F}^{(1)},(\mathcal{F}^{(1)}_{t}),\mathbb{P}^{(1)}\big{)}$
be another filtered probability space accommodating $Y$; $\forall t\geq 0$,
$\forall A\in\mathcal{F}^{(0)}$, let $Q_{t}(A,\cdot)$ be a conditional
probability measure on $\left(\Omega^{(1)},\mathcal{F}^{(1)}\right)$.

All the stochastic dynamics above can be described on the filtered extension
$\left(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P}\right)$, where

$\left\\{\begin{array}[]{l}\Omega=\Omega^{(0)}\times\Omega^{(1)}\\\
\mathcal{F}=\mathcal{F}^{(0)}\otimes\mathcal{F}^{(1)}\\\
\mathcal{F}_{t}=\bigcap_{s>t}\big{(}\mathcal{F}^{(0)}_{s}\otimes\mathcal{F}_{s}^{(1)}\big{)},\hskip
14.22636pt\widetilde{\mathcal{F}}_{t}=\bigcap_{s>t}\big{(}\mathcal{F}^{(0)}\otimes\mathcal{F}_{s}^{(1)}\big{)}\\\
\mathbb{P}\left(A\times\mathrm{d}\omega\right)=\mathbb{P}^{(0)}(A)\cdot\otimes_{t\geq
0}Q_{t}(A,\,\mathrm{d}\omega),\,\forall
A\in\mathcal{F}^{(0)}\end{array}\right.$ (A.4)

Let

$\epsilon_{t}(\omega)=Y_{t}(\omega)-X_{t}$

and the conditional covariance process of noise can be written as

$\gamma_{t}=\int_{\Omega^{(1)}}\epsilon_{t}(\omega)\,\epsilon_{t}(\omega)^{\mathrm{T}}\,Q_{t}(\cdot,\mathrm{d}\omega)$
(A.5)

######  Assumption A-$\gamma$ (noise).

$\forall t\in\mathbb{R}^{+}$,

$\int_{\Omega^{(1)}}\epsilon_{t}(\omega)\,Q_{t}(\cdot,\,\mathrm{d}\omega)=0$

$\forall t\neq s$, $\forall A\in\mathcal{F}^{(0)}_{s\wedge t}$

$\int_{\Omega^{(1)}\times\Omega^{(1)}}\epsilon_{t}(\omega)\,\epsilon_{s}(\omega)^{\mathrm{T}}\,Q_{t}(A,\,\mathrm{d}\omega)\,Q_{s}(A,\,\mathrm{d}\omega)=0$

furthermore,

$\gamma_{t}=\gamma_{0}+\int_{0}^{t}b^{(r)}_{s}\,\mathrm{d}s+\int_{0}^{t}\sigma^{(r)}_{s}\,\mathrm{d}W_{s}+\int_{(0,t]\times
E}\delta^{(r)}(s,x)\,\mathfrak{p}(\mathrm{d}s,\,\mathrm{d}x)$

for the same $\tau_{m}$, $\Gamma_{m}$ in assumption A-$\nu$ and $\forall q\leq
8$

$t\in[0,\tau_{m}]\Longrightarrow\left\\{\begin{array}[]{l}\|b^{(r)}_{t}\|+\|\sigma^{(r)}_{t}\|+\mathbb{E}_{\Omega^{(1)}}\|\epsilon_{t}(\omega)\|^{q}\leq
m\\\ \|\delta^{(r)}(t,x)\|^{2}\wedge 1\leq\Gamma_{m}(x)\end{array}\right.$

##  Appendix B Theoretical preliminaries for proofs

We use the notation and model described in Section 2 and Appendix A in the
main article. In the sequel, the constant $K$ changes across lines but remains
finite; $K_{q}$ is a constant depending on $q$; $a_{n}\asymp b_{n}$ means both
$a_{n}/b_{n}$ and $b_{n}/a_{n}$ are bounded for large $n$; $\mathbb{E}(\cdot)$
denotes the expectation operator on $(\Omega^{(0)},\mathcal{F}^{(0)})$ or
$(\Omega,\mathcal{F})$; $E(\cdot|\mathcal{H})$ denotes the conditional
expectation operator, with $\mathcal{H}$ being $\mathcal{F}^{(0)}_{t}$,
$\mathcal{F}^{(1)}_{t}$, $\mathcal{F}_{t}$, $\widetilde{\mathcal{F}}_{t}$;
$E^{n}_{i}(\cdot)$ denotes $E(\cdot|\mathcal{F}^{n}_{i})$. Six useful results
are stated below.

I. By a localization argument from section 4.4.1 in [33], without loss of
generality we can assume $\exists$ a constant K, a bounded
$\lambda$-integrable function $\Gamma$ on $E$, a convex compact subset
$\mathcal{S}\in\mathcal{S}^{+}_{d}$ and $\epsilon>0$,
$g\in\mathcal{C}^{3}(\mathcal{S}^{\epsilon})$ where $\mathcal{S}^{\epsilon}$
denotes the $\epsilon$-enlargement of $\mathcal{S}$ (see (3.8) in the main
article), such that

$\left\\{\begin{array}[]{l}\|b\|+\|\sigma\|+\|b^{(c)}\|+\|\sigma^{(c)}\|+\|b^{(r)}_{t}\|+\|\sigma^{(r)}_{t}\|+\mathbb{E}\|\epsilon_{t}\|^{q}\leq
K,\;\forall q\leq 8\\\ \|\delta(t,x)\|^{\nu}\wedge
1\leq\Gamma(x),\;\nu\in[0,1)\\\ \|\delta^{(c)}(t,x)\|^{2}\wedge
1+\|\delta^{(r)}(t,x)\|^{2}\wedge 1\leq\Gamma(x)\\\
c\in\mathcal{S}\end{array}\right.$ (B.1)

II. Define a continuous Itô semimartingale with parameters identical to those
of (2.1) in the main article

$X^{\prime}_{t}=X_{0}+\int_{0}^{t}b_{s}\,\mathrm{d}s+\int_{0}^{t}\sigma_{s}\,\mathrm{d}W_{s}$

Let $Y^{*}=Y-X+X^{\prime}$. Based on (3.2) in the main article, define

$\displaystyle\widehat{c}^{*n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}\left(\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i+h}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i+h}-\widehat{Y}^{*n}_{i+h}\right)$
$\displaystyle\widetilde{c}^{*n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i+h}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i+h}$

In the upcoming derivation, $\|\widehat{c}^{n}_{i}-\widehat{c}^{*n}_{i}\|$ is
tightly bounded provided $\nu_{n}$ is properly chosen. The focus then will be
shifted from $\widehat{c}^{n}_{i}$ to $\widehat{c}^{*n}_{i}$. Ditto for
$\widetilde{c}^{n}_{i}$ and $\widetilde{c}^{*n}_{i}$. The volatility
“estimators” calculated on continuous sample paths are more tractable.

III. By estimates of Itô semimartingale increments, $\forall$ finite stopping
time $\tau$,

$\left\\{\begin{array}[]{ll}\big{\|}E\big{(}X^{\prime}_{\tau+s}-X^{\prime}_{\tau}|\mathcal{F}^{(0)}_{\tau}\big{)}\big{\|}+\\\
\hskip
11.38109pt\big{\|}E\big{(}c_{\tau+s}-c_{\tau}|\mathcal{F}^{(0)}_{\tau}\big{)}\big{\|}+\big{\|}E\big{(}\gamma_{\tau+s}-\gamma_{\tau}|\mathcal{F}^{(0)}_{\tau}\big{)}\big{\|}&\leq
Ks\\\
E\left(\sup_{u\in[0,s]}\left\|X^{\prime}_{\tau+u}-X^{\prime}_{\tau}\right\|^{q}|\mathcal{F}^{(0)}_{\tau}\right)&\leq
Ks^{q/2}\\\
E\left(\sup_{u\in[0,s]}\left\|c_{\tau+u}-c_{\tau}\right\|^{q}+\left\|\gamma_{\tau+u}-\gamma_{\tau}\right\|^{q}|\mathcal{F}^{(0)}_{\tau}\right)&\leq
Ks^{(q/2)\wedge 1}\end{array}\right.$ (B.2)

by Lemma 2.1.7, Corollary 2.1.9 in [33],

$\left\\{\begin{array}[]{l}E\left(\sup_{u\in[0,s]}\|J_{\tau+u}-J_{\tau}\|^{q}|\mathcal{F}^{(0)}_{\tau}\right)\leq
K_{q}\,sE\big{[}\widehat{\delta}(q)_{\tau,s}|\mathcal{F}^{(0)}_{\tau}\big{]}\\\
E\left[\sup_{u\in[0,s\wedge
1]}\left(\frac{\|J_{\tau+u}-J_{\tau}\|}{s^{w}}\wedge
1\right)^{q}|\mathcal{F}^{(0)}_{\tau}\right]\leq K\,s^{(1-w\nu)(q/\nu\wedge
1)}a(s)\end{array}\right.$ (B.3)

where $\widehat{\delta}(q)_{t,s}\equiv
s^{-1}\int_{t}^{t+s}\int_{E}\|\delta(u,x)\|^{q}\,\lambda(\mathrm{d}x)\,\mathrm{d}u$
and $a(s)\to 0$ as $s\to 0$.

IV. Let
$\varphi_{n}(t)=\sum_{h=1}^{l_{n}-1}\varphi^{n}_{h}\mathds{1}_{((h-1)\Delta_{n},h\Delta_{n}]}(t)$.
For a generic process $U$, define

$U^{n}_{t,s}=\int_{t}^{t+s}\varphi_{n}(u-t)\,\mathrm{d}U_{u}$ (B.4)

This quantity is useful in analyzing $\accentset{{\cc@style\underline{\mskip
10.0mu}}}{U}^{n}_{i}$, with $U$ being $X$ or $Y$.

