Refined Cramér Type Moderate Deviation Theorems for General Self-normalized Sums with Applications to Dependent Random Variables and Winsorized Mean

Refined general self-normalized moderate deviation

A]Lan<EMAIL_ADDRESS>A, B]Qi-Man Shao[label=t2<EMAIL_ADDRESS>A]Jiasheng Shi[label=e3<EMAIL_ADDRESS>
[A]Department of Statistics, The Chinese University of Hong Kong, e2,e3
[B]Department of Statistics and Data Science, Southern University of Science and Technology, t2

t2The research is partially supported by National Nature Science Foundation of China NSFC 12031005, Shenzhen Outstanding Talents Training Fund, Hong Kong Research Grants Council GRF-14302515 and 14304917.

L. Gao, Q.-M. Shao and J. Shi

Let $\{(X_i, Y_i)\}_{i=1}^n $ be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér type moderate deviation theorem for the general self-normalized sum $\sum_{i=1}^n X_i / (\sum_{i=1}^n Y_i^2)^{1/2}$, which unifies and extends the classical [7] theorem and the self-normalized Cramér type moderate deviation theorems by [20] as well as the further refined version by [31]. The advantage of our result is evidenced through successful applications to weakly  dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramér type moderate deviation theorems for one-dependent random variables, geometrically $\beta$-mixing random variables and causal processes under geometrical moment contraction.  As an additional application, we also derive the Cramér type moderate deviation theorems for self-normalized winsorized mean.

[Primary ]62E17
[; secondary ]62J07

Moderate deviation
Self-normalized sums
Dependent random variables
Winsorized mean

§ INTRODUCTION

Let $ X_1,X_2,\cdots,X_n$ be independent random variables with
$ \e X_i  = 0  $ and $  \ \e  X_i^2   < \infty  \ \mbox{for} \  i \geq 1 $. Set
$B_n^2 = \sum_{i = 1}^n \e  X_i^2 $,

\begin{gather*}
S_n=\sum_{i=1}^{n}X_i, \quad V_n^2=\sum_{i=1}^{n}X_i^2,  \quad \bar{X} = \f 1n  \sum_{i = 1}^n X_i   \\
\mbox{and} \quad \hat{\sigma}_n^2  = \frac { 1}{n - 1 }  \sum_{ i = 1 }^n (X_i - \bar{X})^2 .
\end{gather*}

The self-normalized sum is defined by $ S_n / V_n $ and is closely related to the widely-used Student's $t$ statistic $ t_n  = S_n /( \sqrt{n} \, \hat{\sigma}_n) $ in the sense that

\begin{align*}
\P ( t_n \geq x  ) = \P  ( S_n / V_n \geq  x [ n / ( n + x^2 - 1  ) ]^{1/2} ) .
\end{align*}

Therefore, to investigate the distribution of Student's t statistic is equivalent to consider that of the less complex self-normalized statistic.

The past three decades have witnessed the flourishing development of asymptotic theory for self-normalized sums of independent random variables. Regarding the sufficient and necessary conditions for the self-normalized central limit theorem, we refer to [15] and [27] for independent and identically distributed (i.i.d.) random variables and general non-i.i.d. random variables, respectively. Specifically, for the i.i.d. case, the former paper showed that $ S_n / V_n $ is asymptotically standard normal if and only if the common distribution is in the domain of  attraction of the normal law. Basically, there are two ways to measure the accuracy in normal approximation. One method is the absolute error, which concerns the celebrated Berry-Esseen bound and Edgeworth expansion. See [3], [2] and [32] for Berry-Esseen bounds and [16] for Edgeworth expansion. Another method is the relative error, which estimates the ratio of the tail probabilities, typically including the Cramér type moderate deviations.  [26] established the following self-normalized Cramér type moderate deviation result for i.i.d. random variables. If $ \e | X_1 |^3 < \infty $, then

\begin{align*}
\frac { \P ( S_n > x V_n  ) } {  1 - \Phi (x) }  \longrightarrow 1
\end{align*}

holds uniformly for $ 0 \leq x \leq o ( n^{ 1 / 6  } ) $, where $ \Phi (x) $ is the standard normal distribution function. [20] further extended the result to general independent random variables. In particular, they obtained that if $ \e | X_i |^{ 3 }  < \infty$ for $ i \geq 1  $, then there exists an absolute constant $ A > 0  $, such that

\begin{equation}
\bigg| \f{\P(S_n > x V_n  )}{1-\Phi(x)} - 1 \bigg| \leq  A (1+x)^3 \sum_{i=1}^{n}\e |X_i|^3/B_n^3  \label{classical}
\end{equation}

holds uniformly for $ 0 \leq x \leq  B_n (\sum_{i=1}^{n}\e |X_i|^3)^{ - 1/3}$.

In addition, [31] corrected the skewness in normal approximation and proved that if $ \e | X_i |^3  < \infty $ for $ i \geq 1 $, then there exist positive constants $ A_0 $ and $ C_0   $ such that

\begin{align}
& \frac{\P\big(S_n>xV_n+c B_n \big)}{ [ 1-\Phi(x+c) ] \Psi_x } \label{refinedcase} \\
&\hspace{1cm} =   e^{O_1 \Delta_{n,x}}  \bigg[ 1+ O_2 \Big( (1+x)\sum_{i=1}^{n}\e |X_i|^3 /B_n^3\Big) \bigg], \nn
\end{align}

holds uniformly for $ | c | \leq x / 5 $ and for all $  0 < x \leq  \f 1  3  B_n ( \max_i \e | X_i |^3 )^{ -1/3 } $ and $ x \leq C_0  B_n^3 / \sum_{i = 1}^n \e | X_i |^3 $, where $ | O_1 | \leq A_0 $, $ | O_2 | \leq A_0 $ and

\begin{align*}
\Psi_x&=\exp\bigg[ \gamma^2\Big(\f{4\gamma}{3}-2\Big)x^3 \sum_{i=1}^{n}\e X_i^3 /B_n^3 \bigg],\\
\Delta_{n,x}&= ( 1 + x )^3 B_n^{-3} \sum_{i=1}^{n}\e\big[  |X_i|^3 \mathbbm{1}( (1 + x) |X_i| \geq B_n) \big] \nn \\
& \hspace{0.5cm} + (1 + x)^4 B_n^{-4} \sum_{i=1}^{n}\e\big[|X_i|^4 \mathbbm{1}( ( 1 + x ) |X_i|\leq B_n ) \big] ,
\end{align*}

with $\gamma = \frac 1 2  ( 1 + c  / x ) $. Especially, if $ X_1, \ldots, X_n $ are i.i.d. random variables with $ \e X_1 ^4  < \infty $, then (<ref>) implies there exist positive constants $A_0$ and $C_0$ depending on $ \e X_1^2 $ and $ \e X_1^4 $ such that

\begin{align*}
& \frac { \P ( S_n > x V_n  ) } {  1 - \Phi ( x  )    } =  \exp \left  \{ -   \frac {  x^3 \e [ X_1 ^3 ] } {   3 \sqrt n ( \e [ X_1^2  ] )^{ 3/2 } }  \right\}  \left [   1 +  O_1  \left( \frac {   1 + x     } { \sqrt {n}  }     +  \frac {  ( 1 + x )^4    } { n }  \right) \right],   \label{W2011}
\end{align*}

uniformly in $ 0  < x \leq  C_0  n^{ 1 / 4 }  $, where $ | O_1 | \leq A_0 $.
Observe that in the i.i.d. case, the classical self-normalized Cramér type moderate deviation presented in (<ref>) gives a convergence rate of $ (1 + x )^3 / \sqrt{n} $ and the corresponding range of convergence $ x = o (n^{ 1/6 }) $. Thus, by specifying the skewness correction term $ \Psi_x $, (<ref>) can improve the result of [20] in terms of both the convergence rate and the range of convergence when the higher fourth moments exist.

It is worth mentioning that the moment conditions for self-normalized Cramér type moderate deviation theorems are much weaker than those in the classical theorems for standardized sums. As a result, to account for robustness against heavy-tailed data, the self-normalized sum would be recommended in real-world applications. We refer to [8] for a systematic introduction to the theory and  statistical applications of self-normalized statistics.

Due to its rigorous control on the ratio of tail probabilities, the self-normalized Cramér type moderate deviation has been successfully applied in high-dimensional statistical analysis, including large-scale multiple testing [13, 22, 23], signal detection [9], classification [11] and feature screening [5] among others.

Most of the existing works have focused on the classical self-normalized sum, that is, $ \sum_{i = 1}^n X_i / (\sum_{i = 1}^n X_i^2 )^{1/2} $ for independent random variables $\{X_i\}_{i = 1}^n $. Yet, in some scenarios, the sequence used for normalizing in the denominator could be different from the numerator, which occurs for a variety of commonly used studentized nonlinear statistics such as the studentized U-statistic and the studentized L-statistics. Therefore, investigations into general self-normalized processes beyond the classical form are imperative. [29] attempted to extend the Cramér type moderate deviation theorem to a more general setting, that is, $ ( \sum_{i = 1}^n X_i + D_{1n} ) / ( ( \sum_{i = 1}^n X_i^2 ) (1 + D_{2n})  )^{1/2}$, where the remainders $ D_{1n} $ and $ D_{2n}$ are measurable functions of $ \{X_i\}_{i = 1}^n $ but negligible. Our present work will establish a fundamental framework in Theorem <ref> on the Cramér type moderate deviation for a more general self-normalized form of $ \sum_{i = 1}^n X_i / (\sum_{i = 1}^n Y_i^2 )^{1/2} $, where $ \{(X_i, Y_i)\}_{i = 1}^n $ is a sequence of independent bivariate random vectors and $X_i$ and $ Y_i $ could be different from each other. It is worthwhile to mention that our Theorem <ref> can cover not only the classical Cramér type moderate deviation for standardized sums by [7], but also the self-normalized counterparts by [20] and [31].

Our investigation into the general self-normalized sum is also motivated by seeking to develop sharper self-normalized Cramér type moderate deviation results for weakly dependent random variables. Though Cramér type moderate deviation theory has been well studied for independent random variables, the theory for dependent data remains largely underdeveloped. The biggest challenge is that the classical theory for independent random variables cannot be directly applied due to dependence.  [6] made the first attempt to develop the theory for self-normalized sums of dependent random variables with geometrically decaying dependence. However, their result can be further improved by applying our framework on the general self-normalized sum.

The key observation is that after dividing the weakly dependent random variables into consecutive big blocks and small blocks, its self-normalized sum can be approximated by a general self-normalized sum of independent bivariate random vectors. Therefore, Cramér type moderate deviation theorems for the self-normalized sums of weakly dependent random variables can be established based on our fundamental theory on the general self-normalized sum. More details will be presented in Section <ref>.

The rest of the paper is organized as follows. Our framework on general self-normalized Cramér type moderate deviation is presented in Section <ref>. Section <ref> shows applications to the self-normalized sums of weakly dependent random variables under one-dependence,  geometrically $\beta$-mixing condition, and geometric moment contraction.
Section <ref> presents an additional application to studentized winsorized mean that naturally takes the form of a general self-normalized sum.

Section <ref> is devoted to proofs of the theorems in Sections <ref>–<ref>. Other technical proof details  are included in the Supplementary Material.

§ MAIN RESULTS

Let $(X_1, Y_1), (X_2, Y_2), \ldots, (X_n, Y_n)$ be independent bivariate random vectors satisfying

\begin{equation}
\e X_i=0 \;\; {\rm for}\;\; i \geq 1 \quad   {\rm and}\quad \sum_{i=1}^n \e X_i^2= 1 = \sum_{i=1}^n \e{Y_i}^2. \label{maincon1}
\end{equation}

We remark that for the convenience of presentation, $\{ X_i\} $ and $\{Y_i\}$ are standardized so $\sum_{i=1}^n
\e X_i^2 =1= \sum_{i=1}^n \e{Y_i}^2$. In other words, one should think of $X_i$ as $X_{n,i}$ and similarly $Y_i$ as $ Y_{n, i} $.

\begin{equation}
S_n = \sum_{i=1}^n X_i ,\quad V_n^2 = \sum_{i=1}^n Y_i^2 \quad {\rm and} \quad T_n=\f{S_n}{V_n}.
\label{form}
\end{equation}

We first propose an exponential moment condition as follows.
Suppose there exists some constant $ c_0  \geq  0 $ such that for $x > 0 $ satisfying (<ref>),

\begin{equation}
\e e^{ \min\left\{\frac{X_i^2}{ Y_i^2+c_0 \e Y_i^2 }, \, 2 x X_i \right\} } < \infty.  \label{maincon2}
\end{equation}

The above moment condition links $ X_i $ with $ Y_i $ and shows how they interact with each other. In particular, it is automatically satisfied for the classical self-normalized sum with $Y_i=X_i$.

The following notations  will be used throughout the paper. Define

\begin{align}
L_{3,n} = \sum_{i=1}^n \big( \e |X_i|^3 + \e |Y_i|^3 \big),  \qquad \qquad \qquad \qquad \qquad \qquad  \label{Def}
\end{align}

\begin{align*}
\de_{x,i} &= (1 + x )^3 \Big( \e \big[ |X_i  |^3 \mathbbm{1} (  | ( 1 + x ) X_i|> 1 )\big] + \e \big[  |Y_i  |^3 \mathbbm{1} ( | ( 1 + x  ) Y_i|>1 ) \big] \Big) \nn \\
&~~ + ( 1 + x )^4 \Big( \e \big[ |X_i |^4 \mathbbm{1} (  | ( 1 + x  ) X_i| \leq  1 )\big] + \e \big[  |Y_i  |^4 \mathbbm{1} ( | ( 1 + x  ) Y_i| \leq 1 ) \big] \Big), \nn \\
r_{x,i} &= \e\left[ \exp \Big\{  \min\Big( \frac{X_i^2}{ Y_i^2+c_0 \e Y_i^2 }, 2xX_i \Big) \Big\} \mathbbm{1} ( | ( 1 + x ) X_i|>1 )\right],  \nn \\
R_{x,i} & = \de_{x,i} + r_{x,i},  \quad \de_x =\sum_{i=1}^n \de_{x,i},  \quad  r_x=\sum_{i=1}^n r_{x,i}, \quad \mbox{and} \quad  R_x = \delta_x + r_x. \nn
\end{align*}

Assume (<ref>) and (<ref>) are satisfied. In addition, $ \e | X_i |^3 < \infty $ and $ \e | Y_i |^3 < \infty $ for $ i \geq 1 $. Then there exist absolute positive constants $ 0 < c_1\leq 1/4 $ and $ A > 0  $ such that

\begin{equation}
\P  (S_n \ge x V_n+c  )  =  [ 1 - \Phi (x+c) ] \, \Psi_x^* \, e^{ O_1 R_x } ( 1+O_2 (1+x) L_{3,n} ),
\label{main-re}
\end{equation}

\begin{equation*}
\Psi_x^*=\exp \Big\{  x^3 \Big(  \f43 \ga^3 \sum_{i=1}^n \e X_i^3 - 2 \ga^2 \sum_{i=1}^n \e [X_i Y_i^2] \Big) \Big\} \quad {\rm and}\quad \ga = \f 12 ( 1 + \f cx ),
\end{equation*}

uniformly for $ |c| \le x/5 $ and for all $ x > 0 $ satisfying

\begin{gather}
(1 + x ) L_{3,n} \le c_1 , \qquad x^{ -2  } R_x  \le c_1 ,  \label{G1}\\
\mb{and} \qquad x \le {\f{1}{4} \land \f{1}{2\sqrt{c_0}}\over \big[  \max_i (  \e|X_i|^3 +\e|Y_i|^3 ) \big]^{1/3}}, \quad \max_i r_{x,i} \le c_1 , \label{G2}
\end{gather}

where $|O_1| \leq A $ and $ | O_2  |  \leq A $.

Theorem <ref> unifies the classical standard and self-normalized Cramér type moderate deviation theorems as well as the refined version by [31]. The assumption (<ref>)

is satisfied for a wide class of statistics, including the block sums of weakly dependent random variables and the self-normalized winsorized mean. More details will be provided in the proof of Theorems <ref>–<ref> and <ref>.