V. For $p\in\mathbb{N}^{+}$, $l,m=0,1$, by (4.1) in the main article and
Riemann summation,

$\sum_{h=i}^{i+pl_{n}-2}\,\sum_{h^{\prime}=h+1}^{i+pl_{n}-1}\phi_{l}\Big{(}\frac{h^{\prime}-h}{l_{n}}\Big{)}\phi_{m}\Big{(}\frac{h^{\prime}-h}{l_{n}}\Big{)}=l_{n}^{2}\left(p\Phi_{lm}-\Psi_{lm}\right)+O(pl_{n})$
(B.5)

VI. By Jensen’s inequality and Doob’s maximal inequality, we have the
following lemma:

######  Lemma 1.

Let $Z_{i},i=1,\cdots,M$ be random variables,
$\mathcal{H}_{i}=\sigma(Z_{1},\cdots,Z_{i})$ be the $\sigma$-algebra generated
by $Z_{1},\cdots,Z_{i}$, then

$\mathbb{E}\left(\sup_{m=1,\cdots,M}\bigg{\|}\sum_{i=1}^{m}\left[Z_{i}-E\left(Z_{i}|\mathcal{H}_{i}\right)\right]\bigg{\|}\right)\leq
K\left(\sum_{i=1}^{M}\mathbb{E}\left(\|Z_{i}\|^{2}\right)\right)^{1/2}$

##  Appendix C Derivation for Theorem 1

###  C.1 Properties of spot estimator: I. jumps

By Assumption A-$\nu$, A-$\gamma$, (B.1), (B.2), (B.3), and
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i}=\psi_{n}^{-1/2}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i\Delta_{n},(l_{n}-1)\Delta_{n}}$ from (B.4), we know that
under the tunning (3.9)

$\left\\{\begin{array}[]{lcl}E^{n}_{i}\left(\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\big{\|}^{q}\right)&\leq&K_{q}\Delta_{n}^{q/2},\;\forall
q\leq 8\\\ E^{n}_{i}\left(\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\big{\|}^{q}\right)&\leq&K_{q}\Delta_{n}^{(q/2)\wedge(q/4+1/2)},\;\forall
q\leq 8\\\
E^{n}_{i}\left(\big{\|}\widehat{Y}^{*n}_{i}\big{\|}^{q}\vee\big{\|}\widehat{Y}^{n}_{i}\big{\|}^{q}\right)&\leq&K_{q}\Delta_{n}^{q},\;\forall
q\leq 4\\\
E^{n}_{i}\Big{[}\Big{(}\frac{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i}\|}{\Delta_{n}^{w}}\wedge
1\Big{)}^{q}\Big{]}&\leq&K_{q}\Delta_{n}^{[1/2-(w-1/4)\nu]\times[1\wedge(q/\nu)]}\,a_{n},\;\forall
q\leq 8\end{array}\right.$ (C.1)

for some $a_{n}\to 0$. We can write

$\left\|\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,\mathrm{T}}_{i}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\|\leq\nu_{n}\\}}-\widehat{Y}^{n}_{i}\big{)}-\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i}-\widehat{Y}^{*n}_{i}\big{)}\right\|\leq\sum_{r=1}^{3}\eta^{n,r}_{i}$

where

$\displaystyle\eta^{n,1}_{i}$ $\displaystyle=$
$\displaystyle\Big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,\mathrm{T}}_{i}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\|\leq\nu_{n}\\}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\|\leq\nu_{n}\\}}\Big{\|}$ (C.2)
$\displaystyle\eta^{n,2}_{i}$ $\displaystyle=$
$\displaystyle\big{\|}\widehat{Y}^{n}_{i}-\widehat{Y}^{*n}_{i}\big{\|}$
$\displaystyle\eta^{n,3}_{i}$ $\displaystyle=$
$\displaystyle\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\big{\|}^{2}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\|>\nu_{n}\\}}$

Let $u_{n}=\nu_{n}/\Delta_{n}^{1/2}$,
$Z^{n}_{i}=\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\|/\Delta_{n}^{1/2}$,
$Q^{n}_{i}=(\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i}\|/\Delta_{n}^{1/2})\wedge 1$,
$V^{n}_{i}=(\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i}\|/\Delta_{n}^{\rho})\wedge 1$, we have

$\Delta_{n}\eta^{n,1}_{i}\leq
u_{n}^{-2/(1-2\rho)}(Z^{n}_{i})^{2+2/(1-2\rho)}+(1+Z^{n}_{i})\left[Q^{n}_{i}+u_{n}^{2}(V^{n}_{i})^{2}\right]$

By successive conditioning and (B.3), there is a sequence $a_{n}\to 0$ such
that

$E^{n}_{i}\big{[}(\eta^{n,1}_{i})^{q}\big{]}\leq K_{q}\,\Delta_{n}^{2\rho
q+1/2-(\rho-1/4)\nu}a_{n}$

Analyzing $\eta^{n,2}_{i}$ with (B.2), (B.3), analyzing $\eta^{n,3}_{i}$ with
Cauchy-Schwarz inequality, Markov’s inequality, (C.1), we can get the
following lemma:

######  Lemma 2.

Assume (B.1), (3.9), Assumption A-$\nu$ and A-$\gamma$, then
$\exists\,a_{n}\to 0$ such that $\forall q\leq 4$,

$E^{n}_{i}\left(\|\widehat{c}^{n}_{i}-\widehat{c}^{*n}_{i}\|^{q}\right)\leq
K_{q}\left(a_{n}\Delta_{n}^{1/2-(\rho-1/4)\nu-(1-2\rho)q}+\Delta_{n}^{1/2}\right)$

###  C.2 Properties of spot estimator: II. continuous part

####  C.2.1 variables

“If there is a rifle handing on the wall in act one, it must be fired in the
next act. Otherwise it has no business being there”, said the Russian
playwright Anton Chekhov. Define

$\displaystyle\accentset{{\cc@style\underline{\mskip 10.0mu}}}{C}^{n}_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{\psi_{n}}\sum_{h=1}^{l_{n}-1}(\varphi^{n}_{h})^{2}\Delta^{n}_{i+h}C,\,\,C_{t}=\int_{0}^{t}c_{s}\,\mathrm{d}s$
$\displaystyle D^{n}_{i}$ $\displaystyle=$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{i}-c^{n}_{i}\Delta_{n}$ $\displaystyle\Gamma^{n}_{h}$
$\displaystyle=$
$\displaystyle\Gamma^{n}_{h,h},\,\,\Gamma^{n}_{h,h^{\prime}}=\frac{1}{\psi_{n}}\sum_{v=h\vee
h^{\prime}}^{h\wedge
h^{\prime}+l_{n}-1}(\varphi^{n}_{v-h+1}-\varphi^{n}_{n-h})(\varphi^{n}_{v-h^{\prime}+1}-\varphi^{n}_{v-h^{\prime}})\gamma^{n}_{v}$
$\displaystyle R^{n}_{i}$ $\displaystyle=$
$\displaystyle\widehat{Y}^{*n}_{i}-\Gamma^{n}_{i}$
$\displaystyle\zeta^{n}_{i}$ $\displaystyle=$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{i}-\Gamma^{n}_{i}$

given $p\in\mathbb{N}^{+}$, define

$\displaystyle\zeta(W,p)^{n}_{i}$ $\displaystyle=$
$\displaystyle\sum_{h=i}^{i+pl_{n}-1}\big{[}(\sigma^{n}_{i}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h})\cdot(\sigma^{n}_{i}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h})^{\mathrm{T}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{h}\big{]}$ $\displaystyle\zeta(X,p)^{n}_{i}$
$\displaystyle=$
$\displaystyle\sum_{h=i}^{i+pl_{n}-1}\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,\mathrm{T}}_{h}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{h}\big{)}$ $\displaystyle\zeta(X,p)^{\prime n}_{i}$
$\displaystyle=$
$\displaystyle\sum_{h=i}^{i+pl_{n}-2}\,\sum_{h^{\prime}=h+1}^{i+pl_{n}-1}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,\mathrm{T}}_{h^{\prime}}\phi_{1}\Big{(}\frac{h^{\prime}-h}{l_{n}}\Big{)}$
$\displaystyle\zeta(p)^{n}_{i}$ $\displaystyle=$
$\displaystyle\sum_{h=i}^{i+pl_{n}-1}\zeta^{n}_{h}$

Let $m(n,p)=\left\lfloor\frac{k_{n}-l_{n}}{(p+1)l_{n}}\right\rfloor$,
$a(n,p,h)=1+h(p+1)l_{n}$, $b(n,p,h)=a(n,p,h)+pl_{n}$, then the estimation
error of $\widehat{c}^{*n}_{i}$ can be decomposed as

$\beta^{n}_{i}\coloneqq\widehat{c}^{*n}_{i}-c^{n}_{i}=\xi^{n,0}_{i}+\xi^{n,1}_{i}+\xi^{n,2}_{i}+N(p)^{n}_{i}+M(p)^{n}_{i}$
(C.3)

where

$\displaystyle\xi^{n,0}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{n}-l_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}c^{n}_{i+h}-c^{n}_{i}$
$\displaystyle\xi^{n,1}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}D^{n}_{i+h}$
$\displaystyle\xi^{n,2}_{i}$ $\displaystyle=$
$\displaystyle\frac{-1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}R^{n}_{i+h}$
$\displaystyle N(p)^{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\Big{(}\sum_{h=0}^{m(n,p)-1}\zeta(1)^{n}_{i+b(n,p,h)}+\sum_{h=m(n,p)(p+1)l_{n}}^{k_{n}-l_{n}}\zeta^{n}_{i+1+h}\Big{)}$
$\displaystyle M(p)^{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=0}^{m(n,p)-1}\zeta(p)^{n}_{i+a(n,p,h)}$

####  C.2.2 bounds on $\|\xi^{n,r}_{i}\|$

By Assumption A-$\nu$, (B.1), (B.2),

$\left\\{\begin{array}[]{lcl}\left\|E\left(\xi^{n,0}_{i}|\mathcal{F}^{(0),n}_{i}\right)\right\|&\leq&Kk_{n}\Delta_{n}\\\
E\left(\|\xi^{n,0}_{i}\|^{q}|\mathcal{F}^{(0),n}_{i}\right)&\leq&K_{q}(k_{n}\Delta_{n})^{(q/2)\wedge
1},\,q\geq 0\end{array}\right.$ (C.4)

combined with (3.9),

$\left\\{\begin{array}[]{lcl}\left\|E\left(\xi^{n,1}_{i}|\mathcal{F}^{(0),n}_{i}\right)\right\|&\leq&K\Delta_{n}^{1/2}\\\
E\left(\|\xi^{n,1}_{i}\|^{q}|\mathcal{F}^{(0),n}_{i}\right)&\leq&K_{q}\Delta_{n}^{[(q/2)\wedge
1]/2},\,q\in\mathbb{N}^{+}\end{array}\right.$ (C.5)

By Assumption A-$\gamma$,

$\begin{array}[]{l}\left\|E^{n}_{i}\big{(}\xi^{n,2}_{i}\big{)}\right\|\leq
K\Delta_{n}^{-1}\\\
E^{n}_{i}\big{(}\|\xi^{n,2}_{i}\|^{q}\big{)}\leq\left\\{\begin{array}[]{l}K\,k_{n}^{-1/2},\,q=1;\\\
K_{q}\left(k_{n}^{-q+1}+k_{n}^{-q}\Delta_{n}^{-q/2+1}\right),q=2,3,4\end{array}\right.\end{array}$
(C.6)

####  C.2.3 estimates of $\zeta(X,p)^{n}_{i}$ & $\zeta(X,p)^{\prime n}_{i}$

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{i}=(\psi^{n})^{-1}C^{n}_{i\Delta_{n},(l_{n}-1)\Delta_{n}}$
in view of (B.4), hence by (B.1)

$\|\accentset{{\cc@style\underline{\mskip 10.0mu}}}{C}^{n}_{i}\|\leq
K\Delta_{n}$ (C.7)

According to (3.1) in the main article we have
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{i}=-\psi_{n}^{-1/2}\sum_{h=0}^{l_{n}-1}(\varphi^{n}_{h+1}-\varphi^{n}_{h})(X^{n}_{i+h}-X^{n}_{i})$,
then by (B.2)

$E\left(\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{i}\|^{q}|\mathcal{F}^{(0),n}_{i}\right)\leq
K_{q}\,\Delta_{n}^{q/2}$ (C.8)

Adopt the argument for (5.21) in [12] in the multivariate setting, we have

$\begin{array}[]{lcl}E\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n,\mathrm{T}}_{h^{\prime}}|\mathcal{F}^{(0),n}_{i}\big{)}&=&\frac{l_{n}\Delta_{n}}{\psi_{n}}\phi_{0}\big{(}\frac{|h^{\prime}-h|}{l_{n}}\big{)}\mathbf{I}+O_{p}(l_{n}^{-1/2}\Delta_{n})\\\
E\big{(}\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h}\|^{2m}|\mathcal{F}^{(0),n}_{i}\big{)}&=&\Delta_{n}^{m}(2m-1)!!+O_{p}(l_{n}^{-1}\Delta_{n}^{m}),\,\,m\in\mathbb{N}^{+}\end{array}$
(C.9)

Let
$U^{n}_{i}(p)=\sum_{h=i}^{i+pl_{n}-1}(\sigma^{n}_{i}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h})(\sigma^{n}_{i}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{W}^{n}_{h})^{\mathrm{T}}$,
$S^{n}_{i}(p)=\sum_{h=i}^{i+pl_{n}-1}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{h}$, then

$\zeta(W,p)^{n,jk}_{i}\zeta(W,p)^{n,lm}_{i}=U^{n}_{i}(p)^{jk}U^{n}_{i}(p)^{lm}+S^{n}_{i}(p)^{jk}S^{n}_{i}(p)^{lm}\\\
-U^{n}_{i}(p)^{jk}S^{n}_{i}(p)^{lm}-U^{n}_{i}(p)^{lm}S^{n}_{i}(p)^{jk}$ (C.10)

By (B.5), (C.7), (C.9), (C.10), and through similar arguments in section 5.3
of [12] with a modification for multi-dimension, and exploit the connection
between $\zeta(W,p)^{n}_{i}$ and $\zeta(X,p)^{n}_{i}$, in view of (4.2) in the
main article, we have the following lemma:

######  Lemma 3.

Assume Assumption A-$\nu$, (B.1), $l_{n}$ satisfies (3.9), then

$\displaystyle
E\left(\|\zeta(X,p)^{n}_{i}\|^{4}|\mathcal{F}^{(0),n}_{i}\right)$
$\displaystyle\leq$ $\displaystyle Kp^{4}\Delta_{n}^{2}$
$\displaystyle\left\|E\left[\zeta(X,p)^{n}_{i}|\mathcal{F}^{(0),n}_{i}\right]\right\|$
$\displaystyle\leq$ $\displaystyle Kp\Delta_{n}$ $\displaystyle
E\left[\zeta(X,p)^{\prime n}_{i}|\mathcal{F}^{(0),n}_{i}\right]$
$\displaystyle=$
$\displaystyle\frac{\theta^{2}l_{n}}{\psi_{n}}(p\Phi_{01}-\Psi_{01})\,c^{n}_{i}+p^{2}\,O_{p}(\Delta_{n}^{1/4})$
$\displaystyle
E\left[\zeta(X,p)^{n,jk}_{i}\zeta(X,p)^{n,lm}_{i}|\mathcal{F}^{(0),n}_{i}\right]$
$\displaystyle=$
$\displaystyle\frac{2\theta^{4}}{\psi_{n}^{2}}(p\Phi_{00}-\Psi_{00})\,\Sigma(c^{n}_{i})^{jk,lm}+p^{2}\,O_{p}(\Delta_{n}^{5/4})$

####  C.2.4 estimates of $\zeta(p)^{n}_{i}$

For $i\leq h,h^{\prime}\leq i+pl_{n}-1$, by (B.5)

$\displaystyle\Gamma^{n}_{h,h^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{\psi_{n}l_{n}}\phi_{1}\Big{(}\frac{|h^{\prime}-h|}{l_{n}}\Big{)}\gamma_{i}+O_{p}(\Delta_{n}^{5/4})$
(C.11)
$\displaystyle\sum_{h=i}^{i+pl_{n}-2}\,\sum_{h^{\prime}=h+1}^{i+pl_{n}-1}\Gamma^{n,jk}_{h,h^{\prime}}\Gamma^{n,lm}_{h,h^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{\psi_{n}^{2}}\left(p\Phi_{11}-\Psi_{11}\right)\gamma^{n,jk}_{i}\gamma^{n,lm}_{i}+p^{2}\,O_{p}(\Delta_{n}^{5/4})$

Let $\xi^{n,j_{1}\cdots j_{q}}_{h_{1}\cdots
h_{q}}=\prod_{v=1}^{q}\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,j_{v}}_{h_{v}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j_{v}}_{h_{v}}\big{)}$. By Assumption A-$\gamma$,

$E\left[\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,j}_{h}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\big{)}^{q}\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,k}_{h^{\prime}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h^{\prime}}\big{)}^{r}|\widetilde{\mathcal{F}}^{n}_{h\wedge
h^{\prime}-1}\right]=\left\\{\begin{array}[]{ll}0&q+r=1\\\
\Gamma^{n,jk}_{h,h^{\prime}}&q=r=1\\\
O_{p}\big{(}l_{n}^{-7/2}\big{)}\mathds{1}_{\\{|h-h^{\prime}|\leq
l_{n}\\}}&q+r=3\\\ O_{p}(l_{n}^{-8})&q+r=8\end{array}\right.\\\
E\left[\xi^{n,jklm}_{hhh^{\prime}h^{\prime}}|\widetilde{\mathcal{F}}^{n}_{h\wedge
h^{\prime}}\right]=\Gamma^{n,jk}_{h}\Gamma^{n,lm}_{h^{\prime}}+\Gamma^{n,jl}_{h,h^{\prime}}\Gamma^{n,km}_{h,h^{\prime}}+\Gamma^{n,jm}_{h,h^{\prime}}\Gamma^{n,kl}_{h,h^{\prime}}+O_{p}\left(l_{n}^{-5}\right)$
(C.12)

then by (C.8), (C.12)