The following corollary is a straightforward application of Theorem <ref> to the classical standardized sum of independent random variables. The proof will be given in Section <ref> in the Supplementary Material.

Let $ X_1, \ldots, X_n$ be i.i.d. random variables with $ \e X_i = 0 $ and $   \e [ X_i^2  ]  = \sigma^2 $. Denote $ S_n = \sum_{i = 1}^n X_i $. If there exists a positive constant $t_0  $ such that $ \e e^{ t X_1 } < \infty $ for $ | t  | \leq t_0 $, then we have

\begin{align}
\frac { \P ( S_n > x  \sigma \sqrt n  ) } { 1 - \Phi (x) }  = \exp \Big\{ \frac { x^3   \e X_1^3 } { 6 \sigma^3 \sqrt{n}  } \Big \} \Big[   1 + O \Big(\frac { (1 + x)^4   } {  n  } + \frac   { 1 + x  } { \sqrt n  }\Big) \Big] \label{classical-sum}
\end{align}

holds uniformly for $ 0 < x \leq   O ( n^{1/4})  $.

§ APPLICATIONS TO WEAKLY DEPENDENT RANDOM VARIABLES

The Cramér type moderate deviation theory has been well studied for independent random variables, yet there are few results available for dependent data. The novel work of [6] made the first attempt to develop the theory for self-normalized sums of weakly dependent random variables satisfying the geometrically $\beta$-mixing condition or geometric moment contraction. In this section, we will further improve their results by applying our fundamental framework on the general self-normalized sum. Before that, we will start with a Cramér type moderate deviation theorem for self-normalized sums of one-dependent random variables. The reason to first investigate under one-dependence is two-fold. First, one-dependence is the simplest scenario of dependency and the result for one-dependent random variables can be applied to  $m$-dependence, where $m$ could also go to infinity. Second, many weakly dependent random variables can be approximated by some one-dependent random variables. A typical example includes the block sums of random variables satisfying geometric moment contraction, which will be presented in Section <ref>. Therefore, the following Theorem <ref> under one-dependence lays the foundation for Theorem <ref> under geometric moment contraction.

§.§ Cramér type moderate deviation under one-dependence

Let $\xi_1, \xi_2, \ldots$ be one-dependent random variables, which means for $ i, j \geq 1 $, $ \xi_i $ is independent of $ \xi_j $ if $ | j - i | \geq 2$. Put

\begin{equation}
S_n = \sum_{i=1} ^{n} \xi_i, \quad  V_n^2 = \sum_{i=1}^n \xi_i^2, \quad  \rho_n = \frac{\sum_{i=1}^{n-1} \e \xi_i\xi_{i+1} }{ \sum_{i=1}^{n} \e \xi_i^2  }. \label{notationonedependentbasic}
\end{equation}

Note that $  | \rho_n | \leq 1/2 $. Moreover, in many applications where weakly dependent sequence can be approximated by some one-dependent random variables, the covariances $ \e \xi_i \xi_{i + 1} $ are negligible compared to the variables $ \e \xi_i^2$ due to weak dependence, hence  $\rho_n \to 0 $ as $n \to \infty $. Therefore, $ \rho_n $ can be moved to the remainders and the limiting distribution will still be standard normal. One can find more details in the proof of Theorem <ref>, which obtains Cramér type moderate deviation result under geometric moment contraction by applying Theorem <ref>.

Under existence of the fourth moment, we have the following theorem for self-normalized sums of one-dependent random variables.

Assume that $ \e \xi_i = 0 $, $ \e  \xi_i ^4 \le a_1^4 $ and $ \e  \xi_i^2 \ge a_2^2 $ for $1 \leq i \leq n$ and $    \rho_n \geq \rho $ for some $ \rho > - 1/2  $. Denote $ a= a_1/a_2 $. Then there exist positive numbers  $a_0$ and $A(\rho)$ depending on $ \rho $ such that

\begin{equation}
\P (S_n > x V_n) = \Big[ 1 - \Phi \big( \f{x}{\sqrt{1+2\rho_n}} \big) \Big] \Big( 1 + O_1 a^4 \f{(1+x)^2}{n^{1/4}} \Big)
\label{re-dep}
\end{equation}

holds uniformly for $ x \in ( 0, a_0 a^{-2}  n^{\f18} ) $, where $|O_1|\leq A(\rho)$.

The proof of Theorem <ref> relies on the big-block-small-block technique and an application of Theorem <ref>. The main idea is to approximate the self-normalized sum of one-dependent random variables by a general self-normalized sum of independent random vectors based on the big blocks. In more details, let the length of big blocks be $ l = [n^{\alpha}] $ for $ 0 < \alpha < 1 $, where $ [x] $  denotes the integer part of $ x $ for any $x > 0$, and the length of small blocks be only 1. Denote $  k = [n / (l + 1)] $. For $ 1 \leq j \leq k $, we define the $j$-th big block by

\begin{equation}
H_j = \{i: (j - 1) (l + 1) + 1 \leq i \leq j (l +1) -1 \}
\end{equation}

and the sums over $j$-th big block by

\begin{equation}
X_j = \sum_{i \in H_j} \xi_i ~~\mbox{and}~~ Y_j^2 = \sum_{ i \in H_j } \xi_i^2.
\end{equation}

Observe that by this construction, the big-block sums $ \{ X_j  \}_{j = 1}^{ k} $ and $ \{ Y_j  \}_{j = 1 }^k $ are both sequences of independent random variables, because $ \{\xi_i \}_{  i = 1 }^n  $ are one-dependent and the adjacent big blocks $ H_j $ and $H_{j+1 } $ are separated by a random variable $ \xi_{j (l + 1)}$. Since the big blocks contain $ (1 - n^{ - \alpha}) $ proportion of the random variables in $ \{\xi_i \}_{i = 1}^n   $, we can approximate the self-normalized sum $ S_n / V_n  $ of $ \{ \xi_i  \}_{ i = 1}^n $ by the general self-normalized sum $ \sum_{j = 1}^k X_j / ( \sum_{j = 1}^k Y_j^2 )^{1/2} $.  The crucial quantity $r_{x,j}$ can be separated into two self-normalized sums of independent random variables due to one-dependence and thus can be bounded by using Lemma A.5 in the Supplementary Material.
Therefore, Theorem <ref> can be proved by applying Theorem <ref> and calculating the error terms involved. More details of the proof will be provided in Section <ref>.

Compared with the classical result (<ref>) for independent data, one-dependence results in a narrower zone of convergence and a slower convergence rate. Moreover, (<ref>) can be easily extended to general $m$-dependent random variables, where $m$ could depend on $n$ and go to infinity. Indeed, if $ Z_1, \ldots, Z_n  $ are $ m  $-dependent and suppose $ b =  n / m $ is an integer for simplicity, we define $ \xi_j = \sum_{i = 1 + ( j - 1 ) m }^{ j m  }  Z_i $ for $ 1 \leq j \leq b $, then $ \xi_1, \ldots, \xi_b $ are one-dependent random variables and Theorem <ref> can be applied.

§.§ Cramér type moderate deviation under $ \beta $-mixing

In time series, asymptotic independence conditions such as mixing conditions are usually proposed to replace independence, among which $ \beta $-mixing is an important dependent structure and has been connected with a large class of time series models including ARMA models, GARCH models and certain Markov processes. This subsection provides a Cramér type moderate deviation theorem for block-normalized sums of geometrically $\beta$-mixing random variables, which improves the result by [6].

Let $ \{  X_i  \}_{i = 1}^n $ be a sequence of random variables. Let $ \sigma_{ - \infty }^{t }    $ and $ \sigma_{ t + m }^{ \infty } $ be $ \sigma $-fields generated by $ \{ X_i \}_{ 1 \leq  i \leq t} $ and $ \{ X_i \}_{ i \geq t + m } $, respectively. The $\beta$-mixing coefficient is given by

\begin{align}
\beta( m ) := \sup \limits_{t} \e \sup \{  | \P ( B | \sigma_{ - \infty }^t  )  - \P (B) | : B \in \sigma_{ t + m   }^{ \infty }      \}.   \label{beta-coeff}
\end{align}

We say $ \{  X_i  \}_{ i \geq 1}  $ is geometrically $\beta$-mixing if $\beta (m) $ admits an exponentially decaying rate, that is, there exist positive numbers $ a_1 $, $ a_2 $ and $ \tau $ such that

\begin{equation}
\beta ( m ) \leq a_1 e^{ - a_2 m^{ \tau } }.     \label{g-beta}
\end{equation}

To account for dependence, the block technique is naturally used to estimate the variance of sums of dependent random variables (see [6]). Set $ l = [ n^{\alpha} ] +1 $ for $ 0 < \alpha < 1  $ and $ k = [ n / l ] $ . For $ 1 \leq j \leq k  $, define the $ j $-th block and corresponding $ j $-th block sum by

\begin{align}
H_j = \{ i: l (j-1) + 1 \leq i \le  l j  \}   \quad  \mbox{and} \quad  Y_j = \sum_{ i \in H_j } X_i, \label{blocksum}
\end{align}

respectively. The block-normalized sum is then defined by

\begin{equation}
T_k  = \frac {  \sum_{ j = 1 }^k Y_j  }   { \sqrt{ \sum_{ j = 1 }^k Y_j^2 } }.  \label{block-selfsum}
\end{equation}

Let $ \{ X_i\}_{i = 1}^n $ be a $\beta$-mixing sequence satisfying (<ref>). Assume $ \e  X_i  = 0 $ and there exist positive numbers $ \mu_1  $ and $ \mu_2 $ such that $ \e |X_i|^{r} \le \mu_1^{r} $ for $r > 4$ and $ \e (\sum_{i= s }^{  s + t } X_i )^2 \ge \mu_2^2 t $ for all $i\geq 1, s \geq 0 , t \geq 1 $. Then, for $ 0 < \alpha < 1 $ and $ \tau > 0 $, there exist positive numbers $ A $ and $ d_0 $ depending on $a_1, a_2, \mu_1, \mu_2, \alpha $, $ \tau $ and $r$ such that,

\begin{equation}
\Big | \frac {  \P ( T_k \geq x  ) }  {   1- \Phi(x)  } - 1  \Big | \leq   A  \Big(    \f{ ( 1 + x) ^2  }{ n^{\alpha }  } + \f{ ( 1 + x )^2  \log n } {n^{  \min\{(1 - \alpha)/4, \, \alpha \tau/2 \} }}  \Big)  \label{blocksum1}
\end{equation}

uniformly in $ 0 \leq x \leq  d_0 \min \{   n^{ \alpha / 2  }, ( \log n  )^{ - 1/2  } n^{  \min\{  ( 1 - \alpha ) / 8,\, \alpha \tau / 4 \} }  \}  $.

As for the choice of $ \alpha $, for any given $ \tau $, we can always choose $ \alpha $ such that $  1 - \alpha  \le  2 \alpha \tau $, that is, $ \alpha \ge  \f{1}{1+2\tau} $. To optimize the convergence rate and the range of $x$ in (<ref>),

(i) when $ \tau \ge 2  $, let $ \alpha = \f15 $, then

\begin{equation}
\f{ \P ( T_k \ge x ) }{ 1 - \Phi(x) } = 1 + O \Big( \f{( 1 + x )^2 \log n }{ n^{1/5} }  \Big)  \label{sce1}
\end{equation}

uniformly for $  x \in (0, d_1 (\log n)^{-1/2} n^{1/10} )$;

(ii) when $ \tau < 2  $, let $ \alpha =\f{1}{1+2\tau} $, then

\begin{equation}
\f{ \P ( T_k \ge x ) }{ 1 - \Phi(x) } = 1 + O \Big( \f{( 1 + x )^2 \log n }{ n^{\f{\tau}{2(1+2\tau)} }} \Big) \label{sce2}
\end{equation}

uniformly for $  x \in (0, d_1  (\log n)^{-1/2} n^{\f{\tau}{4(1+2\tau)} } )$.

We now compare our result with Theorem 4.2 in [6]. They proved that given $ \e |X_i|^{r} < \infty $ for $r > 3$ and the same assumption (<ref>),

\begin{align}
\Big|\f{ \P ( T_k \ge x ) }{ 1 - \Phi(x) } - 1 \Big| \leq A \Big[ \frac { ( 1  + x)^2  }{ n^{\alpha} } + \frac { (1 + x)^{5/4} } {n^{ (1 - \alpha ) /8  }   } \Big] \label{remark}
\end{align}

uniformly in $  0 \leq x \leq  d_0 ( \min \{ ( \log n )^{ - 4/5 } n^{( 1 - \alpha )/10 }, n^{ \alpha \tau  /2 }, n^{\alpha /2 }\} )  $. (We have to mention that the original version of Theorem 4.2 in [6] missed the error term $ \frac { (1 + x)^2 } { n^{\alpha} } $ and the corresponding condition $ x \leq d_0 n^{\alpha / 2 } $.)
When $ (1 - \alpha) > 2 \alpha \tau  $, our results might be worse for some choices of $\alpha$. However,  when $ (1 - \alpha ) \leq 2 \alpha \tau $, our results improve theirs in terms of both the convergence rate and the corresponding range of $x$.
The improvement is achieved by applying our framework for general self-normalized sum and correcting the bias to normal approximation by specifying the skewness term $ \Psi_x^*  $.

The proof of Theorem <ref> again builds on the big-block-small-block technique. Recall that for $ 1 \leq j \leq k $, $ Y_j = \sum_{i \in H_j} X_i $ is a block sum defined in (<ref>). We first apply big-block-small-block technique to separate the sequence $ \{ Y_j  \}_{j = 1}^k  $ into consecutive big blocks and small blocks. Let the size of big-blocks be $ m_1 = [ n^{\alpha_1} ] $ for $ 0 < \alpha_1 < 1 - \alpha $ and the size of small-blocks be only $1$. Denote $ k_1 = [ k / ( m_1 +1 ) ]   $. For $ 1 \leq u \leq k_1 $, define the $u$-th big block by

\begin{align}
I_u = \{ j: (m_1 + 1 ) ( u - 1 ) + 1 \leq j \leq (m_1 + 1 ) u - 1  \},
\end{align}

and the sums over $u$-th big block by

\begin{align}
\zeta_u = \sum_{j \in I_u} Y_j, \quad \eta_u^2 = \sum_{j \in I_u} Y_j^2.
\end{align}

Then the self-normalized sum $ T_k = \sum_{j = 1}^k Y_j / (\sum_{j = 1}^k Y_j^2) ^{1/2}$ can be approximated by the general self-normalized sum $ \sum_{u = 1}^{k_1} \zeta_u / (\sum_{u = 1}^{k_1} \eta_u^2)^{1/2} $ constructed on the big blocks. However, unlike the big-block sums under one-dependence, the big-block sums $ \{ (\zeta_u, \eta_u)   \}_{u = 1}^{k_1} $ are not independent under $ \beta $-mixing assumption in (<ref>), but weakly dependent. Therefore, Theorem <ref> can not be directly applied. Note that the adjacent random vectors $ ( \zeta_u, \eta_u ) $ and $ (\zeta_{u+1}, \eta_{u+1} )$ depend on $  \{Y_j\}_{  j = (m_1 + 1 ) (u - 1)+ 1 }^{  (m_1 + 1 ) u - 1 } $ and $  \{Y_j \}_{  (m_1 + 1 ) u + 1  }^{(m_1 + 1 ) (u + 1) - 1 }$, respectively. Since $ Y_j $ defined in (<ref>) is a block sum of $ \{X_i\} $ and by the $ \beta$-mixing assumption on $ \{X_i\} $, we can see the $\beta$-mixing dependence coefficient between $ ( \zeta_u, \eta_u )  $ and $  (\zeta_{u+1}, \eta_{u+1} ) $ is bounded by $ O(  e^{ - a_2 n^{ \alpha \tau} } ) $, which converges to 0 as $ n \to \infty $. According to Lemma <ref> [4] presented in Section <ref>, the weakly dependent random vectors $ \{ (\zeta_u, \eta_u)\}_{u = 1}^{k_1} $ can be replaced with independent random vectors $ \{ ( \tilde{\zeta}_u, \tilde{\eta}_u ) \}_{u = 1}^{k_1} $ that have the same marginal distributions, with probability $ 1 - O (k_1 e^{ - a_2 n^{\alpha \tau}}) $. Moreover, the crucial quantity $r_{x,u}$ can be approximated by two self-normalized sums of independent random variables due to the $\beta$-mixing assumption (see Lemma 5.3) and the block technique. Consequently, our main result in Theorem <ref> can be applied to the general self-normalized sum $ \sum_{u = 1}^{k_1} \tilde{\zeta}_u / (\sum_{u = 1}^{k_1} \tilde{\eta}_u^2)^{1/2} $. Detailed proof will be given in Section <ref>.

§.§ Cramér type moderate deviation for causal processes under geometric moment contraction (GMC)

The GMC (see [35], [17] and [33, 34]) is satisfied by many non-linear time series models including various GARCH models that are commonly used in statistics, econometrics and engineering. In this subsection, we present a Cramér type moderate deviation theorem for block normalized sums of random variables satisfying GMC.

Let $ \{ \varepsilon_t\}_{ t \in \mathbbm{Z}  } $ be i.i.d. random variables and define $ \sigma $-fields $  \mathscr{F}_t = \sigma ( \ldots, \varepsilon_{t-1}, \varepsilon_t )  $. Suppose that $ \{ X_i = G_i ( \mathscr{F}_i ) \}_{i \geq 1} $ is a causal process with $ G_i ( \cdot )  $ being a measurable function such that $ X_i $ is well-defined. Let $ \{ \varepsilon_{t}^*\}_{ t \in \mathbbm{Z} } $ be an independent copy of $ \{ \varepsilon_{t} \}_{ t \in \mathbbm{Z} } $ and we similarly define $  \mathscr{F}_t^* = \sigma ( \ldots, \varepsilon_{t-1}^*, \varepsilon_t^* )   $.

(GMC). Assume that $ \e  | X_i |^r  < \infty $ for all $ i  \geq 1  $ with $ r > 2 $. Define the functional dependence measure by

\begin{equation}
\Delta_r ( n  ) =  \sup \limits_i  \Vert X_i - G_i ( \mathscr{F}_{i - n}^*, \varepsilon_{ i - n + 1 }, \ldots, \varepsilon_{ i }  ) \Vert_r, \label{Delta_n}
\end{equation}

where $ \Vert \cdot  \Vert_r  = (  \e  | \cdot |^r   )^{ 1 / r } $. We say $ \{ X_i  \}_{i \geq 1} $ satisfies GMC if there exist positive constants $ a_1  $, $ a_2   $ and $ 0<\tau\leq 1  $ such that

\begin{equation}
\Delta_r ( n  )  \leq   a_1 e^{ - a_2 n^{\tau} } .    \label{GMC}
\end{equation}

Note that the GMC property (<ref>) implies $ \{ X_i  \}_{  i \geq 1 } $ forgets the past $ \mathscr{F}_{0} = \sigma  ( \ldots, \vp_{-1}, \vp_0 ) $ geometrically fast.

Define another functional dependence measure as

\begin{equation}
\theta_r ( n ) =  \sup \limits_i  \Vert  X_ i -  G_i ( \ldots, \veps_{i - n - 2}, \veps_{i - n - 1}, \veps_{i - n }^*, \veps_{i - n + 1} , \ldots, \veps_{ i } )  \Vert_r.   \label{theta_r}
\end{equation}

The property (<ref>) is equivalent to $ \theta_r (n) \leq    a_1' e ^{ - a_2' n^{\tau} }   $ for some positive constants $ a_1' $ and $ a_2' $.

\begin{align*}
\Delta_r ( n ) & \leq \sup_i \Vert X_i - G_t(\ldots,\epsilon_{i-n-1},\varepsilon_{i-n}^*,\varepsilon_{i-n+1},\ldots,\varepsilon_i)\parallel_r \nn\\
& \quad +\sup_i  \Vert G_i (\ldots,\varepsilon_{i-n-1},\varepsilon_{i-n}^*,\varepsilon_{i-n+1},\ldots,\epsilon_i )\nn \\
&\qquad \qquad \; -G_i (\mathscr{F}_{i-n}^*,\varepsilon_{i-n+1},\ldots,\varepsilon_i )\parallel_r \nn\\
& = \theta_r ( n ) + \sup_i \parallel X_i - G_i (\mathscr{F}_{i-n-1}^*,\epsilon_{i-n},\ldots,\epsilon_i )\parallel_r \leq \sum_{ m = n }^{\infty} \theta_r (m),
\end{align*}

and in the same manner,

\begin{equation*}
\theta_r (n) \leq \Delta_r ( n ) + \Delta_r ( n + 1  ).
\end{equation*}

Observing that $ \int_{n }^{\infty} e^{ - a_2 x^{ \tau } }d x  \leq A e^{ - a_2 n^{ \tau } /2  }   $ for some constant $ A $, it is easy to see (<ref>) is equivalent to $ \theta_r (n) \leq a_1' e^{ - a_2' n^{\tau} }   $.

Now we assume $\{ X_i \}_{i = 1}^n $ is a sequence of random variables with

\begin{equation}
\e X_i =0\;,\quad \quad \e|X_i|^4 < \infty\;, \label{GMCc1}
\end{equation}

for all $i\geq 1$. Write $S_{k,m}=\sum_{i=k+1}^{k+m}X_i$. Assume there exists a positive number $ \omega_1$ such that for any $ k \geq 0, m \geq 1 $,

\begin{equation}
\e(S_{k,m}^2)\geq  \omega_1^2 m.    \label{GMCc2}
\end{equation}

As with the procedure for $\beta$-mixing random variables, we construct the block-normalized sum for random variables satisfying GMC. Let the block size $ m= [ n ^{\alpha}  ] $ for $0<\alpha<1$ and $k= [ n / m ] $. For $ 1 \leq j \leq k  $, define the $ j $-th block and the $ j $-th block sum by

\begin{equation*}
H_j= \{ i : m(j-1) + 1 \leq i \leq m j \} \quad \mb{and} \quad Y_j = \sum_{ i \in H_j } X_i.
\end{equation*}

The block-normalized sum is then given by

\begin{equation}
T_k = \f { \sum_{ j = 1 }^k Y_j } { \sqrt{ \sum_{ j =1 }^k Y_j^2  } }.
\end{equation}

Assume $ \{ X_i\}_{  i = 1 }^n $ is a causal process satisfying (<ref>),  (<ref>) and (<ref>). Then we have for $ 0 < \alpha < 1 $ and $ \tau > 0   $, there exist positive numbers $ A $ and $ d_0 $ depending on $ a_1, a_2, \omega_1 , \alpha $ and $ \tau $ such that

\begin{align}
\P ( T_k \geq  x  ) =   [ 1 - \Phi (x) ] \Big( 1 + O_1 ( \frac { 1 + x^2 } { n^{ \alpha } } + \frac { 1 + x^2  } { n^{ (1 - \alpha ) / 4  } } ) \Big) \label{GMC_result}
\end{align}

uniformly in $ 0 \leq x \leq d_0 \min \{    n^{ \alpha \tau / 2  } ,n^{ \alpha / 2 },  n^{ ( 1 - \alpha  )/ 8 } \} $, where $ | O_1 |  \leq A $.

Corollary 4.3 in [6] stated that (<ref>) also holds for the self-normalized block sum of GMC random variables. Compared with their result, our convergence rate and the associated converging range of $x$ significantly improve theirs.

The main idea of our proof for Theorem <ref> is to approximate $ \{Y_j \}_{j = 1 }^k $ by one-dependent random variables and then apply Theorem <ref>. Define

\begin{equation}
\tilde{Y}_j  = \e ( Y_j | \varepsilon_{l}, m(j - 2) + 1 \leq l \leq m j )
\end{equation}

\begin{equation}
\tilde{T}_k = \frac {  \sum_{j = 1}^k \tilde{Y}_j } { ( \sum_{j = 1}^k \tilde{Y}_j^2 )^{1/2} },
\end{equation}

where $ m = [n^{\alpha}] $. Since $ Y_j  = \sum_{i \in H_j } X_i$ belongs to $ \mathscr{F}_{mj} $ and depends weakly on $ \mathscr{F}_{m (j - 2)} $ by the GMC assumption (<ref>), it is intuitive that $ \tilde{Y}_j $ is close to $Y_j$. In particular, we can prove $ \Vert \tilde{Y}_j - Y_j \Vert_r  \leq  a_1 m e^{ - a_2 m \tau} $. Therefore, the self-normalized sum $ T_k $ of $ \{Y_j\}_{j = 1}^k  $ can be well approximated by the self-normalized sum $ \tilde{T}_k $ of $ \{\tilde{Y}_j \}_{j = 1}^k $. Moreover, since $ \{\varepsilon_t \}_{t \in \mathbbm{Z}} $ are i.i.d. random variables, it is easy to see $ \{\tilde{Y}_j \}_{j = 1}^k $ are one-dependent. Consequently,  Theorem <ref> can be proved by applying Theorem <ref> and controlling the errors caused by the approximation by one-dependent random variables. We will present the detailed proof in Section <ref>.

§ APPLICATION TO HIGH-DIMENSIONAL STATISTICAL INFERENCE PROBLEM

One of the core issues in analyzing high-dimensional sparse linear regression models is to adjust the tuning parameters coming along with the estimating methods. The problem has long attracted tremendous attention, and great effort has been made towards standard methods of tackling high-dimensional statistical inference problems such as LASSO and a pivotal modification version introduced in <cit.>, called the LASSO$^{1/2}$ (Square-root LASSO). LASSO$^{1/2}$ stands out for its computational efficiency in high-dimensional statistic inference problem due to its pivotal nature in the sense that LASSO$^{1/2}$ neither relies on knowledge of the standard deviation $\sigma$ nor does it need to pre-estimate $\sigma$, and thereby has been widely accepted and successfully adopted in applications like post-selection inference problems (<cit.>, see also <cit.> and <cit.>) and de-biased LASSO estimation (<cit.>, <cit.>).

In this section,  under the setting of random design high-dimensional sparse linear regression models, we make a recommendation on the choice of tuning parameter $\lambda$, which leads the estimation noise to be dominated by the penalty level with a high probability (e.g, $1-\alpha$), and obtain an asymptotic bound on the estimation error accordingly. Throughout this chapter, we allow $\alpha$ to depend on $n$ and possibly go to $0$.

Here, compared with <cit.>, whose discussion focused on fixed design settings with i.i.d errors, we emphasize that our result is not a direct corollary of it in the sense that we reduced the bounded condition required (Condition SN in Supplementary appendix of <cit.>), and derived the results with a uniform rate (i.e., uniform in design matrix $X$) in the random design case. Hence, we are able to handle the heteroscedasticity across the error term without pre-estimating the variance when all terms of the random design matrix are homoscedastic. Last but not least, we extended the range for dimension $p=p(n)$ when the confidence level $\alpha=\alpha(n)\rightarrow 0$, i.e., allows a wider range for $p$ as $\log(p/\alpha)=o(n^{1/3})$ when errors are i.i.d, which is so far the largest possible known  range.

§.§ Choice of tuning parameter in LASSO$^{1/2}$ method in random design case

Consider a fixed design high-dimensional sparse linear regression model formed by outcome $y_i$ and $p$-dimensional regressor $x_i$,

\begin{equation*}
y_i = x_i' \beta_0 + \epsilon_i, \quad i=1,2,\cdots,n,
\end{equation*}

with noise $\epsilon_i\sim F_i$ being independent random variables, denote

\begin{equation*}
\e (\epsilon_{i}) = 0 ,\quad \e (\epsilon_{i}^2) =\sigma_{\epsilon_i}^2.
\end{equation*}

We have $\beta_0\in\mathbb{R}^P$ being the parameter of interests and $\sigma_{\epsilon_i}^2>0$ being the unknown standard deviation. The regressors $x_i=(x_{ij},j=1,\cdots,p)'\in \mathbb{R}^P$, by denoting $y=(y_1,\cdots,y_n)'$, $X=(x_1,\cdots,x_n)'\in \mathcal{M}_{n\times p}$ being independent of the noise $\epsilon=(\epsilon_{1},\cdots,\epsilon_{n})'$, we have,

\begin{equation}
y=X\beta_0+ \epsilon. \label{model}
\end{equation}

For each item $x_{ij}$ of the design matrix, we denote its mean and second moment as $\mu_{x_{ij}}$ and $m_{ x_{ij}}^2$ respectively.

The LASSO$^{1/2}$ estimator to this regression problem, denoted as $\hat{\beta}$, is defined as $\hat{\beta}$, which satisfies

\begin{equation}
\hat{\beta} \in \arg \min_{\beta\in \mathbb{R}^p} \{\hat{Q}(\beta)\}^{1/2}+\frac{\lambda}{n}\Vert \beta \Vert_1,
\end{equation}

\begin{equation*}
\hat{Q}(\beta)=\frac{1}{n} \sum_{ i=1 }^n (y_i-x_i'\beta)^2 \quad {\rm and}\quad \Vert \beta \Vert_1 = \sum_{ j=1 }^p |\beta_j|.
\end{equation*}

A rather subtle issue is adjusting the tuning parameter, which usually associates with the penalty level in high-dimensional statistical methods. This is a tradeoff between the estimation bias and the estimation noise as well as the computational efficiency. One commonly accepted criterion or principle in adjusting the tuning parameter is that we want it to be small enough to control the estimation bias while large enough that the estimation noise is dominated by the penalty term and thereby maintain efficiency. As the score $\tilde{S}$, which is the gradient of the loss function evaluated at the true parameter value $\beta=\beta_0$, summarizes the estimation noise for LASSO$^{1/2}$,

\begin{equation}
\tilde{S} := \bigtriangledown \hat{Q}^{1/2}(\beta_0)= \frac{\bigtriangledown \hat{Q}(\beta_0)}{2\{ \hat{Q}(\beta_0)  \}^{1/2}} = \frac{\frac{1}{n}\sum_{ i=1 }^n x_i \epsilon_{i}}{\left( \frac{1}{n}\sum_{ i=1 }^n \epsilon_{i}^2  \right)^{1/2}} \in \mathbb{R}^p,
\end{equation}

so an intuitive choice would be setting the tuning parameter $\lambda/n$ at the smallest penalty level that dominates the estimation noise, i.e, choose $\lambda$ such that

\begin{equation}
\lambda \geq c \Lambda, \quad {\rm where}\quad \Lambda=n \Vert \tilde{S} \Vert_{\infty},
\end{equation}

with a high probability (e.g, $1-\alpha$), where $c>1$ is a theoretical constant of an instructive work by <cit.>, which was initially proposed for the choice of the penalty level of LASSO. We present the condition needed first.

* Noise obeys $\e (|\epsilon_{i}|^3)<+\infty$ for $1\leq i\leq n$.

* Dimension $p$ associates with confidence level $\alpha$ satisfy that $\log(p/\alpha)=o(n^{1/3}/\ell^{2/3})$ for constant $\ell$ defined in (<ref>).

* Design matrix $X$ is independent of the noise $\epsilon$ with independent, homoscedastic and sub-gaussian rows $\{ x_i \}_{1\leq i\leq n}$, i.e.,

\begin{gather}
\e x_{ij}^2 \triangleq m_{x_{ij}}^2 = 1, \;\; {\rm and}\;\;  \e e^{x_{ij}^2 /m_{ij}^2 } = \e e^{x_{ij}^2} \leq K_1\;\; {\rm for\; some\; } K_1.
\end{gather}

To illustrate the dimension condition $(ii)$, for instance, if $ (x_{ij},\epsilon_i) $ are i.i.d or bounded random variables for $ i = 1, \ldots, n $, we have $\ell$ is bounded by an absolute constant that does not depend on $n$, then $(ii)$ of condition (<ref>) simply becomes $\log(p/\alpha)=o(n^{1/3})$, which is so far the best possible range for $p$ for a similar result to the best of our knowledge.