$E(\zeta^{n}_{h}|\widetilde{\mathcal{F}}^{n}_{h})=\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}{\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}}^{\mathrm{T}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n}_{h},\,\,\,\|E^{n}_{h}\left(\zeta^{n}_{h}\right)\|\leq
K\Delta_{n}^{3/2},\,\,\,E^{n}_{h}\left(\|\zeta^{n}_{h}\|^{4}\right)\leq
K\Delta_{n}^{4}\\\
E\big{(}\zeta^{n,jk}_{h}\zeta^{n,lm}_{h^{\prime}}|\widetilde{\mathcal{F}}^{n}_{h\wedge
h^{\prime}}\big{)}=\sum_{r=1}^{3}\vartheta(r)^{n,jk,lm}_{h,h^{\prime}}\\\
+\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n,jk}_{h}\big{)}\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,l}_{h^{\prime}}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,m}_{h^{\prime}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n,lm}_{h^{\prime}}\big{)}$

where

$\displaystyle\vartheta(1)^{n,jk,lm}_{h,h^{\prime}}$ $\displaystyle=$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,l}_{h^{\prime}}\xi^{n,km}_{hh^{\prime}}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,m}_{h^{\prime}}\xi^{n,kl}_{hh^{\prime}}$ $\displaystyle\hskip
85.35826pt+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,l}_{h^{\prime}}\xi^{n,jm}_{hh^{\prime}}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,m}_{h^{\prime}}\xi^{n,jl}_{hh^{\prime}}$
$\displaystyle\vartheta(2)^{n,jk,lm}_{h,h^{\prime}}$ $\displaystyle=$
$\displaystyle\xi^{n,jklm}_{hhh^{\prime}h^{\prime}}-\xi^{n,jk}_{hh}\Gamma^{n,lm}_{h^{\prime}}-\xi^{n,lm}_{h^{\prime}h^{\prime}}\Gamma^{n,jk}_{h}+\Gamma^{n,jk}_{h}\Gamma^{n,lm}_{h^{\prime}}$
$\displaystyle\vartheta(3)^{n,jk,lm}_{h,h^{\prime}}$ $\displaystyle=$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\xi^{n,klm}_{hh^{\prime}h^{\prime}}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h}\xi^{n,jlm}_{hh^{\prime}h^{\prime}}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,l}_{h^{\prime}}\xi^{n,jkm}_{hhh^{\prime}}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,m}_{h^{\prime}}\xi^{n,jkl}_{hhh^{\prime}}$

Let
$\Upsilon^{n,jk,lm}_{h,h^{\prime}}=\Theta(\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,\mathrm{T}}_{h^{\prime}},\Gamma^{n}_{h,h^{\prime}})^{jk,lm}$
in light of (4.2) in the main article, then

$E\left(\zeta^{n,jk}_{h}\zeta^{n,lm}_{h^{\prime}}|\widetilde{\mathcal{F}}^{n}_{h\wedge
h^{\prime}}\right)=\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,j}_{h}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,k}_{h}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n,jk}_{h}\big{)}\big{(}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,l}_{h^{\prime}}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n,m}_{h^{\prime}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{C}^{n,lm}_{h^{\prime}}\big{)}\\\
+\Upsilon^{n,jk,lm}_{h,h^{\prime}}+\Sigma(\Gamma^{n}_{h,h^{\prime}})^{jk,lm}+O_{p}\big{(}l_{n}^{-5}+(\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h}\|+\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{X}^{n}_{h^{\prime}}\|)l_{n}^{-7/2}\big{)}$

hence

$\zeta(p)^{n,jk}_{i}\zeta(p)^{n,lm}_{i}=\zeta(X,p)^{n,jk}_{i}\zeta(X,p)^{n,lm}_{i}\\\
+\sum_{h=i}^{i+pl_{n}-2}\sum_{h^{\prime}=h+1}^{i+pl_{n}-1}\Big{[}\Upsilon^{n,jk,lm}_{h,h^{\prime}}+\Upsilon^{n,lm,jk}_{h,h^{\prime}}+\Sigma(\Gamma^{n}_{h,h^{\prime}})^{jk,lm}+\Sigma(\Gamma^{n}_{h,h^{\prime}})^{lm,jk}\Big{]}\\\
+\sum_{h=i}^{i+pl_{n}-1}\Big{[}\Upsilon^{n,jk,lm}_{h,h^{\prime}}+\Sigma(\Gamma^{n}_{h,h^{\prime}})^{jk,lm}\Big{]}+p^{2}\,O_{p}(\Delta_{n}^{5/4})$

then by (C.8), (C.11)

$E\left[\zeta(p)^{n,jk}_{i}\zeta(p)^{n,lm}_{i}|\widetilde{\mathcal{F}}^{n}_{i}\right]=\zeta(X,p)^{n,jk}_{i}\zeta(X,p)^{n,lm}_{i}+\\\
\frac{2}{\psi_{n}l_{n}}\Theta(\zeta(X,p)^{\prime
n}_{i},\gamma^{n}_{i})^{jk,lm}+\frac{2}{\psi_{n}^{2}}(p\Phi_{11}-\Psi_{11})\,\Sigma(\gamma^{n}_{i})^{jk,lm}+p^{2}\,O_{p}(\Delta_{n}^{5/4})$

According to these results and Lemma 3, one can get the following lemma:

######  Lemma 4.

Assume Assumption A-$\nu$, A-$\gamma$, (B.1), $l_{n}$ satisfies (3.9), then

$\displaystyle E[\zeta(p)^{n}_{i}|\widetilde{\mathcal{F}}^{n}_{i}]$
$\displaystyle=$ $\displaystyle\zeta(X,p)^{n}_{i}$
$\displaystyle\|E^{n}_{i}[\zeta(p)^{n}_{i}]\|$ $\displaystyle\leq$
$\displaystyle Kp\Delta_{n}$ $\displaystyle
E^{n}_{i}(\|\zeta(p)^{n}_{i}\|^{q})$ $\displaystyle\leq$ $\displaystyle
K_{q}\,p^{\lfloor q/2\rfloor\vee 1}\Delta_{n}^{q/2},\hskip 5.69054ptq=1,2,3,4$

moreover

$\left|E^{n}_{i}\left[\zeta(p)^{n,jk}_{i}\zeta(p)^{n,lm}_{i}\right]-(p+1)\theta\Delta_{n}\,\Xi(c^{n}_{i},\gamma^{n}_{i};p)^{jk,lm}\right|\leq
K\Delta_{n}^{5/4}$

where

$\Xi(x,z;p)=\frac{p\Phi_{00}-\Psi_{00}}{(p+1)\Phi_{00}}\,\Sigma(x)+\frac{p\Phi_{01}-\Psi_{01}}{(p+1)\Phi_{01}}\,\Theta(x,z)+\frac{p\Phi_{11}-\Psi_{11}}{(p+1)\Phi_{11}}\,\Sigma(z)$
(C.13)

Let $p\asymp\Delta_{n}^{-1/12}$, based on Lemma 4,

$\displaystyle\left\|E^{n}_{i}\left[M(p)^{n}_{i}\right]\right\|$
$\displaystyle\leq$ $\displaystyle K\Delta_{n}^{1/2}$
$\displaystyle\left\|E^{n}_{i}\left[N(p)^{n}_{i}\right]\right\|$
$\displaystyle\leq$ $\displaystyle Kp^{-1}\Delta_{n}^{1/2}$ $\displaystyle
E^{n}_{i}\left[\|M(p)^{n}_{i}\|^{q}\right]$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}K_{q}\big{(}k_{n}\Delta_{n}^{1/2}\big{)}^{-q/2},&q=1,2,4\\\
K\big{(}k_{n}\Delta_{n}^{1/2}\big{)}^{-2},&q=3\end{array}\right.$ (C.16)
$\displaystyle E^{n}_{i}\left[\|N(p)^{n}_{i}\|^{q}\right]$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}Kp\big{(}k_{n}\Delta_{n}^{1/2}\big{)}^{-1},&q=1\\\
K_{q}p^{-q/2}\big{(}k_{n}\Delta_{n}^{1/2}\big{)}^{-q/2},&q=2,4\\\
Kp^{-1}\big{(}k_{n}\Delta_{n}^{1/2}\big{)}^{-2},&q=3\end{array}\right.$ (C.20)

####  C.2.5 estimates of $\beta^{n}_{i}$

We need to define more variables:

$\begin{array}[]{ll}\zeta(p)^{n}_{i,h}=\zeta(p)^{n}_{i+a(n,p,h)}&A(p)^{n}_{i+v}=\sum_{h=v}^{v+pl_{n}-1}(c^{n}_{i+h}-c^{n}_{i})\Delta_{n}\\\
D(p)^{n}_{i+v}=\sum_{h=v}^{v+pl_{n}-1}D^{n}_{i+h}&R(p)^{n}_{i+v}=\sum_{h=v}^{v+pl_{n}-1}R^{n}_{i+h}\end{array}$
(C.21)

we have

Table 2: Estimates of ingredients under (3.9) scaling properties | $E(\|\cdot\|^{2}|\mathcal{F}^{n}_{i})$ | $\|E(\cdot|\mathcal{F}^{n}_{i})\|$
---|---|---
$R(p)^{n}_{i+v}$ | $p\Delta_{n}^{3/2}$ | $p\Delta_{n}^{3/2}$
$D(p)^{n}_{i+v}$ | $p\Delta_{n}^{3/2}$ | $p\Delta_{n}$
$A(p)^{n}_{i+v}$ | $p^{2}\Delta_{n}^{2}\big{(}p\Delta_{n}^{-1/2}+v\big{)}$ | $p\Delta_{n}^{3/2}\big{(}p\Delta_{n}^{-1/2}+v\big{)}$
$\zeta(p)^{n}_{i,h}$ | $p\Delta_{n}$ | $p\Delta_{n}$

Define

$\alpha(p)^{n}_{i,h}=-R(p+1)^{n}_{i+a(n,p,h)}+D(p+1)^{n}_{i+a(n,p,h)}+A(p+1)^{n}_{i+a(n,p,h)}+\zeta(p+1)^{n}_{i,h}$