If Condition <ref> is satisfied. For taking choice of $\lambda=c\sqrt{n}\Phi^{-1}(1-\alpha/2p)=c\Lambda(1-\alpha)$, with probability at least $1-\alpha-o(\alpha)$, we have $\lambda\geq c\Lambda$. Additionally,

\begin{equation}
\Lambda_F(1-\alpha)\leq \Lambda(1-\alpha)(1+o(1)), \label{optimalfixHD}
\end{equation}

where $\Lambda_F(1-\alpha)$ is the exact $(1-\alpha)$-th quantile of $\Lambda$.

As one may notice, to reduce the moment condition on the noise term $\epsilon$, Condition <ref> has restrictions on the sub-gaussian norm of $x_{ij}$. We propose the following Condition as a tradeoff choice.

* Noise $\{\epsilon_{i}\}_{i\geq 1}$ is a sequence of independent r.v's with sub-gaussian norm $\Vert \epsilon_{i} \Vert_{\psi_2}<\infty$ that does not depend on $n$, and $\sum_{ i = 1 }^n \sigma_{ \epsilon_i }^2 \asymp(n)$.

* Dimension $p$ associates with confidence level $\alpha$ satisfy that $\log(p/\alpha)=o(n^{1/3}/\ell^{2/3})$ for constant $\ell$ defined in (<ref>).

* Design matrix $X$ is independent of the noise $\epsilon$ with independent, homoscedastic, sub-gaussian rows $\{ x_i \}_{1\leq i\leq n}$. Sub-gaussian norm $\Vert x_{ij} \Vert_{\psi_2}$ does not depend on $n$, and for simplicity, we again assume $m_{ij}^2=1$ for all $i,j$.

Under Condition <ref> instead of Condition <ref>, Theorem <ref> holds.

For a special case when the noises are homoscedastic (i.e., $\sigma_{ \epsilon_i }^2 = \sigma_{ \epsilon }^2$), Theorem <ref> holds under Condition <ref> but without requiring $x_{ij}$ to be homoscedastic (i.e., $m_{ij}=1$). This would cover fixed design cases where $x_{ij}$'s obey the bounded condition, $\sup_{n \geq 1} \max_{1 \le i \le n} \Vert x_i \Vert_{\infty} < +\infty $.

§.§ Asymptotic bounds on LASSO$^{1/2}$ estimation error

For asymptotic bounds on the estimation error, one can inherit the proof from Theorem $1$ of <cit.> combined with our Theorem <ref> (or Corollary <ref>) to obtain the same bound with improved conditions, where the improvement is essential in the sense that we allow the confidence level $\alpha$ to depend on $n$ and possibly go to $0$, and we allow $\log(p/\alpha)=o(n^{1/3})$. The proof is omitted for redundancy and we present the results first by introducing the following variant of restricted eigenvalue condition comes from <cit.>. One comprehensive discussion can be found in <cit.>.

For $\hat{\beta}$ being an estimator of $\beta_0$ associates with an observation of the random design matrix $X$, the prediction norm is defined as $\Vert \hat{\beta} - \beta_0 \Vert_{2,n} := [(\hat{\beta} - \beta_0)^T (X^TX/n) (\hat{\beta} - \beta_0) ]^{1/2}$.

For $S_0=supp(\beta_0)=\{j: \beta_{0,j}\neq 0\}$ with cardinality $s_0$, we define $\mu_{S_0}\in \mathcal{M}_{p\times 1}$ by letting $\mu_{S_0,j}=\mu_j$ if $j\in S_0$ and $\mu_{S_0,j}=0$ if $j\notin S_0$, and we similarly define $\mu_{S_0^c}\in \mathcal{M}_{p\times 1}$ as well.

Associating with a constant $r>1$ and an observation of the random design matrix, the restricted eigenvalues are defined as

\begin{equation*}
\phi(r,X)=\min_{\Vert \mu_{S_0^c} \Vert_1 \leq r \Vert \mu_{S_0} \Vert_1} \frac{s_0^{1/2} \Vert \mu \Vert_{2,n} }{\Vert \mu_{S_0} \Vert_1},\quad \tilde{\phi}(r,X)=\min_{\Vert \mu_{S_0^c} \Vert_1 \leq r \Vert \mu_{S_0} \Vert_1} \frac{\Vert \mu \Vert_{2,n} }{\Vert \mu\Vert_2}.
\end{equation*}

[restricted eigenvalue condition]
Associating with a constant $r>1$ and an observation of the random design matrix, there exist finite constants $N_0>0$ and $\phi_0>0$, s.t., $\phi(r,X)\geq \phi_0$ and $\tilde{\phi}(r,X)\geq \phi_0$ for all $n\geq N_0$.

Given $X$ has been observed, assume restricted eigenvalue condition and Condition <ref> (or Condition <ref>) holds, then with probability at least $1-\alpha-o(\alpha)$,

\begin{equation*}
\Vert \hat{\beta}-\beta_0 \Vert_{2} \leq A\sigma_{ \epsilon } \frac{\lambda s_0^{1/2}}{n},
\end{equation*}

where $A=8\Big( \phi(r,X)\tilde{\phi}(r,X)(1-\lambda^2s_0/(n^2\phi(r,X)^2))   \Big)^{-1}$ is a constant. If we adjust the definition of the restricted eigenvalues to be taken as the minimum over design matrix $X$, i.e., $\phi(r):=\min_{X} \phi(r,X)$, $\tilde{\phi}(r):=\min_{X} \tilde{\phi}(r,X)$, then without observing the design matrix, the asymptotic bound for estimation error holds.

§ APPLICATIONS TO SELF-NORMALIZED WINSORIZED MEAN

Although the sample mean has always been a prominent unbiased estimator for  a location parameter, it has the troubling disadvantage of being heavily influenced by gross outliers. Yet, robustness is often a desirable property, especially in real-world applications. Thus robust alternatives, typically including the trimmed mean ([25]), the winsorized mean ([10], [18]), and the Huber estimator ([18, 19]), are imperative to make more reliable statistical inference for unknown parameters. Suppose we have i.i.d. observations $ Y_1, Y_{2}, \ldots, Y_n $ with common distribution $Y$ and

\mu =  \e [Y] \quad \mbox{and} \quad \sigma^2 = \var (Y).

For a thresholding parameter $ \tau > 0 $ that determines the tradeoff between bias and robustness, the winsorized mean is defined by

\begin{equation} \label{winsor}
\hat{\mu}_{W} = n^{-1}  \sum_{i = 1}^n  f  (Y_i),
\end{equation}

\begin{equation} \label{f}
f  (x) =   x \mathbbm{1} ( | x | \leq \tau  )  + \tau \mathbbm{1} (  x > \tau ) - \tau \mathbbm{1} ( x < - \tau ).
\end{equation}

The trimmed mean is defined by

\begin{equation}
\hat{\mu}_{T} = c_{\tau, n}^{-1}  \sum_{i = 1}^n  Y_i \mathbbm{1} \{ | Y_i | \leq \tau \},    \label{trimmed}
\end{equation}

where $ c_{\tau, n} = \sum_{i=1}^n  \mathbbm{1} \{ | Y_i | \leq \tau \} $.

Moreover, the Huber loss ([18]) is given by

\begin{equation}
\ell_{\tau} (u) = \left\{
\begin{aligned}
& \frac 12 u^2   \quad & \mathrm{if}~| u | \leq \tau, \\
& \tau | u | - \frac 12 \tau^2 \quad  & \mathrm{if}~ | u | > \tau,
\end{aligned}
\right.
\end{equation}

which is a compromise between square loss and absolute loss. The Huber estimator is then defined as

\begin{equation}
\hat{\mu}_{H} = \mathop{\argmin} \limits_{\mu \in \mathbbm{R}} \sum_{i = 1}^n \ell_{\tau} (Y_i - \mu). \label{huber}
\end{equation}

These robust estimators are common in reducing the impact of outliers and are all asymptotically equivalent to the sample mean when the associated tuning parameter $\tau$ tends to infinity. Compared with the Huber estimator, the trimmed mean and the winsorized mean have explicit formulas and therefore are easier to be applied in real-world applications.
It is well-known that these robust estimators are asymptotically normal under some regularity conditions.

Recently [36] obtained a Cramér-type moderate deviation theorem for the Huber estimator when allowing the tuning parameter $\tau$ to diverge with the sample size $n$ in some regime, and they applied the result to establish theoretical guarantees for the false discovery rate in multiple testing procedure for population means.
However, the statistic they investigated depends on the unknown variance, which needs to be well estimated in practice.

§.§ Cramér-type moderate deviation for self-normalized winsorized mean and trimmed mean

In this section, we will provide Cramér-type moderate deviation theorems for the self-normalized winsorized mean defined in (<ref>) and self-normalized trimmed mean defined in (<ref>), as an application of our main Theorem <ref>.
The self-normalized winsorized mean and trimmed mean are asymptotically pivotal statistics in the sense that their asymptotic distributions do not depend on unknown parameters as $(n, \tau ) \to (\infty, \infty)$, therefore they can be directly used in the multiple testing of population means with theoretical justification.
In addition, we will see our results for self-normalized winsorized mean and trimmed mean outperform that for the Huber estimator established in [36].

Since the winsorized mean and trimmed mean have explicit expressions as presented in (<ref>) and (<ref>), we can easily construct the studentized counterparts by plugging in the sample variance. The studentized winsorized mean is given by

\begin{equation} \label{S}
S_{\tau, n} = \frac { \sum_{i=1}^n  (f(Y_i  ) - \mu) } { \sqrt{    \sum_{i = 1}^n ( f (Y_i) - \hat{\mu}_{W} )^2 } }
\end{equation}

and the studentized trimmed mean is given by

\begin{equation}
U_{\tau, n} = \frac { \sum_{i \in \mathcal{N}} (Y_i \IF \{ |Y_i| \leq \tau \} - \mu ) } { \sqrt{ \sum_{i \in \mathcal{N}}  (Y_i \IF \{ | Y_i | \leq \tau \} - \hat{\mu}_T ) ^2 }  },
\end{equation}

where $ \mathcal{N} = \{1 \leq i \leq n: | Y_i | \leq \tau  \} $.
Observe that

\begin{align}
S_{\tau, n} & =  \frac { \sum_{i = 1}^n  ( f (Y_i) -\mu ) } {  \sqrt{ \sum_{i=1}^n (  f(Y_i )  - \mu + \mu -  \hat{\mu}_W  )^2 } } \nn \\
& =  \frac {  \sum_{i = 1}^n  ( f (Y_i) -\mu )  } {  \sqrt{ \sum_{i=1}^n (  f(Y_i )  - \mu )^2   - \frac 1 n   \big( \sum_{i = 1}^n ( f (Y_i) - \mu ) \big)^2  } }  = \frac { S_{\tau, n} ^* } { \sqrt{ 1 - \frac 1n ( S_{\tau, n}^* )^2 } },  \nn
\end{align}

where $ S_{\tau, n}^* $ is the self-normalized winsorized mean defined as

\begin{equation} \label{self-winsor}
S_{\tau, n}^* =    \f { \sum_{i=1}^n ( f(Y_i ) - \mu )  } { \sqrt{ \sum_{i=1}^n ( f (Y_i ) - \mu )^2 } }.
\end{equation}

Similarly, we have for the studentized trimmed mean that

\begin{equation*}
U_{\tau, n} = \frac { U^*_{\tau, n} } { \sqrt{1 - \frac {1} { c_{\tau, n}} (U^*_{\tau, n})^2 } } ,
\end{equation*}

where $U_{\tau, n}^*$ is the self-normalized trimmed mean defined as

\begin{equation} \label{self-trimmed}
U_{\tau, n}^* = \frac { \sum_{i \in \mathcal{N}} ( Y_i \IF \{ | Y_i | \leq \tau \} - \mu)   } { \sqrt{\sum_{i \in \mathcal{N}} ( Y_i \IF \{ | Y_i | \leq \tau \} - \mu)^2 } }
\end{equation}

and $c_{\tau, n} = \sum_{i = 1}^n \IF \{ | Y_i | \leq \tau \}$. Since the function $ x / (1 - \f 1n x^2)^{1/2} $ is an increasing function for $ 0 < x < n^{1/2} $, we have

\begin{equation*}
\P ( S_{\tau, n}  >  x )   = \P \Big( S_{\tau, n}^* > \frac {  x } { \sqrt{1 + \f { x^2 } { n } } } \Big)
\end{equation*}

\begin{equation*}
\P ( U_{\tau, n} > x   ) = \P \Big( U_{\tau, n}^* > \frac {  x } { \sqrt{1 + \f { x^2 } { c_{\tau, n} } } }  \Big)
\end{equation*}

Therefore, to investigate the limiting properties of $ S_{\tau, n} $ and $U_{\tau, n}$ is equivalent to investigate that for the simpler self-normalized statistics $ S_{\tau, n}^* $ and $U_{\tau, n}^*$, respectively.

Before stating our Cramér type moderate deviation results, let us first present how the self-normalized winsorized mean and trimmed mean connect with the general self-normalized sum investigated in our main Theorem <ref>. First for the self-normalized winsorized mean, though (<ref>) presents the form of a self-normalized sum of independent random variables for $ S_{\tau, n}^* $, the expectation of $ f (Y) - \mu $ is slightly deviated from $0$ and needs to be calibrated. Denote

\begin{equation*}
\tilde{\mu} =  \e  f (Y) , \quad \sigma_1^2 = \e ( f (Y) - \tilde{\mu} )^2, \quad \sigma_2^2  = \e ( f (Y) - \mu )^2.
\end{equation*}

Then we can write

\begin{align}
S_{\tau, n}^*&  = \frac {  \frac { \sum_{i = 1}^n ( f (Y_i) - \tilde{\mu} ) } { \sqrt n \sigma_1 }  - \frac { \sqrt{n} (\mu - \tilde{\mu} ) } { \sigma_1  } } { \sqrt {  \frac { \sum_{i = 1} ^n  ( f (Y_i) - \mu )^2 } { n \sigma_{2} ^2 }  }  } \cdot \frac{ \sigma_1 }  { \sigma_2 } \nn \\
& = \frac { S_n - c } { V_n } \cdot \frac{ \sigma_1 }  { \sigma_2 },  \label{general-self}
\end{align}

where $ S_n  $, $ V_n  $ and $c$ are denoted by

\begin{gather*}
S_n  =  \sum_{i = 1}^n   \frac {   f (Y_i) - \tilde{\mu}  } { \sqrt n \sigma_1 }  , \quad V_n^2 =   \sum_{i = 1} ^n  \frac { ( f (Y_i) - \mu )^2 } { n \sigma_{2} ^2 }\\
\mbox{and} \quad c =  \frac { \sqrt{n} (\mu - \tilde{\mu} ) } { \sigma_1  } .
\end{gather*}

\begin{equation*}
\P ( S_{\tau, n}^* > x  ) = \P \Big( \frac { S_n - c } { V_n } >  \frac { \sigma_2 } { \sigma_1 }  x \Big).
\end{equation*}

Note that the random variables involved in $ S_n $ and $ V_n $ are different, which means the existing results for classical self-normalized sums cannot be directly applied.

As for the self-normalized trimmed mean, note that in the numerator, $ \sum_{i \in \mathcal{N}} ( Y_i \IF \{ | Y_i | \leq \tau \} - \mu) $ in (<ref>) is equal to $ \sum_{i = 1}^n (Y_i - \mu ) \IF \{ |Y_i | \leq \tau  \} $. Similarly in the denominator, $  \sum_{i \in \mathcal{N}} ( Y_i \IF \{ | Y_i | \leq \tau \} - \mu)^2   $ is equal to $ \sum_{i = 1}^n [ (Y_i - \mu) \IF \{ | Y_i | \leq \tau \} ]^2 $. Thus we have

\begin{equation*}
\begin{split}
U_{\tau, n}^* & = \frac { \sum_{i = 1}^n (Y_i - \mu ) \IF \{ |Y_i | \leq \tau  \}  } { \sqrt{ \sum_{i = 1}^n [ (Y_i - \mu) \IF \{ | Y_i | \leq \tau \} ]^2 } }
\end{split}
\end{equation*}

\begin{align*}
\mu_0 &= \e [ (Y - \mu) \IF\{ | Y_i  | \leq \tau \} ], \quad    \sigma_{3} ^2 = \e [ (Y - \mu) \IF\{ | Y_i  | \leq \tau \} -  \mu_0]^2, \\
& \mbox{and}\quad \sigma_{4}^2 = \e [ (Y - \mu)^2 \IF\{ | Y_i  | \leq \tau \} ].
\end{align*}

Similar to the self-normalized winsorized mean, we can obtain

\begin{equation*}
\P (U_{\tau, n}^* > x ) = \P \Big( \frac { S_n^{\circ}  - \delta } { V_n^{\circ} } > \frac { \sigma_4 } { \sigma_3 } x \Big),
\end{equation*}

\begin{align*}
S_n^{\circ} &= \sum_{i = 1}^n \frac { (Y_i - \mu) \IF \{ | Y_i | \leq \tau \} - \mu_0 } { \sqrt n \sigma_3 }, \quad  (V_n^{\circ} )^2 = \sum_{i = 1}^n \frac { [ (Y_i - \mu) \IF \{ |Y_i| \leq \tau  \} ]^2 } { n \sigma_4^2 } \\
&\hspace{2cm} \mbox{and} \quad \delta = \frac { \sqrt n \mu_0 } { \sigma_3 }.
\end{align*}

Consequently, our result for general self-normalized sums in Theorem <ref> can be directly applied to $  S_{\tau, n}^*   $ and $ U_{\tau, n}^* $ to derive the following bias-corrected Cramér-type moderate deviation theorems for the self-normalized winsorized mean and trimmed mean under the fourth moment.

Assume $  \e [ Y ^4 ] < \infty $. Then there exist an absolute positive constant $ c_1 $ and positive constants $ c_2 $ and $ A $ depending on $\sigma$, $ \e [ |Y|^3 ]$ and $ \e [Y^4 ] $,  such that for

\begin{equation}
\tau \geq c_1 n^{1/6} \max \{ (  \e  [ Y ^4 ]  )^{1/2} / \sigma, ( \e [ Y ^4 ]   / \sigma )^{1/3} \} ,  \label{cond-2}
\end{equation}

it holds that

\begin{align}
\P ( S_{\tau, n} ^* > x ) & =  [ 1 - \Phi (x) ] \exp \Big\{ - \f  { x^3 \e (Y - \mu)^3  } { 3  \sqrt n  \sigma^3 }  \Big \}  \nn \\
& \quad \times \Big[ 1 + O_1  \Big(  \frac { (1 + x ^4 )  } { n  }     +  \frac { (1 + x  ) \sqrt n  } { \tau^3  } + \frac { (1 + x ) } { \sqrt n  } \Big)  \Big]  \label{re-2}
\end{align}

uniformly for $ x \in (0, c_2  \min \{ n^{1/4},   \tau^3 n^{- 1/2 }   \}  ) $,  where $ O_1 $ is a  bounded quantity satisfying $ | O_1 | \leq A $. Similar result holds for $ \P ( S_{\tau, n} ^* < - x ) $.

Under the conditions of Theorem <ref>, the same result as (<ref>) holds for $ U_{\tau, n}^* $.

Observe that under the fourth moment, the general framework Theorem <ref> enables us to pin down the bias-corrected term $ \exp \{ - \f  { x^3 \e (Y - \mu)^3  } { 3  \sqrt n  \sigma^3 }  \} $ which depends on the skewness of the underlying distribution. After correcting this skewness in normal approximation, the convergence rate and the converging range significantly improve that  given in Theorem <ref>, where only third moment is assumed.

The choice of $ \tau  $ should be determined by taking both convergence rate and robustness of estimator into account. We observe from (<ref>) that the ratio $ \P ( S_{\tau, n}^*  > x ) / [ (  1 - \Phi (x) )   \exp  \{ - \f  { x^3 \e (Y - \mu)^3  } { 3  \sqrt n  \sigma^3 }    \}   ]$ converges to  1  for $x \in (0,  o ( \min \{  n^{1/4} , \tau^3 n^{- 1/2 } \} ) )$. The widest possible range $ x \in (0,  o (n^{1/4}) ) $ can be achieved by choosing $  \tau \geq O ( n^{1/4})  $. When $ \tau \leq O (n^{1/3}) $, the larger $ \tau $ is, the faster rate of convergence and wider range of $x$ can be obtained. Yet, once $ \tau  $ exceeds $ O (n^{1/3}) $, our result reduces to the bias-corrected Cramér-type moderate deviation for the classical self-normalized sample mean (see Theorem 1.1 in [31]), which is reasonable because the winsorized mean and trimmed mean are asymptotically equivalent to the sample mean as $ \tau \to \infty $. It is worth mentioning that when $ O (n^{1/6}) \leq \tau \leq O (n ^{1/3}) $, the ratio $  {\P ( S_{\tau, n} ^*  > x  ) }  / \big(  [ 1 - \Phi(x)  ] \exp \{ - \f  { x^3 \e (Y - \mu)^3  } { 3  \sqrt n  \sigma^3 }   \}  \big) $ converges to $1$ at the rate of $ O ( (1 + x )^4 n^{ - 1 } + (1 + x) \sqrt n \tau^{ - 3} ) $ uniformly for $x \in o(\min\{ n^{1/4}, \tau^3 n^{-1/2} \})$. In this regime of $\tau$, though the convergence rate of winsorized mean and trimmed mean could be slightly slower than that of the classical self-normalized sample mean and the ranges of $x$ for convergence could be narrower, the winsorized mean and trimmed mean provide  robuster inference. We will provide the proof of Theorem <ref> in Section <ref> and the proof of Theorem <ref> in Section <ref>.

Theorem 2.3 in [36] is closely related to ours. They established a Cramér-type moderate deviation result for Huber estimator $ \hat{\mu}_H $ defined in (<ref>) by using a Bahadur representation for the Huber estimator. Theorem 2.1 in their paper reveals that $ \hat{\mu}_H - \mu   $ is asymptotically close to $ n ^{- 1} \sum_{i = 1}^n f ( Y_i - \mu ) $, where $ f(\cdot) $ is defined by (<ref>). Therefore, it is easy to see that the Huber estimator $ \hat{\mu}_{H} $ is close to the winsorized mean $ \hat{\mu}_{W} = n^{ - 1 } \sum_{i=1}^n f (Y_i) $ as $ \tau \to \infty $. Theorem 2.3 in [36] for the Huber estimator can be restated as follows. The notation $ a_n \ll b_n  $ means $ a_n = o (b_n) $ as $n \to \infty$.

Assume $ \e | Y |^3 < \infty $. [36] proved for $ n^{1/4} \ll \tau \ll n^{1/2} $ that

\begin{align}
&  \frac { \P (   { \sqrt n \sigma^{-1} | \hat{\mu}_H - \mu | } > x  ) } { 2 (1 - \Phi (x)) }  \nn\\
& = 1 + O(1) \Big\{ \frac { ( \sqrt {\log n}  + x )^3  } { \sqrt n } + \frac { 1 + x } {n^{3/10}}  + \frac { ( 1 + x ) \sqrt n } { \tau^2  }  + e ^{ - O ( \frac {n} {\tau^2 } ) }\Big \} \label{zhou}
\end{align}

uniformly for $ 0 \leq x = o ( \min\{ \sqrt{ n} / \tau, \tau^2 / \sqrt n \} ) $.

Compared to their condition $ n^{1/4} \ll \tau \ll n^{1/2} $, our condition $ \tau \geq O ( n^{1/6}) $ is less restrictive. Moreover, when $ \tau \gg n^{1/4} $, both of our convergence rate and the associated   converging range of $x$ improve theirs.

Our improvement mainly relies on the explicit formula of the self-normalized winsorized mean presented in (<ref>) and our fundamental result for general self-normalized sum established in Theorem <ref>. In addition, since the higher moment $ \e [Y^4 ] < \infty $ is assumed, after correcting the bias in normal approximation, the convergence rate and the associated range of $x$ could be significantly improved.

The common downside of our bias-corrected result in Theorem <ref> and the normal approximation for Huber estimator by [36] in Remark <ref> is that the limiting distributions depend on unknown parameters. In real-world applications, if reliable estimations for the unknown parameters are unavailable, we can directly use normal approximation for the self-normalized winsorized mean presented in the following theorem, where only third moment is required and the limiting distribution does not depend on any unknown parameters.

Assume $  \e | Y |^3 < \infty $. Then there exist absolute positive constants $ c_1, c_2 $ and $ A $ such that for

\begin{equation}
\tau \geq c_1 n^{1/4} \max \{ \e | Y |^3 / \sigma^2, ( \e | Y |^3  / \sigma )^{1/2} \} ,  \label{cond}
\end{equation}

it holds that

\begin{align}
\frac {\P ( S_{\tau, n} ^*  > x  ) } { 1 - \Phi(x) } = 1 + O_1  \Big( \f { (1 +x^3) \e | Y|^3  } { \sigma^3 \sqrt{n} } + \frac { (1 + x )\sqrt{n} \e | Y|^3  } { \sigma \tau^2 }   \Big), \label{re1}
\end{align}

uniformly for $ x \in (0, c_2 \min \{  n^{1/6} \sigma^3 /  \e | Y |^3   , \tau^2 \sigma / (\sqrt{n} \e | Y |^3 ) \} ) $, where $ O_1 $ is a bounded quantity satisfying $ | O_1 | \leq A $. Similar result holds for $ \P ( S_{\tau, n} ^* < - x ) $.

Under the conditions of Theorem <ref>, the same result as (<ref>) holds for $U_{\tau, n}^*$

It can be observed that the convergence rate and the converging range of $x$ also outperform the results of [36] shown in Remark <ref>, and our condition on $\tau$ is less restrictive. We relegate the proof of Theorem <ref> to Section <ref> and the proof of Theorem <ref> in Section <ref> in the Supplementary Material.

§.§ Simultaneous confidence intervals

Cramér type moderate deviation results are useful in providing theoretical guarantees for a wide spectrum of statistical applications, including the multiple testing procedure and multiple confidence intervals for ultra-high dimensional parameters. For an illustrative example, we will construct simultaneous confidence intervals for the means under the following ultra-high mean model by using the studentized winsorized mean estimator defined in (<ref>). We consider

\begin{equation*}
{\bf Z}_i =   \bm {\mu}+ \bm{\epsilon}_i, \quad i=1,\cdots,n,
\end{equation*}

where $\{ {\bf Z}_1,\ldots, {\bf Z}_n \}$ are i.i.d. observations, $\bm \mu=( \mu_1,\cdots, \mu_p)^T\in \mathbb{R}^p$, and $\{\bm \epsilon_1, \ldots, \bm \epsilon_i \}$ are i.i.d. errors. Denote $\bm \Sigma =  \cov(\bm \epsilon_i ) := (\Sigma_{i j})_{p \times p}$. Assume there exist constants $C_1,C_2$ such that $ \max_{ 1 \leq j  \leq p }\e |Z_{ij}|^3 \leq C_1$ and $\min_{1\leq j\leq p} \Sigma_{jj}  \geq C_2$.

Assume the dimensionality $p$, the significance level $ \alpha $ and the thresholding parameter $\tau$ satisfying $\log ( p / \alpha)= o(n^{1/3})$ and $\tau \gg n^{1/3}$. Then for $\alpha\in (0,1)$, and $t_0$ satisfying the equation

\begin{equation*}
\f{t_0}{1+t_0^2/ n }= \Phi^{-1} \big(1-\f{\alpha}{2p} \big),
\end{equation*}

we have

\begin{equation*}
\f{\sum_{ i = 1 }^n f (Z_{ij}) }{n} \pm \f{ t_0 }{n} \sqrt{\sum_{ i = 1 }^n \big( f (Z_{ij})  \big)^2 -\f{1}{n} \Big[ \sum_{ i = 1 }^n f (Z_{ij})   \Big]^2  } \triangleq (L_j, U_j ), \quad 1 \leq j \leq p
\end{equation*}

are the $1-\alpha-o(1)$ simultaneous confidence intervals for $\big( \mu_j \big)_{j=1}^p $, where $f(\cdot)$ is the function defined in (<ref>).

The proof of Theorem <ref> will be provided in Section <ref> in the Suppplementary Material.

§ PROOFS

In this section, we present proofs of Theorem <ref>, Theorems <ref>–<ref> and Theorems <ref>. Throughout the rest of this section, $A$ and $C$ denote positive absolute constants that may take different values at each appearance.

§.§ Proof of Theorem <ref>

We prove the theorem for the two scenarios $0 < x \leq 3$ and $x > 3$, respectively. First, we prove it for $0 < x \leq 3$. For this range, it is sufficient to prove a Berry-Esseen bound as the following proposition will show. The proof of Proposition <ref> is postponed to Section <ref> the Supplementary Material.

For $ 0 < x \leq 3 $, there exists an absolute constant $A> 0$ such that

\begin{align}
\big| \P (S_n > x V_n + c )  -  [1 - \Phi (x+c)] \big| \leq  A  L_{3,n}.
\end{align}

Note that  $ 1 - \Phi (3.6)  \leq 1 - \Phi (x + c) \leq 1$ for $0 < x \leq 3$ and $ |c | \leq x /5 $. Thus, it follows from Proposition <ref> that for $0 < x \leq 3$,

\begin{align}
\P ( S_n > x V_n + c  ) = [1 - \Phi (x+c)]  (1 + O (1 + x) L_{3, n} ). \label{eq1}
\end{align}

Moreover, it holds for $0 < x \leq 3$ satisfying (<ref>) that

\begin{align*}
| (\Psi_x^*) ^{-1} - 1 | \leq  A  x^3 L_{3, n}  \leq  A L_{3, n} ,
\end{align*}

which combining with (<ref>) entails that

\begin{align*}
\P ( S_n > x V_n + c  ) = [1 - \Phi (x+c)] \Psi_x^*  (1 + O (1 + x) L_{3, n} ) .
\end{align*}

Consequently, we have

\begin{align*}
\P ( S_n > x V_n + c  ) = [1 - \Phi (x+c)] \Psi_x^* e^{O_1 R_x} (1 + O (1 + x) L_{3, n} ),
\end{align*}

where the quantity  $ |O_1| \leq A$ for some absolute constant. This completes the proof for $ 0 < x \leq 3 $.

Next we deal with the case $ x >  3 $. By applying the elementary inequality

\begin{equation}
1 + s / 2 - s^2 \leq  ( 1 + s  )^{1/2}  \leq 1 +  s/ 2   , \label{ele-ineq}
\end{equation}

for $ s = V_n^2 - 1 $, we obtain

\begin{align*}
\frac {1} {2} (V_n^2 + 1) - (V_n^2 - 1)^2   \leq  V_n \leq \frac {1} {2} (V_n^2 + 1).
\end{align*}

Therefore, plugging in the above upper and lower bounds yields

\begin{equation}
\P \( S_n > x V_n + c \) \ge \P \( 2x S_n - x^2 V_n^2 \ge  x^2 + 2xc \) \label{lower-bdd}
\end{equation}

\begin{align}
&\P \( S_n > x V_n + c \) \label{upper-bdd}\\
& \leq \P \(   2x S_n - x^2 V_n^2 \ge x^2 + 2xc - x \De_{n}    \) \nn \\
& \quad +  \P \(  S_n > x V_n + c, \, |V_n^2-1| >  x^{ -1 } ( 1 \lor 6 R_x^{1/2} ) \), \nn
\end{align}

where $\De_{n} = \min   \{ 2x \( V_n^2-1 \)^2,  x^{-1} ( 2 \lor 72 R_x ) \} $ and the notation $  a \lor b $ means the maximum of $a$ and $b$. The upper bound holds because

\begin{align*}
&\P \( S_n > x V_n + c,\,|V_n^2-1|\le  x^{ -1 } ( 1 \lor 6 R_x^{1/2} )  \)  \\
&\leq \P \( 2x S_n - x^2 V_n^2 \ge x^2+ 2xc - 2x^2 \( V_n^2 - 1 \)^2 ,\, |V_n^2-1|\le  x^{ -1 } ( 1 \lor 6 R_x^{1/2} ) \) \nn \\
& \leq \P \( 2x S_n - x^2 V_n^2 \ge x^2 + 2xc - x \De_{n} \). \nn
\end{align*}

The following Propositions <ref>–<ref> draw an outline of the proof for the case $x > 3$. Their proofs are relegated to Sections <ref>–<ref> in the Supplementary Material.