By Table 2 and Cauchy-Schwarz inequality,

$E^{n}_{i}\left(\left|\alpha(p)^{n,jk}_{i,h}\alpha(p)^{n,lm}_{i,h}-\zeta(p+1)^{n,jk}_{i,h}\zeta(p+1)^{n,lm}_{i,h}\right|\right)\\\
\leq K\big{(}p^{2}\Delta_{n}^{5/4}+p^{3/2}\Delta_{n}^{3/2}v^{1/2}\big{)}$
(C.22)

Given $j,k,l,m=1,\cdots,d$, in the light of (C.3) and (C.21), by Table 2 we
have

$\left|E^{n}_{i}\big{(}\beta^{n,jk}_{i}\beta^{n,lm}_{i}\big{)}-(k_{n}\Delta_{n}^{1/2})^{-1}\Xi(c^{n}_{i},\gamma_{i})^{jk,lm}\right|=\sum_{r=1}^{5}\mu^{n,r}_{i}+p\,O_{p}((k_{n}^{2}\Delta_{n})^{-1})$

where

$\displaystyle\mu^{n,1}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-1}E^{n}_{i}\left(\left|\alpha(p)^{n,jk}_{i,h}\alpha(p)^{n,lm}_{i,h}-\zeta(p+1)^{n,jk}_{i,h}\zeta(p+1)^{n,lm}_{i,h}\right|\right)$
$\displaystyle\mu^{n,2}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-2}\sum_{h^{\prime}=h+1}^{m(n,p)-1}\left|E^{n}_{i}\left[\alpha(p)^{n,jk}_{i,h}\alpha(p)^{n,lm}_{i,h^{\prime}}+\alpha(p)^{n,lm}_{i,h}\alpha(p)^{n,jk}_{i,h^{\prime}}\right]\right|$
$\displaystyle\mu^{n,3}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-1}E^{n}_{i}\left(\left|\zeta(p+1)^{n,jk}_{i,h}\zeta(p+1)^{n,lm}_{i,h}\right.\right.$
$\displaystyle\hskip
102.42992pt\left.\left.-(p+2)\theta\Delta_{n}\,\Xi\big{(}c^{n}_{i+a(n,p,h)},\gamma^{n}_{i+a(n,p,h)};p+1\big{)}^{jk,lm}\right|\right)$
$\displaystyle\mu^{n,4}_{i}$ $\displaystyle=$
$\displaystyle\frac{(p+2)\theta}{(k_{n}-l_{n})^{2}\Delta_{n}}\sum_{h=0}^{m(n,p)-1}\left|E^{n}_{i}\left[\Xi\big{(}c^{n}_{i+a(n,p,h)},\gamma^{n}_{i+a(n,p,h)};p+1\big{)}^{jk,lm}\right.\right.$
$\displaystyle\hskip
216.24094pt\left.\left.-\Xi(c^{n}_{i},\gamma^{n}_{i};p+1)^{jk,lm}\right]\right|$
$\displaystyle\mu^{n,5}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}^{1/2}}\left|\frac{(p+2)\theta}{(k_{n}-l_{n})\Delta_{n}^{1/2}}\left\lfloor\frac{k_{n}}{(p+1)l_{n}}\right\rfloor-\frac{(k_{n}-l_{n})}{k_{n}}\right|\cdot\left|\Xi(c^{n}_{i},\gamma^{n}_{i};p+1)^{jk,lm}\right|$
$\displaystyle\hskip
130.88268pt+\frac{1}{k_{n}\Delta_{n}^{1/2}}\left|\Xi(c^{n}_{i},\gamma^{n}_{i};p+1)^{jk,lm}-\Xi(c^{n}_{i},\gamma^{n}_{i})^{jk,lm}\right|$

Use Table 2 and (C.22) to get bounds on $\mu^{n,r}_{i},\,r=1,2,3,4,5$; combine
(C.3), (C.4), (C.5), (C.6), (C.2.4), we get the following lemma:

######  Lemma 5.

Assume (B.1), (3.9) and Assumption A-$\nu$, A-$\gamma$, given
$p\in\mathbb{N}^{+}$,

$\displaystyle\|E^{n}_{i}(\beta^{n}_{i})\|$ $\displaystyle\leq$ $\displaystyle
Kk_{n}\Delta_{n}$ $\displaystyle E^{n}_{i}(\|\beta^{n}_{i}\|^{q})$
$\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}K_{q}\Big{[}(k_{n}\Delta_{n})^{(q/2)\wedge
1}+(k_{n}\Delta_{n}^{1/2})^{-q/2}\Big{]},&q=1,2,4\\\
Kk_{n}\Delta_{n},&q=3\end{array}\right.$

additionally

$\left|E^{n}_{i}\big{(}\beta^{n,jk}_{i}\beta^{n,lm}_{i}\big{)}-(k_{n}\Delta_{n})^{-1/2}\Xi(c^{n}_{i},\gamma^{n}_{i})^{jk,lm}\right|\leq
K\big{[}k_{n}\Delta_{n}+p^{-1}(k_{n}\Delta_{n}^{1/2})^{-1}\big{]}$

###  C.3 Structure

Define

$\displaystyle\lambda(x,z)$ $\displaystyle=$
$\displaystyle\sum^{d}_{j,k,l,m=1}\partial^{2}_{jk,lm}g(x)\times\Xi(x,z)^{jk,lm}$
$\displaystyle\eta^{n}_{i}$ $\displaystyle=$
$\displaystyle\lambda(\widehat{c}^{n}_{i},\widehat{\gamma}^{n}_{i})-\lambda(\widehat{c}^{*n}_{i},\widehat{\gamma}^{n}_{i})$

As $\Delta_{n}^{-1/4}|a^{n}_{t}-1|<k_{n}\Delta_{n}^{3/4}\to 0$, letting
$a^{n}_{t}=1$ doesn’t not affect the asymptotic analysis. By Cramér-Wold
theorem, we can suppose $g$ is $\mathbb{R}$-valued. We have

$\Delta_{n}^{-1/4}\big{[}\widehat{S}(g)^{n}-S(g)\big{]}=\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,0}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,1}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,2}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,3}+\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,4}$ (C.24)

where

$\displaystyle\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}^{n,0}_{t}$
$\displaystyle=$
$\displaystyle\Delta_{n}^{-1/4}\Bigg{[}\sum_{i=0}^{N^{n}_{t}-1}\int_{ik_{n}\Delta_{n}}^{(i+1)k_{n}\Delta_{n}}g(c^{n}_{ik_{n}})-g(c_{s})\,\mathrm{d}s-\int_{N^{n}_{t}k_{n}\Delta_{n}}^{t}g(c_{s})\,\mathrm{d}s\Bigg{]}$
$\displaystyle\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}^{n,1}_{t}$
$\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\Big{[}g(\widehat{c}^{n}_{ik_{n}})-g(\widehat{c}^{*n}_{ik_{n}})-(2k_{n}\Delta_{n}^{1/2})^{-1}\eta^{n}_{ik_{n}}\Big{]}$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,2}_{t}$ $\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\sum^{d}_{j,k=1}\partial_{jk}g(c^{n}_{ik_{n}})\Bigg{[}\sum_{r=0}^{2}\xi^{n,r,jk}_{ik_{n}}+N(p)^{n,jk}_{ik_{n}}\Bigg{]}$
$\displaystyle\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}^{n,3}_{t}$
$\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\Big{[}g(\widehat{c}^{*n}_{ik_{n}})-g(c^{n}_{ik_{n}})-\sum^{d}_{j,k=1}\partial_{jk}g(c^{n}_{ik_{n}})\beta^{n,jk}_{ik_{n}}$
$\displaystyle\hskip
187.78818pt-(2k_{n}\Delta_{n}^{1/2})^{-1}\lambda(\widehat{c}^{*n}_{i},\widehat{\gamma}^{n}_{i})\Big{]}$
$\displaystyle\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,4}_{t}$ $\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\sum^{d}_{j,k=1}\partial_{jk}g(c^{n}_{ik_{n}})\times
M(p)^{n,jk}_{ik_{n}}$

###  C.4 Asymptotic negligibility

First of all, we need to get bounds on
$\partial^{r}g(\widehat{c}^{n}_{i}),\,r\leq 3$, where $\partial^{r}g$ denotes
the $r$-th order partial derivatives. Let
$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}=(k_{n}-l_{n})^{-1}\sum_{h=1}^{k_{n}-l_{n}+1}c^{n}_{i}$
and $I^{n}_{t}=\\{0,1,\cdots,N^{n}_{t}-1\\}$, note that
$|I^{n}_{t}|\asymp(k_{n}\Delta_{n})^{-1}$, according to Lemma 2, there is a
sequence $a_{n}\to 0$ such that

$\mathbb{E}\Big{(}\sup_{i\in
I_{n}}\|\widehat{c}^{n}_{i}-\widehat{c}^{*n}_{i}\|\Big{)}\leq
K\left(a_{n}\Delta_{n}^{\kappa-1/2-(\rho-1/4)\nu-(1-2\rho)}+\Delta_{n}^{\kappa-1/2}\right)$

Note $\widehat{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}=\xi^{n,1}_{i}+\xi^{n,2}_{i}+N(p)^{n}_{i}+M(p)^{n}_{i}$,
by (C.5), (C.6), (C.2.4) and $\kappa<3/4$,
$E^{n}_{i}\big{(}\|\widehat{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|^{4}\big{)}\leq K\Delta_{n}^{2\kappa-1}$, so

$\mathbb{E}\Big{(}\sup_{i\in
I_{n}}\|\widehat{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|^{4}\Big{)}\leq K\Delta_{n}^{3\kappa-2}$

hence by (3.9) and Markov’s inequality

$\sup_{i\in I_{n}}\|\widehat{c}^{n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|=o_{p}(1)$ (C.25)

According to (B.1) and convexity, $\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\in\mathcal{S}$. By (C.25),
$\widehat{c}^{n}_{i}\in\mathcal{S}^{\epsilon}$ when $n$ is sufficiently large.
Therefore by (3.8) in the main article, for asymptotic analysis we can assume

$\|\partial^{r}g(\widehat{c}^{n}_{i})\|\leq K,\hskip 14.22636pt\forall
r=0,1,2,3,\,\forall i\in I^{n}_{t}$ (C.26)

Through an almost identical argument for Lemma 4.4 in [21],

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,0}\overset{u.c.p.}{\longrightarrow}0$ (C.27)

Define function $g_{n}$ on $\mathcal{S}^{+}_{d}\times\mathcal{S}^{+}_{d}$ as
$g_{n}(x,z)=g(x)-(2k_{n}\Delta_{n}^{1/2})^{-1}\lambda(x,z)$. Based on (C.26),

$\|g_{n}(x,z)-g_{n}(y,z)\|\leq K\|x-y\|\\\
+K(k_{n}\Delta_{n}^{1/2})^{-1}\|x-y\|\left(\|x\|^{2}+\|z\|^{2}+\|x-y\|^{2}+\|z\|\|x-y\|\right)$

so $\|g_{n}(x,z)-g_{n}(y,z)\|\leq K\|x-y\|$ when $n$ is sufficiently large. By
Lemma 2

$\mathbb{E}\left(\sup_{u\in[0,t]}\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,1}_{u}\big{\|}\right)\leq
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\left\|g_{n}(\widehat{c}^{n}_{ik_{n}},\widehat{\gamma}^{n}_{ik_{n}})-g_{n}(\widehat{c}^{*n}_{ik_{n}},\widehat{\gamma}^{n}_{ik_{n}})\right\|\\\
\leq
Kt\left(a_{n}\Delta_{n}^{1/4-(\rho-1/4)\nu-(1-2\rho)}+\Delta_{n}^{1/4}\right)$

Since $\rho>\frac{3-\nu}{4(2-\nu)}$, $1/4-(\rho-1/4)\nu-(1-2\rho)>0$, we have
the following lemma:

######  Lemma 6.

Assume Assumption A-$\nu$, A-$\gamma$, (3.9), (C.26) then

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,1}\overset{u.c.p.}{\longrightarrow}0$

Given $e^{n}_{i}\in\mathbb{R}^{d\times d}$, consider the process

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{*n}_{t}=k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\sum^{d}_{j,k=1}\partial_{jk}g(c^{n}_{ik_{n}})\times
e^{n,jk}_{ik_{n}}$

suppose $e^{n}_{i}$ satisfies

$\left\\{\begin{array}[]{lcl}\left\|E^{n}_{i}\left(e^{n}_{i}\right)\right\|&\leq&K\Delta_{n}^{1/4}a_{n}\\\
E^{n}_{i}\left(\|e^{n}_{i}\|^{2}\right)&\leq&K(k_{n}\Delta_{n}^{1/2})^{-1}b_{n}\end{array}\right.$
(C.28)

where $a_{n},b_{n}\to 0$. Since $\partial g$ is bounded by (B.1),

$\mathbb{E}\left(\sup_{u\in[0,t]}\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{*n}_{u}\big{\|}\right)\leq
Kk_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\mathbb{E}\left(\left\|E^{n}_{ik_{n}}\left(e^{n}_{ik_{n}}\right)\right\|\right)\\\
+Kk_{n}\Delta_{n}^{3/4}\mathbb{E}\left(\sup_{s\in[0,t]}\left\|\sum_{i=0}^{N^{n}_{s}-1}\left[e^{n}_{ik_{n}}-E^{n}_{ik_{n}}\left(e^{n}_{ik_{n}}\right)\right]\right\|\right)$

by Lemma 1,

$\mathbb{E}\left(\sup_{s\in[0,t]}\left\|\sum_{i=0}^{N^{n}_{s}-1}\left[e^{n}_{ik_{n}}-E^{n}_{ik_{n}}\left(e^{n}_{ik_{n}}\right)\right]\right\|\right)\leq
K\left(\sum_{i=0}^{N^{n}_{t}-1}\mathbb{E}\left(\|e^{n}_{ik_{n}}\|^{2}\right)\right)^{1/2}$

note that $k_{n}\Delta_{n}N^{n}_{t}\asymp t$, we have

$\mathbb{E}\left(\sup_{u\in[0,t]}\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{*n}_{u}\big{\|}\right)\leq K\left(ta_{n}+\sqrt{tb_{n}}\right)\to
0$

To show the asymptotic negligibility of
$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}(p)^{n,2}$, we need to
show $e_{i}$ satisfies (C.28) in each of the following 4 cases:

  * ①

when $e^{n}_{i}=\xi^{n,0}_{i}$, by (C.4),
$a_{n}=k_{n}\Delta_{n}^{3/4},\,b_{n}=(k_{n}\Delta_{n}^{3/4})^{2}$;

  * ②

when $e^{n}_{i}=\xi^{n,1}_{i}$, by (C.5), $a_{n}=\Delta_{n}^{1/4}$,
$b_{n}=k_{n}\Delta_{n}$;

  * ③

when $e^{n}_{i}=\xi^{n,2}_{i}$, by (C.6), $a_{n}=\Delta_{n}^{3/4}$,
$b_{n}=\Delta_{n}^{1/2}$;

  * ④

when $e^{n}_{i}=N(p)^{n}_{i}$, by (C.2.4), $a_{n}=p^{-1}\Delta_{n}^{1/4}$,
$b_{n}=p^{-1}$.

Hence we have the following lemma:

######  Lemma 7.

Assume Assumption A-$\nu$, A-$\gamma$, (3.9), (C.26), and let
$p\asymp\Delta_{n}^{-1/12}$, then

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,2}\overset{u.c.p.}{\longrightarrow}0$

Let $\chi^{n}_{i}=\widehat{\gamma}^{n}_{i}-\gamma^{n}_{i}$. By (B.2), the
choice of $m_{n}$ and Jensen’s inequality

$E^{n}_{i}(\|\chi^{n}_{i}\|^{q})\leq K_{q}\Delta_{n}^{q/4},\,q=1,2$ (C.29)

Let
$\iota^{n}_{i}=\lambda(\widehat{c}^{*n}_{i},\widehat{\gamma}^{n}_{i})-\lambda(c^{n}_{i},\gamma^{n}_{i})$,
then by (4.2) and (4.3) in the main article

$\|\iota^{n}_{i}\|\leq
K\left(\|\beta^{n}_{i}\|+\|\chi^{n}_{i}\|+\|\beta^{n}_{i}\|^{2}+\|\beta^{n}_{i}\|\|\chi^{n}_{i}\|+\|\chi^{n}_{i}\|^{2}\right)$

hence by Lemma 5, (3.9), (C.29),

$E^{n}_{i}(\|\iota^{n}_{i}\|)\leq Kk_{n}^{-1/2}\Delta_{n}^{-1/4}$ (C.30)

We can rewrite $\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}^{n,3}$ as

$\accentset{{\cc@style\underline{\mskip 10.0mu}}}{S}^{n,3}=G^{n}+H^{n}$

where

$\displaystyle G^{n}_{t}$ $\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\left[s^{n}_{ik_{n}}+\frac{1}{2k_{n}\Delta_{n}^{1/2}}\iota^{n}_{ik_{n}}+E^{n}_{ik_{n}}\left(v^{n}_{ik_{n}}\right)\right]$
$\displaystyle H^{n}_{t}$ $\displaystyle=$ $\displaystyle
k_{n}\Delta_{n}^{3/4}\sum_{i=0}^{N^{n}_{t}-1}\left[v^{n}_{ik_{n}}-E^{n}_{ik_{n}}\left(v^{n}_{ik_{n}}\right)\right]$
$\displaystyle s^{n}_{i}$ $\displaystyle=$ $\displaystyle
g(c^{n}_{i}+\beta^{n}_{i})-g(c^{n}_{i})-\sum^{d}_{j,k=1}\partial_{jk}g(c^{n}_{i})\beta_{i}^{n,jk}-\frac{1}{2}\sum^{d}_{j,k,l,m=1}\partial^{2}_{jk,lm}g(c^{n}_{i})\beta_{i}^{n,jk}\beta_{i}^{n,lm}$
$\displaystyle v^{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum^{d}_{j,k,l,m=1}\partial^{2}_{jk,lm}g(c^{n}_{i})\left[\beta_{i}^{n,jk}\beta_{i}^{n,lm}-(k_{n}\Delta_{n}^{1/2})^{-1}\Xi(c^{n}_{i},\gamma^{n}_{i})^{jk,lm}\right]$

By Lemma 5, (C.26), (C.30), if we let $p\asymp\Delta_{n}^{-12}$,

$\mathbb{E}\left(\sup_{s\in[0,t]}\|G^{n}_{s}\|\right)\leq
Kt\left[k_{n}\Delta_{n}^{3/4}+(k_{n}\Delta_{n}^{2/3})^{-1}\right]$ (C.31)

and

$E^{n}_{i}(\|v^{n}_{i}\|^{2})\leq
K\sum^{d}_{j,k,l,m=1}E^{n}_{i}\left(\left|\beta_{i}^{n,jk}\beta_{i}^{n,lm}-\frac{1}{k_{n}\Delta_{n}^{1/2}}\Xi(c^{n}_{i},\gamma^{n}_{i})^{jk,lm}\right|^{2}\right)\\\
\leq K\left[k_{n}\Delta_{n}+(k_{n}\Delta_{n}^{1/2})^{-2}\right]$

then Lemma 1 implies

$\mathbb{E}\left(\sup_{s\in[0,t]}\|H^{n}_{s}\|\right)\leq
Kk_{n}\Delta_{n}^{3/4}\left(\sum_{i=0}^{N^{n}_{t}-1}\mathbb{E}(\|v^{n}_{ik_{n}}\|^{2})\right)^{1/2}\\\
\leq
K\sqrt{t}\left[k_{n}\Delta_{n}^{3/4}+(k_{n}\Delta_{n}^{1/2})^{-1/2}\right]$
(C.32)