There exists an absolute constant $A$ such that

\begin{align}
\P \big( 2x S_n - &x^2V_n^2 \ge x^2 + 2xc \big) \label{prop1-re} \\
=& \[1-\Phi (x+c)\]  \Psi_{x}^*   e^{O_1 R_x} \{ 1+O_2(1+x)L_{3,n}  \},
\nn
\end{align}

for $ x > 3$ satisfying (<ref>) and (<ref>) and $|c| < x/5$, where $ |O_1| \leq A$ and $ |O_2|  \leq A$.

There exist absolute constants $A_1$ and $ A_2 $ such that

\begin{align}
\P \big( 2x S_n - &x^2V_n^2 \ge x^2 + 2xc - x\Delta_{n} \big) 	\label{prop2-re} \\
\le& \[1-\Phi (x+c)\]  \Psi_{x}^*  e^{A_1 R_x} \{ 1+A_2(1+x)L_{3,n} \},
\nn
\end{align}

for $ x > 3$ satisfying (<ref>) and (<ref>), and $|c| < x/5$.

There exist absolute constants $A_1$ and $ A_2 $ such that

\begin{align}
& \P \(S_n  \ge x V_n+c, ~ |V_n^2-1|>  x^{ - 1} ( 1 \lor 6 R_x^{1/2} )  \) \label{prop3-re} \\
& \hspace{4cm} \le A_1 R_x    [ 1 - \Phi( x + c) ]  \Psi_x^* e^{ A_2 R_x } \nn ,
\end{align}

for $ x > 3$ satisfying (<ref>) and (<ref>), and $|c| < x/5 $.

We obtain by substituting the results in Propositions <ref>–<ref> into (<ref>) that

\begin{align*}
&\P \( S_n > x V_n + c \) \\
& \leq \[1-\Phi (x+c)\]  \Psi_{x}^*  e^{A_1 R_x} \{ 1+A_1 R_x + A_2(1+x)L_{3,n}  \} \\
& \leq \[1-\Phi (x+c)\] \Psi_{x}^*  e^{A R_x} \{ 1  + A (1+x)L_{3,n}  \}
\end{align*}

which together with the result in Proposition <ref> yields the desired result (<ref>) for $x > 3$. The proof is completed.

§.§ Proof of Theorem <ref>

The main idea is to apply the big-block-small-block technique to construct a general self-normalized sum based on an independent sequence to which our main result Theorem <ref> can be applied. Denote $  B_n^2 = \sum_{ i = 1 }^n \e \xi_i^2  $. We first apply Berry-Esseen bound for sum of one-dependent random variables to cope with the case $ 0 \leq  x \leq O  ( \sqrt{ \log n } ) $. Note that

\begin{align*}
&  \left| \P ( S_n \geq x V_n  )  - \Big[  1 - \Phi \Big( \frac  { x }  { \sqrt{ 1 + 2 \rho_n } }  \Big)  \Big]  \right| \nn \\
& \hspace{0.5cm} \leq \left |  \P \Big( S_n \geq x B_n ( 1 - n^{ - 1/3 } )^{ 1/2 }  \Big)  - \Big[  1 - \Phi \Big( \frac  { x }  { \sqrt{ 1 + 2 \rho_n } }  \Big)  \Big]  \right|  \nn \\
&   \hspace{0.5cm}  \quad + \left |  \P \Big( S_n \geq x B_n ( 1 +  n^{ - 1/3 } )^{ 1/2 }  \Big)  - \Big[  1 - \Phi \Big( \frac  { x }  { \sqrt{ 1 + 2 \rho_n } }  \Big)  \Big]  \right|  \nn \\
&   \hspace{0.5cm} \quad +  \P \Big( | V_n^2 - B_n^2 | > n^{ - 1/3 }  B_n^{ 2 } \Big)  \\
&   \hspace{0.5cm} : = E_1 + E_2 + E_3.
\end{align*}

Recalling the definition of $\rho_n$ in (<ref>), we have $ \var (S_n) = (  1 + 2 \rho_n ) B_n^2  $. By noticing the assumptions

\begin{align*}
\e \xi_i^4 \leq a_1^4, \quad \e \xi_2^2 \geq a_2^2, \quad  a = a_1/a_2, \quad   - 1/2 < \rho \leq  \rho_n \leq 1/2
\end{align*}

and applying the Berry-Esseen bound for sums of one-dependent random variables (see [30]), we obtain

\begin{align*}
E_1 & \leq   \frac  { A(\rho)  a^3 } { \sqrt{n}  } +  \Big|  \Phi \Big( \frac  { x }  { \sqrt{ 1 + 2 \rho_n } }  \Big)   -  \Phi \Big( \frac  { x ( 1 - n^{ -  1 / 3  } )^{ 1/2 } }  { \sqrt{ 1 + 2 \rho_n } }  \Big)  \Big| \nn \\
& \leq A ( \rho ) a^3  ( n^{ - 1/2 } + x^2n^{ - 1/3 } ) \leq  A ( \rho ) a^3 (1+x)^2 n^{ - 1/3 },
\end{align*}

where $ A (\rho) $ is a positive constant depending on $ \rho $ and may take different values at each appearance. In the same manner, this above bound applies to $E_2$ as well.
As for $E_3$, it follows by Chebyshev's inequality that

\begin{align*}
E_3 \leq ( n^{ 1/3  } B_n^{ - 2 } )^2 \e [ ( V_n^2 - B_n^2  )^2 ]  \leq  A a^4  n^{ - 1/3 }.
\end{align*}

\begin{align}
\left| \P ( S_n \geq x V_n  )  - \Big[  1 - \Phi \Big( \frac  { x }  { \sqrt{ 1 + 2 \rho_n } }  \Big)  \Big]  \right|  \leq A (\rho) a^4 (1+x)^2 n^{ - 1/3 }, \label{BEboundMdependent}
\end{align}

Moreover, observe that $ 1 - \Phi ( \frac { x } {\sqrt{1 +  2 \rho_n } } ) \geq 1 - \Phi ( \frac { 2 \sqrt 3  } {\sqrt {1 + 2 \rho} } ) $ for $ 0 \leq x  \leq 2 \sqrt 3 $ and

\begin{equation}
1 - \Phi (  \frac { x } {\sqrt{1 +  2 \rho_n } }  )  \geq  A(\rho) x^{-1}  \exp \Big\{ -   \frac { x^2 } {  2(1 + 2 \rho) } \Big\}
\end{equation}

for $x > 2 \sqrt 3 $. As a consequence, there exists a constant $ c_{\rho} $ depending on $ \rho $ such that (<ref>) holds for $ 0 \leq x \leq  c_{\rho}  \sqrt{\log n } $.

Next we turn to the proof for $ x >   c_{\rho} \sqrt{ \log n  } $. Let the length of big blocks be $ l =[n^\alpha] $, and each small block contains only one random variable. Denote $ k = [n / (l + 1 )] $. Without loss of generality, let us assume $ n / (l + 1) $ to be an integer, then the sequence of $\{\xi_i\}_{1 \leq i \leq n } $ can be divided into $ k  $ big blocks and $ k $ small blocks. Observe that  if $n / (l + 1 )$ is not an integer, the sequence of $\{\xi_i\}_{1 \leq i \leq n } $ will be divided into $k + 1$ big blocks and $k$ small blocks,  in which the first $ k  $ big blocks are of size $ l $ and the last one is of size $ n - (l +1 ) [ n / (l + 1 )] $. Although the size of the last big block might be different, our analysis also applies under this scenario.
For $ 1 \leq j \leq k  $, the $j$-th big block and the corresponding block sums are given by

\begin{gather*}
H_j = \{ i: (j-1)( l +1) + 1 \le i \le j( l + 1 ) - 1 \}  ~ ~ \mbox{and} ~~ X_j = \sum_{i \in H_j} \xi_i,   \;\;	Y_j^2 = \sum_{i \in H_j } \xi_i^2,
\end{gather*}

Moreover,  we denote

\begin{gather}
S_{n1} = \sum_{j=1}^{k} X_j, \quad V_{n1}^2 = \sum_{j=1}^{k} Y_j^2, \quad B_{n1}^2 = \e V_{n1}^2,	\label{notationonedependent}\\
S_{n2} = \sum_{j=1}^{k} \xi_{j(l+1)},  \quad  V_{n2}^2 = \sum_{j=1}^{k} \xi_{j(l+1)}^2, \quad  B_{n2}^2 = \e V_{n2}^2.   \label{smallb-def}
\end{gather}

Observe that $ \{ X_j \}_{1 \leq j \leq k} $ and $ \{ \xi_{j (l +1 )} \}_{1 \leq j \leq k } $ both consists of independent random variables. As $ S_{n1} $ is the sum of $ \xi_i 's $ in big blocks and it is the main part of $ \sum_{i = 1}^n \xi_i  $, while $ S_{n2} $ corresponds to the small blocks. The big-block-small block technique splits the sum $ \sum_{i = 1}^n \xi_i  $ into two parts $S_{n1}$ and $ S_{n2} $, each of which is a sum of independent random variables.  Let $ \tau = B_{n2}/x $ and we do truncations $ \hat{\xi_i} = \xi_i \IF ( |\xi_i| \le \tau ) $ only for the $ \xi_i $'s in small blocks, that is, $ i = j ( l + 1 ) $ for $ 1 \leq j \leq k $, so

\begin{align}
\P \( S_n \geq x  V_n  \)
&\le \P (  \hatS_n  \geq x  \hatV_n   ) + \P ( S_n \geq x V_n, \max_{1 \le j \le k} |\xi_{j(l+1)}| > \tau ) , \label{up-bdd}\\
\P (S_n \geq x V_n)
&\ge \P (  \hatS_n  \geq x  \hatV_n    ) - \P (   \hatS_n  \geq x  \hatV_n   , \max_{1 \le j \le k} |\xi_{j(l+1)}| > \tau ),  \label{low-bdd}
\end{align}

where $ \hatS_n = S_{n1} + \hatS_{n2} $, $ \hatV_n^2 = V_{n1}^2 + \hatV_{n2}^2  $ with $ \hatS_{n2} = \sum_{j = 1}^k \hat{\xi}_{ j ( \ell + 1 ) } $ and $ \hat{V}_{n2}^2 =  \sum_{j = 1}^k \hat{\xi}_{ j ( \ell + 1 ) }^2 $.
For a positive number $ d_1 > 0 $, we have the upper bound

\begin{align}
\P ( \hatS_n \geq x \hatV_n ) &\le \P \big(S_{n1} \geq x V_{n1} - d_1 n^{ -\f{ \alpha}{2}  } x B_n\big) \label{uu-bb} \\
& \quad + \P \big(\hatS_{n2} \geq  d_1 n^{ -\f{ \alpha}{2}  }  x B_n  \big) \nn
\end{align}

and the lower bound

\begin{align}
\P ( \hatS_n \geq x \hatV_n ) &\ge \P \big(S_{n1} \geq x V_{n1} +   d_1 n^{ -\f{ \alpha}{2}  }  x B_n\big) \label{ll-bb}\\
&\quad  - \P \big( S_{n1} \geq x V_{n1}, ~V_{n1}^2 < B_n^2 / 4\big) \nn \\
&  \quad - \P \big( \hatS_{n2} < - d_1 n^{ -\f{ \alpha}{2}  } x B_n + x (\hatV_n - V_{n1}), ~V_{n1}^2 > B_n^2 / 4 \big) . \nn
\end{align}

We can obtain the following bounds for the terms involved in (<ref>) and (<ref>). The proofs of Propositions <ref> and <ref> are given in Sections <ref> and <ref> in the Supplementary Material.

There exists an absolute positive constant $ d_0 $ and a constant $A(\rho, d_0)$ depending on $ \rho   $ and $ d_0  $ such that, for $ d_1  =  \kappa_{\rho} a^2 $ with some sufficiently large constant $ \kappa_{\rho} $,

\begin{align}
\P \Big( S_{n 1 }  &\geq x V_{n 1 } + d_1 n^{ - \alpha / 2  } x B_n  \Big) \label{big-block} 	 \\
&  =  \Big[ 1 - \Phi( \f{x}{ \sqrt{ 1 + 2 \rho_n } } ) \Big]  \Big(  1 + O_1 \Big(  \frac { a^4 x^4  } { n^{ 1- \alpha } }+  \frac {  a^2  x^2  } { n^{ \alpha/ 2 } } +\f{a^3x}{n^{\f{1-\alpha}{2}}} \Big)  \Big),  \nn
\end{align}

uniformly for $ x \in ( 2 , d_0 a^{ -1  } \min \{  n^{ \alpha/4 }, n^{ ( 1 - \alpha ) / 4  } \}  ) $, where $ | O_1 | \leq  A(\rho, d_0) $.
A similar result holds for $ \P ( S_{n1} \geq x V_{n1} - d_1 n^{ - \alpha/2  } x B_n ) $. Moreover,

\begin{align}
\P \Big( S_{n1} &\geq x V_{n1}, ~V_{n1}^2 < \f{B_n^2}{4} \Big) \label{error_low1} \\
&\le  \f{ A a^4 x^4} {n^{1-\alpha}} \Big[ 1 - \Phi\Big( \f{x}{ \sqrt{ 1 + 2 \rho_n } } \Big) \Big] \exp {\Big\{ A  a^4 \f{x^4} {n^{1-\alpha}} + A  a^3 \f  {x^3} {\sqrt{n}} \Big\} } , \nn
\end{align}

uniformly for $ x \in ( 2 , d_0 a^{ - 1} n^{\f12 (1-\alpha)}) $.

For $ d_1 =\kappa_{\rho}  a^2   $ with $\kappa_{\rho}\geq 10$, we have

\begin{align}
\P \big(\hatS_{n2} > d_1 n^{ - \alpha/2  }  x B_n  \big)  &  \le \exp  \{  -     \kappa_{\rho} x^2 / 6   \}, \label{error_up}  \\
\P \big( \hatS_{n2} < - d_1 n^{ -  \alpha /2 } x B_n  &  +  x (\hatV_n - V_{n1}), ~V_{n1}^2 >  B_n^2 /4 \big)  \label{error_low2}  \\
&   \le \exp \{  -  \kappa_{\rho}  x^2 /  14    \} . \nn
\end{align}

Note that for $ x > c_{\rho} \sqrt{ \log n  } $ and sufficiently large $ \kappa_{\rho} $,

\begin{align}
\frac { \exp\{  - \kappa_{\rho} x^2 / 6   \}  } { 1 - \Phi \big( \frac { x } { \sqrt {1 + 2 \rho_n} }  \big) }  \leq \exp \{ - \kappa_{\rho} x^2 / 12  \} \leq n^{ - 1/4 }.   \label{error}
\end{align}

By choosing $ \alpha = 1/2 $ and combining (<ref>)-(<ref>), we obtain

\begin{align}
\P \Big(\hatS_n \geq x \hatV_n \Big)
=  \Big[ 1 - \Phi\Big( \f{x}{ \sqrt{ 1 + 2 \rho_n } } \Big) \Big]   \Big( 1 + O \Big( \frac {a ^4  x^2    } { n^{ 1/4 } }  \Big) \Big) \label{trun-re}
\end{align}

uniformly for $ x \in ( c_{\rho} \sqrt{ \log n  } , d_0 a^{ -1  }  n^{ 1/ 8 }  ) $.

In addition, we can bound the error terms in (<ref>) and (<ref>) as follows. The proof of Proposition <ref> is shown in Section <ref> in the Supplementary Material.