According to (C.31) and (C.32), we have the following lemma:

######  Lemma 8.

Assume Assumption A-$\nu$, A-$\gamma$, (3.9), (C.26), then

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}^{n,3}\overset{u.c.p.}{\longrightarrow}0$

###  C.5 Stable convergence in law to a continuous Itô semimartingale

Recall (C.21), let

$\Lambda(p)^{n}_{i,h}=\partial g(c^{n}_{ik_{n}})\odot\zeta(p)^{n}_{ik_{n},h}$
(C.33)

where $\odot$ denotes Hadamard product, i.e.,
$\Lambda(p)^{n,jk}_{i,h}=\partial_{jk}g(c^{n}_{ik_{n}})\odot\zeta(p)^{n,jk}_{ik_{n},h}$.
We can write

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,4}_{t}=\frac{k_{n}}{k_{n}-l_{n}}\sum^{d}_{j,k=1}\Delta_{n}^{-1/4}\sum^{N^{n}_{t}-1}_{i=0}\sum_{h=0}^{m(n,p)-1}\Lambda(p)^{n,jk}_{i,h}$

Let $\mathcal{H}(p)^{n}_{i,h}=\mathcal{F}^{n}_{ik_{n}+a(n,p,h)}$, by Lemma 4,

$\Delta_{n}^{-1/2}\sum^{N^{n}_{t}-1}_{i=0}\sum_{h=0}^{m(n,p)-1}\left\|E\big{[}\zeta(p)^{n}_{ik_{n},h}|\mathcal{H}(p)^{n}_{i,h}\big{]}\right\|^{2}\leq
Ktp\Delta_{n}$

Let $N$ be a bounded martingale orthogonal to $W$ or $N=W^{l}$ for some
$l=1,\cdots,d^{\prime}$, and $\Delta
N(p)^{n}_{i,h}=N^{n}_{ik_{n}+b(n,p,h)}-N^{n}_{ik_{n}+a(n,p,h)}$. The following
four statements about convergence in probability for any indices $j,k,l,m$ can
verify the conditions of theorem IX.7.28 in [34]:

$\displaystyle\Delta_{n}^{-1/4}\sum_{i=0}^{N^{n}_{t}-1}\sum_{h=0}^{m(n,p)-1}\left\|E\left[\Lambda(p)^{n}_{i,h}|\mathcal{H}(p)^{n}_{i,h}\right]\right\|$
$\displaystyle\overset{\mathbb{P}}{\longrightarrow}$ $\displaystyle 0$ (C.34)
$\displaystyle\Delta_{n}^{-1}\sum_{i=0}^{N^{n}_{t}-1}\sum_{h=0}^{m(n,p)-1}E\left[\left\|\Lambda(p)^{n}_{i,h}\right\|^{4}|\mathcal{H}(p)^{n}_{i,h}\right]$
$\displaystyle\overset{\mathbb{P}}{\longrightarrow}$ $\displaystyle 0$ (C.35)
$\displaystyle\Delta_{n}^{-1/4}\sum_{i=0}^{N^{n}_{t}-1}\sum_{h=0}^{m(n,p)-1}\left\|E\big{[}\Lambda(p)^{n}_{i,h}\Delta
N(p)^{n}_{i,h}|\mathcal{H}(p)^{n}_{i,h}\big{]}\right\|$
$\displaystyle\overset{\mathbb{P}}{\longrightarrow}$ $\displaystyle 0$ (C.36)
$\Delta_{n}^{-1/2}\sum_{i=0}^{N^{n}_{t}-1}\sum_{h=0}^{m(n,p)-1}E\left[\Lambda(p)^{n,jk}_{i,h}\,\Lambda(p)^{n,lm}_{i,h}|\mathcal{H}(p)^{n}_{i,h}\right]\\\
\overset{\mathbb{P}}{\longrightarrow}\int_{0}^{t}\partial_{jk}g(c_{s})\partial_{lm}g(c_{s})^{\mathrm{T}}\,\Xi(c_{s},\gamma_{s};p)^{jk,lm}\,\mathrm{d}s$
(C.37)

Under (C.26), one can verify (C.34), (C.35) by the second and third claims of
Lemma 4, respectively. The same argument as that for (5.58) in [12] leads to
(C.36). By the last claim of Lemma 4, the left-hand side of (C.37) equals

$\sum_{i=0}^{N^{n}_{t}-1}\sum_{h=0}^{m(n,p)-1}\partial_{jk}g(c^{n}_{ik_{n}})\partial_{lm}g(c^{n}_{ik_{n}})^{\mathrm{T}}\times\\\
\Xi\big{(}c^{n}_{ik_{n}+a(n,p,h)},\gamma^{n}_{ik_{n}+a(n,p,h)};p\big{)}^{jk,lm}(p+1)l_{n}\Delta_{n}+tp\,O_{p}(\Delta_{n}^{1/4})$

then (C.37) is verified by Riemann summation. By Theorem IX.7.28 in [34] we
have the following lemma:

######  Lemma 9.

Assume Assumption A-$\nu$, A-$\gamma$, (3.9), (C.26), then for $\forall
p\in\mathbb{N}^{+}$,

$\accentset{{\cc@style\underline{\mskip
10.0mu}}}{S}(p)^{n,4}\overset{\mathcal{L}-s(f)}{\longrightarrow}Z(p)$

where $Z(p)$ is a process defined on an extension of the space
$\left(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P}\right)$, such that
conditioning on $\mathcal{F}$ it is a mean-0 continuous Itô martingale with
variance

$\widetilde{E}[Z(p)Z(p)^{\mathrm{T}}|\mathcal{F}]=\int_{0}^{t}\sum^{d}_{j,k,l,m=1}\partial_{jk}g(c_{s})\partial_{lm}g(c_{s})^{\mathrm{T}}\,\Xi(c_{s},\gamma_{s};p)^{jk,lm}\,\mathrm{d}s$

where $\widetilde{E}$ is the conditional expectation operator on the extended
probability space and $\Xi(x,z;p)$ is defined in (C.13).

By (C.27), Lemma 6, 7, 8, 9, and $\Xi(x,z;p)\to\Xi(x,z)$ as $p\to\infty$, we
arrive at the asymptotic result in Theorem 1.

##  Appendix D Derivation for Theorem 2

As a counterpart to (C.1), we have under (3.12)

$\left\\{\begin{array}[]{lcl}E^{n}_{i}\left(\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\big{\|}^{q}\right)&\leq&K_{q}\Delta_{n}^{q/2},\;\forall
q\leq 8\\\ E^{n}_{i}\left(\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\big{\|}^{q}\right)&\leq&K_{q}\Delta_{n}^{(q/2)\wedge[1/2-\delta+(1/2+\delta)q/2]},\;\forall
q\leq 8\\\
E^{n}_{i}\Big{[}\Big{(}\frac{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{J}^{n}_{i}\|}{\Delta_{n}^{w}}\wedge
1\Big{)}^{q}\Big{]}&\leq&K_{q}\Delta_{n}^{[1/2-\delta-(w-1/4-\delta/2)\nu]\times[1\wedge(q/\nu)]}\,a_{n},\;\forall
q\leq 8\end{array}\right.$ (D.1)

for some $a_{n}\to 0$. It follows that

$\big{\|}\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n,\mathrm{T}}_{i}\mathds{1}_{\\{\|\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{n}_{i}\|\leq\nu_{n}\\}}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n}_{i}\cdot\accentset{{\cc@style\underline{\mskip
10.0mu}}}{Y}^{*n,\mathrm{T}}_{i}\big{\|}\leq\eta^{n,1}_{i}+\eta^{n,3}_{i}$

where $\eta^{n,1}_{i}$ and $\eta^{n,3}_{i}$ are defined in (C.2).

Through a similar argument to that in Section C.1, for $1\leq q\leq 4$ there
is a sequence $a_{n}\to 0$ such that under (3.12),

$\displaystyle E^{n}_{i}\big{[}(\eta^{n,1}_{i})^{q}\big{]}$
$\displaystyle\leq$ $\displaystyle
K_{q}\big{[}\Delta_{n}^{2q}+a_{n}\Delta_{n}^{2q\rho+1/2-\delta-(\rho-1/4-\delta/2)\nu}\big{]}$
$\displaystyle E^{n}_{i}\big{[}(\eta^{n,3}_{i})^{q}\big{]}$
$\displaystyle\leq$ $\displaystyle
K_{q,q^{\prime}}\Delta_{n}^{q+q^{\prime}(1-2\rho)/2},\;q^{\prime}>1/(1-2\rho)$

Note
$\|\widetilde{c}^{n}_{i}-\widetilde{c}^{*n}_{i}\|\leq[(k_{n}-l_{n})\Delta_{n}]^{-1}\sum_{h=1}^{k_{n}-l_{n}+1}\big{[}\eta^{n,1}_{i+h}+\eta^{n,3}_{i+h}\big{]}$,
then

$E^{n}_{i}\left(\|\widetilde{c}^{n}_{i}-\widetilde{c}^{*n}_{i}\|^{q}\right)\leq
K_{q}\left(a_{n}\Delta_{n}^{1/2-\delta-(\rho-1/4-\delta/2)\nu-(1-2\rho)q}+\Delta_{n}^{q}\right)$
(D.2)

Define the following quantities

$\displaystyle\xi^{\prime n,2}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})\Delta_{n}}\sum_{h=1}^{k_{n}-l_{n}+1}\Gamma^{n}_{i+h}$
$\displaystyle\beta^{\prime n}_{i}$ $\displaystyle=$
$\displaystyle\xi^{n,0}_{i}+\xi^{n,1}_{i}+\xi^{\prime
n,2}_{i}+N(p)^{n}_{i}+M(p)^{n}_{i}$

where $\xi^{n,0}_{i},\,\xi^{n,1}_{i},\,N(p)^{n}_{i},\,M(p)^{n}_{i}$ are
defined in (C.3). Note $\widetilde{c}^{*n}_{i}-c^{n}_{i}=\beta^{\prime
n}_{i}$.

Under the tuning (3.12),

$\left\\{\begin{array}[]{lcl}\left\|E\left(\xi^{n,1}_{i}|\mathcal{F}^{(0),n}_{i}\right)\right\|&\leq&K\Delta_{n}^{1/2-\delta}\\\
E\left(\|\xi^{n,1}_{i}\|^{q}|\mathcal{F}^{(0),n}_{i}\right)&\leq&K_{q}\,\Delta_{n}^{[(q/2)\wedge
1](1/2-\delta)},\;q\in\mathbb{N}^{+}\end{array}\right.$ (D.3)

By (3.1) and Assumption A-$\gamma$ in the main article,

$\begin{array}[]{l}\left\|E^{n}_{i}\big{(}\xi^{\prime
n,2}_{i}\big{)}\right\|\leq K\Delta_{n}^{2\delta}\\\
E^{n}_{i}\big{(}\|\xi^{\prime n,2}_{i}\|^{q}\big{)}\leq
K_{q}\,\Delta_{n}^{2\delta q}\end{array}$ (D.4)

Based on the same derivations in Section C.2 that lead to Lemma 4, we know

$\displaystyle\left\|E^{n}_{i}\left[N(p)^{n}_{i}\right]\right\|$
$\displaystyle\leq$ $\displaystyle
K\big{(}p^{-1}\Delta_{n}^{1/2-\delta}+pk_{n}^{-1}\Delta_{n}^{-2\delta}\big{)}$
$\displaystyle E^{n}_{i}\left[\|N(p)^{n}_{i}\|^{q}\right]$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}K_{q}\,p^{-q/2}\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-q/2},&q=1,2,4\\\
Kp^{-1}\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-2},&q=3\end{array}\right.$
(D.7)

and

$\displaystyle\left\|E^{n}_{i}\left[M(p)^{n}_{i}\right]\right\|$
$\displaystyle\leq$ $\displaystyle K\Delta_{n}^{1/2-\delta}$ $\displaystyle
E^{n}_{i}\left[\|M(p)^{n}_{i}\|^{q}\right]$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}K_{q}\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-q/2},&q=1,2,4\\\
K\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-2},&q=3\end{array}\right.$
(D.10)

Since $\widetilde{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}=\xi^{n,1}_{i}+\xi^{\prime
n,2}_{i}+N(p)^{n}_{i}+M(p)^{n}_{i}$, based on (D.3), (D.4), (D), (D),

$\mathbb{E}\Big{(}\sup_{i\in
I_{n}}\|\widetilde{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|^{4}\Big{)}\leq K\Delta_{n}^{3\kappa-2-2\delta}$

then use $\|\widetilde{c}^{n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|\leq\|\widetilde{c}^{n}_{i}-\widetilde{c}^{*n}_{i}\|+\|\widetilde{c}^{*n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|$ and (D.2) with $q=1$, under (3.12)

$\sup_{i\in
I_{n}}\|\widetilde{c}^{n}_{i}-\accentset{{\cc@style\underline{\mskip
10.0mu}}}{c}^{n}_{i}\|=o_{p}(1)$

Therefore, just as explained at the beginning of Section C.4, when the p.s.d.
plug-in $\widetilde{c}^{n}_{i}$ and the tuning (3.12) are used, we can still
assume (C.26).

We know by combining (C.4), (D.3), (D.4), (D) and (D),

$\displaystyle\left\|E^{n}_{i}\left(\beta^{\prime n}_{i}\right)\right\|$
$\displaystyle\leq$ $\displaystyle
K\big{(}k_{n}\Delta_{n}+\Delta_{n}^{2\delta}\big{)}$ $\displaystyle
E^{n}_{i}\left(\|\beta^{\prime n}_{i}\|^{q}\right)$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}K_{q}\big{[}(k_{n}\Delta_{n})^{(q/2)\wedge
1}+\Delta_{n}^{[(q/2)\wedge 1](1/2-\delta)}&\\\ \hskip
85.35826pt+\Delta^{2\delta
q}+\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-q/2}\big{]},&q=1,2,4\\\
K\big{[}k_{n}\Delta_{n}+\Delta_{n}^{1/2-\delta}+\big{(}k_{n}\Delta_{n}^{1/2+\delta}\big{)}^{-2}\big{]},&q=3\end{array}\right.$
(D.14)

Now we study the quadratic form of $\beta^{\prime n}_{i}$. Recall the
definitions in (C.21) and the estimates in Table 2. Let $R(p)^{\prime
n}_{i+v}=\sum_{h=v}^{v+pl_{n}-1}\Gamma^{n}_{i+h}$. Analogously when
$l_{n}\asymp\Delta_{n}^{-(1/2+\delta)}$ we have the following estimates

Table 3: Estimates of ingredients under (3.12) scaling properties | $E(\|\cdot\|^{2}|\mathcal{F}^{n}_{i})$ | $\|E(\cdot|\mathcal{F}^{n}_{i})\|$
---|---|---
$R(p)^{\prime n}_{i+v}$ | $p^{2}\Delta_{n}^{1+2\delta}$ | $p\Delta_{n}^{1/2+\delta}$
$D(p)^{n}_{i+v}$ | $p\Delta_{n}^{3/2-3\delta}$ | $p\Delta_{n}^{1-2\delta}$
$A(p)^{n}_{i+v}$ | $p^{2}\Delta_{n}^{2-2\delta}\big{(}p\Delta_{n}^{-1/2-\delta}+v\big{)}$ | $p\Delta_{n}^{3/2-\delta}\big{(}p\Delta_{n}^{-1/2-\delta}+v\big{)}$
$\zeta(p)^{n}_{i,h}$ | $p\Delta_{n}^{1-2\delta}$ | $p^{3/2}\Delta_{n}^{1-2\delta}$

Let

$\alpha(p)^{\prime n}_{i,h}=R(p+1)^{\prime
n}_{i+a(n,p,h)}+D(p+1)^{n}_{i+a(n,p,h)}+A(p+1)^{n}_{i+a(n,p,h)}+\zeta(p+1)^{n}_{i,h}$

Given $j,k,l,m=1,\cdots,d$, by Table 3 we have

$\left|E^{n}_{i}\big{(}\beta^{\prime n,jk}_{i}\beta^{\prime
n,lm}_{i}\big{)}-(k_{n}\Delta_{n}^{1/2+\delta})^{-1}\Sigma(c^{n}_{i})^{jk,lm}\right|=\sum_{r=1}^{5}\mu^{\prime
n,r}_{i}+O_{p}\big{(}p(k_{n}^{2}\Delta_{n}^{1+2\delta})^{-1}\big{)}$

where

$\displaystyle\mu^{\prime n,1}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-1}E^{n}_{i}\left(\left|\alpha(p)^{\prime
n,jk}_{i,h}\alpha(p)^{\prime
n,lm}_{i,h}-\zeta(p+1)^{n,jk}_{i,h}\zeta(p+1)^{n,lm}_{i,h}\right|\right)$
$\displaystyle\mu^{\prime n,2}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-2}\sum_{h^{\prime}=h+1}^{m(n,p)-1}\left|E^{n}_{i}\left[\alpha(p)^{\prime
n,jk}_{i,h}\alpha(p)^{\prime n,lm}_{i,h^{\prime}}+\alpha(p)^{\prime
n,lm}_{i,h}\alpha(p)^{\prime n,jk}_{i,h^{\prime}}\right]\right|$
$\displaystyle\mu^{\prime n,3}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(k_{n}-l_{n})^{2}\Delta_{n}^{2}}\sum_{h=0}^{m(n,p)-1}E^{n}_{i}\left(\left|\zeta(p+1)^{n,jk}_{i,h}\zeta(p+1)^{n,lm}_{i,h}\right.\right.$
$\displaystyle\hskip
85.35826pt\left.\left.-\Delta_{n}^{1/2-\delta}\frac{(p+1)\Phi_{00}-\Psi_{00}}{(p+2)\Phi_{00}}\,\Sigma\big{(}c^{n}_{i+a(n,p,h)}\big{)}^{jk,lm}(p+2)l_{n}\Delta_{n}\right|\right)$
$\displaystyle\mu^{\prime n,4}_{i}$ $\displaystyle=$
$\displaystyle\frac{[(p+1)\Phi_{00}-\Psi_{00}]\,l_{n}}{\Phi_{00}(k_{n}-l_{n})^{2}\Delta_{n}^{1/2+\delta}}\sum_{h=0}^{m(n,p)-1}\left|E^{n}_{i}\left[\Sigma\big{(}c^{n}_{i+a(n,p,h)}\big{)}^{jk,lm}-\Sigma(c^{n}_{i})^{jk,lm}\right]\right|$
$\displaystyle\mu^{\prime n,5}_{i}$ $\displaystyle=$