Under the conditions in Theorem <ref>, we have  for $  x \in (c_{\rho} \sqrt{\log n}, d_0 a^{-1} n^{1/8} )$ that

\begin{equation}
\begin{split}
& \P \Big( \hatS_n \geq x \hatV_n, \max_{1 \le j \le k} |\xi_{j(l+1)}| > \tau \Big) \\
& \quad \leq A(\rho, d_0) \f{ a^ 4 (1+x)^4} {n^{1/2}}\Bigg[ 1 - \Phi\Big( \f{x}{ \sqrt{ 1 + 2 \rho_n } } \Big) \Bigg]
\end{split}
\label{error-bound1}
\end{equation}

\begin{equation}
\begin{split}
&\P \Big( S_n \geq x V_n, \max_{1 \le j \le k} |\xi_{j(l+1)}| > \tau \Big) \\
& \quad \leq A(\rho, d_0)  \f{a^4 (1 + x)^4}{n^{1/2}}\bigg[ 1-\Phi\Big(  \f{x}{\sqrt{1+2\rho_{ n }}}  \Big) \bigg] .
\end{split}
\label{error-bound2}
\end{equation}

Consequently, substituting (<ref>)–(<ref>) into (<ref>) and (<ref>) yields the desired result (<ref>). This completes the proof of Theorem <ref>.

§.§ Proof of Theorem <ref>

The proof for Theorem <ref> again builds on the big-block-small-block technique, and also exploits a lemma in [28] to replace the weakly dependent big blocks and small blocks by independent random variables, respectively.
We begin the proof by introducing three essential lemmas from the literature. Lemma <ref> (Theorem 4.1 of [28]) and Lemma <ref> (Theorem 10.1.b of [21]) concern the bound of moments under weak dependence while Lemma <ref> (Lemma 2.1 of [4]) shows that a $ \beta $-mixing sequence of random variables can be replaced by an independent sequence of random variables in a domain whose measure is at least $ 1 - \sum_{i = 1}^n  \beta^{(i)}$.

(Theorem 4.1 in [28].)
Let $\{X_i, i \ge 1\} $ be a sequence of zero-mean random variables with $ \e | X_i |^r  \le \mu^r$ for $ r > 2 $ and $ \mu >0  $. Assume that mixing condition (<ref>) holds, then

\begin{equation*}
\e \Big[ \Big| \sum_{i=k}^{i=k+m} X_i \Big|^{r'} \Big] \le C m^{r'/2} \mu^{r'}\;,
\end{equation*}

for any $ 2 \le r' < r, m \ge 1$ and $ k\ge 0 $, where $C$ is a constant that depends on $  r', r, a_1, a_2 $ and $ \tau $.

(Theorem 10.1.b of [21].)
Assume $ \{ X_i \}_{ i \geq 1 } $ is a sequence of random variables and $ \beta(n) $ is the $ \beta $-mixing coeffient defined in (<ref>). Denote by $ \sigma_{1}^k $ and $ \sigma_{k + n}^{\infty}$ the $\sigma$-fields generated by $ \{X_i\}_{1 \leq i \leq k} $ and $ \{ X_{i} \}_{i \geq k + n} $, respectively.
For $ X \in L_p ( \sigma_1^k ) $ and $Y \in L_q (\sigma_{k+n}^{\infty}) $ with $ p, q, r \ge 1 $ and $ \f1p +\f1q + \f1r = 1 $, we have

\begin{equation}
| \e X Y - \e X \e Y | \le 8 \beta(n)^{1/r} \| X \|_p \| Y \|_q \;.  \label{esti-cov}
\end{equation}

For two random variables (or vectors) $X$ and $Y$, define

\begin{equation*}
\beta( X, Y) = \frac 12 \sup_A  \Big(  \big( \P_{X, Y} -  \P_X \times \P_Y  \big) (A) -  \big( \P_{X, Y} -  \P_X \times \P_Y  \big) (A^{c}) \Big).
\end{equation*}

(Lemma 2.1 of [4].)
Let $ \{  \xi_i, 1 \le i \le n \} $ be a sequence of random variables on the same probability space and define $ \beta^{(i)} = \beta ( \xi_i,\, (\xi_{i+1}, \dots, \xi_n) ) $. Then the probability space can be extended with random variables $ \tilde{\xi_i} $ distributed as $ \xi_i$ such that $ \{ \tilde{\xi_i} \}_{ 1 \le i \le n}  $ are independent and

\begin{equation}
\P \Big(\xi_i \neq  \tilde{\xi_i}, ~\mbox{for some} ~1 \le i \le n \Big)  \le \beta^{(1)} + \dots + \beta^{(n-1)}.  \label{betaindepapproerror}
\end{equation}

Recall the definition of block sums $\{Y_j\}_{1 \leq j \leq k }$ in (<ref>). We set the size of big blocks as $ m_1 = [n^{\alpha_1}] $ for some $ 0 < \alpha_1 < 1 - \alpha $ and the size of small blocks as $1$. Denote $ k_1 = k / (m_1 + 1 )$, where $k = [n / l]$.  For simplicity of presentation, we assume $k / (m_1 +1 )$ to be an integer, as explained in the proof of Theorem <ref>. The $ u $-th big block is given by

\begin{equation*}
I_u = \Big\{ j : ( m_1 + 1 )( u -1) + 1 \le j \le (m_1 + 1 )u - 1  \Big\}  , \quad {\rm for}\;  1\le u \le k_1 .
\end{equation*}

\begin{gather*}
S_k = \sum_{j = 1}^k Y_j, \quad V_k^2 = \sum_{j = 1}^k Y_j^2, \quad  \xi_u  = \sum_{j \in I_u } Y_j ,\quad  \eta_u^2 = \sum_{j \in I_u } Y_j^2, \\
S_{k,1} = \sum_{ u =1}^{k_1} \xi_u, \quad V_{k,1}^2 = \sum_{ u =1}^{k_1} \eta_u^2 ,  \quad S_{k,2} = \sum_{u=1}^{k_1} Y_{ u (m_1 + 1)} , \quad  V_{k,2}^2 = \sum_{ u =1}^{k_1} Y_{ u (m_1 + 1)}^2,\\
B_n^2 = \sum_{j=1}^k \e Y_j^2,\quad  B_{n,1}^2 = \sum_{ u =1}^{k_1} \sum_{j \in I_u } \e  Y_j^2,\quad  B_{n,2}^2 = \sum_{ u =1}^{k_1} \e Y_{ u (m_1 + 1)}^2 .
\end{gather*}

By Lemma <ref>, it is easy to see that under the mixing condition (<ref>),

\begin{equation}
\f { B_n^2 } { B_{n,1}^2 } = 1 + O\big(\f{k_1}{k} \big) = 1 + O(n^{- \alpha_1}) , \quad   \f { \e S_{k,1}^2 } { B_{n,1}^2 } = 1 + O( n^{- \alpha} ) + O( n^{- \alpha_1} ) . \label{mo-ratio}
\end{equation}

Let $ \hatY_j = Y_j \IF ( |Y_j| \le b ) $, where $ b = B_{n,2}/(1 + x) $. Parallel to one-dependent case, we separate the big blocks and small blocks right after truncating the terms inside the small blocks. Denote

\begin{align*}
&\hatS_{k, 2}  = \sum_{u = 1}^{k_1} \hatY_{ u ( m_1 + 1  ) }, \quad \hatS_{k} = S_{k, 1} + \hatS_{k, 2}\\
{\rm and}\quad  &\hatV_{k,2}^2 =  \sum_{u = 1}^{k_1} \hatY_{ u ( m_1 + 1  ) }^2 , \quad \hatV_{k}^2 = V_{k, 1}^2 + \hatV_{k, 2}^2 .
\end{align*}

It is straightforward that

\begin{align}
\P \big(  S_k \geq x V_k  \big) &\le \P \big( \hatS_k \ge x \hatV_k \big) + \P \Big( S_k \ge x V_k , \max_{1 \le u \le k_1} |Y_{  u (m_1 + 1) }| > b \Big) , \label{trun1}   \\
\P \big(  S_k \geq x V_k \big) & \ge \P \big( \hatS_k \ge x \hatV_k \big)	- \P\Big( \hatS_k \ge x \hatV_k, \max_{1 \le u  \le k_1} |Y_{ u (m_1 + 1) }| > b \Big) . \label{trun2}
\end{align}

Further, for the main term $ \P ( \hatS_k \ge x \hatV_k  ) $, we choose $\vp = d_1 n ^{ - \alpha_1 / 2 } \log n  $ with  a positive number $d_1 > 0$ and obtain

\begin{align}
\P \Big( \hatS_k \ge x \hatV_k  \Big) &\le \P \big( S_{k,1} \ge x V_{k,1} - \vp x B_n \big) + \P\Big( \hatS_{k,2} > \vp x B_n \Big), \label{upboundfortruncatedinbetamixing} \\
\P \Big( \hatS_k \ge x \hatV_k  \Big) & \ge \P \big( S_{k,1} \ge x V_{k,1} + \vp x B_n \big) \label{lowerbound}\\
&\qquad  - \P\Big( S_{k,1} \ge x V_{k,1} , V_{k,1}^2  \le \f14 B_n^2 \Big)  \nn \\
&\qquad - \P\Bigg( \hatS_{k,2} - \f{x \hatV_{k,2}^2 }{ \hatV_k + V_{k,1} } < -\vp x B_n, V_{k,1}^2 > \f14 B_n^2 \Bigg). \nn
\end{align}

The estimate for dominated terms $ \P (  S_{k,1} \ge x V_{k,1} - \vp x B_n ) $ and $ \P (  S_{k,1} \ge x V_{k,1} + \vp x B_n )  $  is presented in the following lemma and the proof will be shown in Section <ref>.

Assume $ \vp = d_1 n^{\alpha_1 / 2 } \log n  $ for a positive number $d_1 > 0$ and $ \alpha_1 \leq \alpha \tau $.  Under the conditions of Theorem <ref>, there exist a positive constant $ c_0 $ depending on $ d_1, \mu_1 / \mu_2, a_1, a_2 , \alpha$ and $\tau$ such that

\begin{align}
&\P  ( S_{k,1} \ge x V_{k,1} \pm \vp x B_n ) \big/ [1- \Phi(x)  ]  \label{main1} \\
=& 1 + O\bigg(\f{(1 + x)^4}{ n^{1- \alpha- \alpha_1} }  + \f{1 + x }{n^{(1 - \alpha - \alpha_1 )/2 }} + \f{( 1 + x )^2 }{ n^{\alpha }  } + \f{ ( 1 + x )^2 \log n } {n^{ \alpha_1/2 }} \bigg)  \nn
%& \P( S_{k,1} \ge x V_{k,1} + \vp x B_n) \nn \\
%&=  [1- \Phi(x)] \big(1 + O(1) (\f{(1 + x)^4}{ n^{1- \alpha- \alpha_1} } + \f{(1 + x )^3}{\sqrt{n}} + \f{1 + x }{n^{(1 - \alpha - \alpha_1 )/2 }} + \f{( 1 + x )^2 }{ n^{\alpha }  } + \f{ ( 1 + x )^2 } {n^{ \alpha_1/2 }} ) \big)   \label{main2}
\end{align}

uniformly for $ x \in (0, c_0 \min \{ n^{( 1 - \alpha - \alpha_1 )/4 }, n^{\alpha \tau/2}, n^{\alpha/2}, (\log n)^{-1/2}n^{\alpha_1/4}\} )$,

For small-block-related error terms, $ S_{k,2} $ and $ V_{k,2}^2 $ can be replaced with the sum of independent random variables by Lemma <ref>. In addition, following a similar proof to Proposition <ref>, we obtain that under $ \vp = d_1 n^{- \alpha_1/2} \log n $ for $ d_1 $ being some positive number depending on $ \mu_1/\mu_2 $, there exist positive numbers $ C_1 $ and $ C_2 $ depending on $a_1$, $a_2$, $ \mu_1 $, $ \mu_2 $, $\alpha$ and $ \tau $ such that

\begin{align}
& \P\Big( \hatS_{k,2} > \vp x B_n \Big)  \label{er1} \\
& \qquad \le \exp  \{- C_1 ( 1 + x  )^2  d_1 \log n   \} +  C_2 \exp  \{ - a_2 n^{\alpha \tau  } / 2   \},    \nn \\
& \P\Big( \hatS_{k,2} - \f{x \hatV_{k,2}^2 }{ \hatV_k + V_{k,1} } < -\vp x B_n, V_{k,1}^2 > \f14 B_n^2 \Big) \label{er2} \\
& \qquad \leq  \P \Big(   \hatS_{k,2} - \f{x \hatV_{k,2}^2 }{ B_n /2 } < -\vp x B_n \Big)\nn \\
& \qquad \le \exp \{- C_1 ( 1 + x  )^2  d_1 \log n   \} +  C_2 \exp \{ - a_2 n^{\alpha \tau  } / 2 \},    \nn
\end{align}

where the error term $C_2 \exp \{ - a_2 n^{\alpha \tau  } / 2 \}  $ is obtained by applying Lemma <ref> to replace $S_{k, 2}$ and $V^2_{k, 2}$ with sum of independent random variables and the fact that (see [4])

\begin{equation}
\beta \big( \{ X_i\}_{i\in \mathcal{J}'}, \{ X_i\}_{ i\in\mathcal{J}'' } \big) \leq \beta \big( \{ X_i\}_{i\leq k}, \{ X_i\}_{ i\geq n + k } \big) \leq \beta( n ), \label{betamixingproperty}
\end{equation}

for any sets $ \mathcal{J}'\subset \{ i\leq k \}$, $\mathcal{J}''\subset \{ i\geq n+k \}$.

Regarding big-block-related error terms, by using Chebyshev's inequality and Taylor expansion, we control the error term in the same manner as the proof of (<ref>). We can obtain,

\begin{align}
\P \Big( S_{k,1} &\geq x V_{k,1}, V_{k,1}^2  \le \f14 B_n^2 \Big)  \label{er3} \\
&\le A_5 \f{(1 + x)^4}{n^{1 - \alpha - \alpha_1 }} \Big( 1 - \Phi \big[x \big(1 + O \big(n^{-\alpha} + n^{- \alpha_1 } \big) \big)\big] \Big) \nn
\end{align}

uniformly for  $ x \in (3, c_0 \min \{ n^{( 1 - \alpha - \alpha_1 )/4 }, n^{\alpha \tau/2} \} )$.
When $ 0 < x \leq 3 $, it follows from Lemmas <ref> and <ref> and Chebyshev inequality that under condition (<ref>),

\begin{align*}
\P \Big( S_{k,1} \geq x V_{k,1}, V_{k,1}^2  \le \f14 B_n^2 \Big) & \leq \P (  V_{k,1}^2  \le \f14 B_n^2 ) \\
&  \leq   a_1 n^{ \alpha_1 }e^{ - a_2 n^{\alpha \tau }  } + \exp \Big \{  - \frac { B_{n,1}^2 - B_n^2 / 4 } { 2 \sum_{u = 1}^{k_1}  \e \eta_u^4 }  \Big \} \\
& \leq a_1 n^{ \alpha_1 }e^{ - a_2 n^{\alpha \tau }  }  + \exp \{ - A_1  n^{ 1 - \alpha - \alpha_1 } \mu_2 / \mu_1  \} \\
& \leq  \frac { A_2 (1 + x)^4 } { n^{ 1 - \alpha - \alpha_1  } } \big[ 1 - \Phi (x) \big].
\end{align*}

Substituting (<ref>), (<ref>), (<ref>), and (<ref>) into (<ref>) and (<ref>) yields

\begin{align}
&\P (\hatS_k \ge x \hatV_k )/ [1- \Phi(x)  ] \label{truncatedresultinbetamixing} \\
=& 1 + O \bigg(\f{(1 + x)^4}{ n^{1- \alpha- \alpha_1} }  + \f{1 + x }{n^{(1 - \alpha - \alpha_1 )/2 }} + \f{( 1 + x )^2 }{ n^{\alpha }  } + \f{ ( 1 + x )^2 \log n } {n^{ \alpha_1/2 }} \bigg)  \nn
\end{align}

uniformly for $ x \in (0, c_0 \min \{ n^{( 1 - \alpha - \alpha_1 )/4 }, n^{\alpha \tau/2}, n^{\alpha/2}, (\log n)^{-1/2}n^{\alpha_1/4}\} )$.

To avoid redundance, we omit the analysis of the truncation errors in (<ref>) and (<ref>) as their proofs share the same fashion with (<ref>) and (<ref>). Consequently, (<ref>) also holds for $\P(S_k \ge x V_k)$. Finally, we need to balance the error terms by choosing $ \alpha_1 $ and seeking the best convergence rate or largest range for convergence. As a result, we choose $ \alpha_1 = (1 - \alpha)/2 $ when $ (1 - \alpha)/2 \le \alpha \tau$, and choose $ \alpha_1 = \alpha \tau $ when $ (1 - \alpha)/2 > \alpha \tau$, and then the desired result follows. This completes the proof for Theorem <ref>.

§.§ Proof of Theorem <ref>

The main idea is to use one-dependent random variables to approximate $ \{Y_j \}_{1 \leq j \leq k } $ and then apply Theorem <ref>. Recall that $ m = [n^{\alpha}] $ and $ k  =[n /m ] $. Let

\begin{equation*}
\tilde{Y}_j=\e(Y_j|\varepsilon_l, mj-2m+1\leq l\leq mj)
\end{equation*}

\begin{equation*}
\tilde{T}_k=\frac{\sum_{j=1}^{k}\tilde{Y}_j}{ ( \sum_{ j = 1 }^k \tilde{Y}_j^2 )^{ 1/2 } }.
\end{equation*}

As $ \{\varepsilon_t \}_{t \in \mathbbm{Z}} $ are i.i.d. random variables, $\{ \tilde{Y}_j \}_{j\geq 1}$ are one-dependent. Note that by conditional Jensen's inequality, for $ 2 \leq r \leq 4  $,

\begin{align*}
&\Big\Vert X_i  - \e  \Big( X_i \Big\vert \vp_{\ell}: m j  - 2 m + 1 \leq \ell \leq i \Big) \Big\Vert_r^r \nn \\
& \hspace{0.8cm} =  \e \left\{  \Big| \e \Big[ X_i - G_k \Big( \mathscr{F}_{ m j  - 2m }^* , \vp_{ m j  - 2m + 1 }, \ldots, \vp_{i} \Big)  \Big\vert \mathscr{F}_i  \Big] \Big|^r  \right\} \nn \\
& \hspace{0.8cm} \leq \e \Big[ \Big| X_i  -  G_k \Big( \mathscr{F}_{ m j  - 2m }^* , \vp_{ m j  - 2m + 1 }, \ldots, \vp_{i}  \Big)  \Big|^r  \Big] \nn \\
& \hspace{0.8cm} \leq  \Big[ \Delta_r ( i - m j + 2 m   ) \Big] ^{r} ,
\end{align*}

which together with the assumption (<ref>) and the fact that $ m ( j - 1) + 1 \leq i \leq m j  $ for $ i \in  H_j $ yields

\begin{align}
\parallel Y_j- \tilde{Y}_j \parallel_r & \leq \sum_{ i \in H_j  } \Big\Vert   X_i - \e \Big(  X_i \Big\vert \vp_{\ell} : i - 2m + 1 \leq \ell \leq i  \Big)  \Big\Vert_r \label{difference} \\
& \leq m a_1 e^{ -  a_2 m^{\tau}} .   \nn
\end{align}

The above bound shows that $ \{Y_j\}_{1 \leq j \leq k } $ can be well approximated by the one-dependent sequence $ \{ \tilde{Y}_j \}_{1 \leq j \leq k} $. We can derive Theorem <ref> by aggregating the following two propositions. The proofs of Propositions <ref> and <ref> will be provided in Sections <ref> and <ref> in the Supplementary Material, respectively.

Under conditions of Theorem <ref>, we have there exists a positive constant $ d_0 $ depending on $\tau, \alpha, w_1, a_1$ and $a_2$ such that

\begin{align} \label{main-p}
\P ( \tilde{T}_k \geq x )
= [ 1 - \Phi (x) ] \Big( 1 + O \Big( \frac { 1 + x^2  }  { n^{ ( 1 - \alpha )/ 4  } } +  \frac { 1 +  x^2  } { n^{ \alpha } } \Big)  \Big),
\end{align}

Uniformly for $ x \in ( 0, d_0 \min \{ n^{ (1 - \alpha)/8 }, n^{\alpha/ 2}\} )   $.

Under conditions of Theorem <ref>, we have for $ x > 0 $,

\begin{align}
\P (T_k \geq x ) &  \leq \P ( \tilde{T}_k \geq x  - C_1 n^{-1} ) + C_2 ( e^{ - a_2 n^{\alpha \tau}}  + e^{ - O (n^{1 - \alpha})}) \label{error_1p} \\
\mbox{and} \quad \P (T_k \geq x ) &  \geq \P ( \tilde{T}_k \geq x + C_1  n^{-1} ) - C_2 ( e^{ - a_2 n^{\alpha \tau}}  + e^{ - O (n^{1 - \alpha})}). \label{error_2p}
\end{align}

Applying Proposition <ref> to $ \P ( \tilde{T}_k \geq x+ O(  n^{-1} ) ) $ yields

\begin{align*}
\P \Big(  \tilde{T}_k &\geq x + O (n^{-1})  \Big)   \\
& =  \Big[ 1 - \Phi \Big( x +O(n^{-1}) \Big)  \Big]  \Big( 1 + O \Big( \frac { 1 + x^2  } { n^{ ( 1 - \alpha  ) / 4   } } + \frac { 1 + x^2  } { n^{ \alpha  } } \Big) \Big) \nn \\
&  =  \big[ 1 - \Phi (x) \big] \Big( 1 + O \Big( \frac { 1 + x^2  } { n^{ ( 1 - \alpha  ) / 4   } } + \frac { 1 + x^2  } { n^{ \alpha  } } \Big) \Big)
\end{align*}

uniformly for $ x \in (0, d_0 \min  \{ n^{ (1 - \alpha)/8 }, n^{\alpha/ 2}\}) $.
Note that the error term $e^{ - a_2 n^{\alpha \tau}}  + e^{ - O (n^{1 - \alpha})}  $ decays at an exponential rate, which is always faster than the polynomial rate. By substituting the above result into (<ref>) and (<ref>) leads to the desired result (<ref>). The proof of Theorem <ref> is completed.

§ ACKNOWLEDGEMENTS

The authors would like to thank the Co-Editors, Associate Editor, and referees for their constructive comments that have helped improve the paper significantly.

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Supplement to “Refined Cramér Type Moderate Deviation Theorems for
General Self-normalized Sums with Applications to Dependent Random Variables
and Winsorized Mean”.

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Shao, Qi-ManQ.-M. Wang, QiyingQ.
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Supplementary Material to “Asymptotic Distributions of High-Dimensional Distance Correlation Inference"

Lan Gao, Qi-Man Shao and Jiasheng Shi

This Supplementary Material contains all technical details of proofs. Section <ref> provides the proofs of all the propositions, Theorems <ref>, <ref> and <ref>, and Corollary <ref>. Section <ref> presents the proofs of lemmas.

§ PROOF OF PROPOSITIONS

§.§ Proof of Proposition <ref>

The proof borrows some techniques from [31]. The main idea is to first truncate the random variables, and then to apply Berry-Esseen bound for U-statistics and Berry-Esseen bound for sum of independent random variables for the upper bound and the lower bound, respectively. Define

\begin{gather*}
\hatX_i = X_i \mathbbm{1} ( | (1 + x ) X_i | \le 1 ) , \quad \hatS_n = \sum_{i=1}^n \hatX_i , \\
\hatY_i = Y_i \mathbbm{1} ( | (1 + x) Y_i | \le 1 )  ,  \quad  {\hatV_n}^2 = \sum_{i=1}^n \hatY_i^2.
\end{gather*}

It is easy to see

\begin{align}
& \big|\P (S_n > x V_n+c) - [ 1 - \Phi(x + c) ] \big| \label{truncation}  \\
& \leq \big|\P (\hatS_n > x \hatV_n+c) - [ 1 - \Phi(x + c) ] \big|  \nn \\
&\qquad + \sum_{i=1}^n \P \( | (1 + x ) X_i| > 1 \) + \sum_{i=1}^n \P \( | ( 1 + x ) Y_i| > 1 \)  \nn  \\
& \leq \big|\P (\hatS_n > x \hatV_n+c) - [ 1 - \Phi(x + c) ] \big| + (1 + x)^3 L_{3,n}. \nn
\end{align}

Let $ \hatB_n^2 = \sum_{i=1}^n \e \hatY_i^2 $ and $ K_i = \hatY_i^2 - \e \hatY_i^2  $, then it follows from the basic inequality (<ref>) that

\begin{align}
\P (\hatS_n > x \hatV_n+c) &\ge \P\( 2 x \hatS_n -x^2 \hatV_n^2 > x^2 + 2 x c  \), \label{lb}
\end{align}

\begin{align}
& \P (\hatS_n > x \hatV_n+c) \label{ub} \\
&\le \P\Bigg( \hatS_n - \f{x}{2 \hatB_n} \( \hatV_n^2 - \hatB_n^2 \) + \f{x}{\hatB_n^3} \( \hatV_n^2 - \hatB_n^2 \)^2 > x \hatB_n + c \Bigg) \nn \\
&= \P \Bigg( \sum_{i=1}^{n} (U_i - \e U_i) + \f{x}{\hatB_n^3} \sum_{i \ne j} K_i K_j \ge  x \hatB_n + c - \sum_{i=1}^{n} \e U_i \Bigg),    \nn
\end{align}

where $ U_i = \hatX_i - \f{x}{2\hatB_n} K_i +\f{x}{\hatB_n^3} K_i^2 $. Observe that $ \sum_{i=1}^{n} (U_i - \e U_i) + \f{x}{\hatB_n^3} \sum_{i \ne j} K_i K_j $ is a U-statistic. We will apply the Berry-Esseen bound for U-statistics established by [1].
Obviously, $ | \hatB_n^2  -  1 | \leq    (1 + x ) L_{3, n} \leq c_1 $ by (<ref>), hence $ 3/4 \leq \hatB_n^2 \leq 5/4 $ for $ c_1 \leq 1/2  $. We obtain that for $ 0 < x \leq 3  $,

\begin{gather*}
\Big| \sum_{i=1}^{n} \e U_i \Big| \leq  (1 + x)^2 \sum_{i = 1}^n \e  |X_i|^3  + \f{x}{\hatB_n^3} \sum_{i = 1}^n \e \hatY_n^4  \le AL_{3,n}, \\
\Big| \sum_{i=1}^{n}\var( U_i )  -  1 \Big|  \leq A L_{3, n}, \\
\sum_{i=1}^n \e |U_i|^3 \le \sum_{i=1}^n A\Big( \e |\hat{X}_i|^3 +\frac{x^3}{\hat{B}_n^3}\e |K_i|^3 + \frac{x^3}{\hat{B}_n^9}\e |K_i|^6   \Big) \le AL_{3,n},\\
\mb{and} \quad  \sum_{ i \neq j } \var( \f{x}{\hatB_n^3} K_iK_j ) = \sum_{ i \ne j } \f{x^2}{\hat{B}_n^6} \e K_i^2 \e K_j^2  \le AL_{3,n}^2.
\end{gather*}

It follows from the Berry-Esseen bound for U-statistics (see Theorem 1 in [1]) that

\begin{align*}
& \bigg|  \P \Big( \sum_{i=1}^{n} (U_i - \e U_i) + \f{x}{\hatB_n^3} \sum_{i \ne j} K_i K_j \ge  x \hatB_n + c - \sum_{i=1}^{n} \e U_i \Big)   - [ 1 - \Phi (x + c )  ] \bigg| \\
& \leq A L_{3, n}  + \Big| [ 1 - \Phi ( x \hatB_n + c - \sum_{i = 1}^n \e U_i ) ] - [ 1 - \Phi(x + c) ]  \Big| \\
& \leq A L_{3, n} + A \Big( ( \hatB_n - 1 ) + \Big| \sum_{i = 1}^n \e U_i \Big| \Big) \leq A L_{3,n },
\end{align*}

which together with (<ref>) yields the upper bound

\begin{equation}
\P (\hatS_n > x \hatV_n+c) - [1 - \Phi(x + c) ] \le A L_{3,n}. \label{ub_xb}
\end{equation}

Regarding the lower bound, denote $ \hat{W_i} = 2x \hatX_i - x^2 \hatY_i^2 $. It is easy to find that for $ 0 <  x \leq 3 $,

\begin{gather*}
\Big | \sum_{i=1}^n \e \hat{W_i}  +  x^2 \Big|  \leq A x L_{3,n}, \quad \Big| \sum_{i=1}^n \var(\hat{W_i}) -  4x^2 \Big| \leq A  x^2 L_{3,n}, \\
\mb{and}  \qquad \sum_{i=1}^n \e | \hat{W_i} |^3 \leq A x^3  L_{3,n}.
\end{gather*}

Consequently, the Berry-Esseen bound for sums of independent random variables implies

\begin{align}
& \big|\P( 2 x \hatS_n -x^2 \hatV_n^2 > x^2 + 2 x c  ) - [1 - \Phi( x + c )] \big|  \\
&   \leq \Big| \P \Big( \frac { \sum_{i = 1}^n  ( \hat{W}_i -  \e  \hat{W}_i  ) } {  (\sum_{i = 1}^n \var(\hat{W}_i) )^{1/2}  } >    x + c - A L_{3, n}   \Big)  \nn \\
&  \hspace{0.7cm}   - [ 1 + \Phi ( x + c - A L_{3, n}) ]  \Big|  +  | \Phi ( x + c - A L_{3, n} ) - \Phi ( x + c ) |  \nn \\
&   \le   \frac { A \sum_{ i = 1 }^n \e | \hat{W}_i |^3 } { \big(\sum_{i = 1}^n \var(\hat{W}_i)  \big)^{3/2}}+ A L_{3,n} \leq A L_{3, n}\nn ,
\end{align}

which together with (<ref>) yields

\begin{equation}
\P (\hatS_n > x \hatV_n+c) - [1 - \Phi(x + c) ] \ge - A L_{3,n}. \label{lb_xb}
\end{equation}

Combining (<ref>), (<ref>) with (<ref>), we obtain that for $ 0 < x \leq 3 $,

\begin{align*}
\big| \P (S_n > x V_n + c )  -  [1 - \Phi (x+c)] \big| \leq  A  L_{3,n}.
\end{align*}

This completes the proof.

§.§ Proof of Propositions <ref>

Before starting to prove Proposition <ref>, we first collect some notations related to the conjugated method, which is the main tool to prove Propositions <ref>–<ref>. For $1 \leq i \leq n$, let $ W_i = 2 x X_i - x^2 Y_i^2 $ and $ (\xi_i, \eta_i) $ be independent random vectors with distribution

\begin{equation}
V_i \(x,y\) = \f { \e  \{  e^{\la W_i } \mathbbm{1} (X_i\le x, \, Y_i\le y ) \} } { \e e^{ \la W_i }}, \label{change-measure}
\end{equation}

Denote $ \WT{W_i} = 2x \xi_i - x^2 \eta_i^2 $. It holds that

\begin{align*}
\e \WT{W_i} & = \frac { \e W_i e^{\lambda W_i}  } { \e e^{ \lambda W_i }  }, \\
\var \WT{W_i} & = \frac {  \e  W_i^2   e^{ \lambda W_i }  } { \e e^{ \lambda W_i }   } - ( \e \WT{W_i} )^2,  \\
\e | \WT{W_i} |^3 & = \frac { \e | W_i |^3  e^{ \lambda W_i }    } { \e e^{ \lambda W_i }   }.
\end{align*}

We establish the expansion of the above moments in the Lemma below. The proof of Lemma <ref> will be given in Section <ref>.

Let $ W_i = 2x X_i - x^2 Y_i^2 $. For $ \f14 \leq  \la \leq  \f34 $ and $ x > 0  $ satisfying (<ref>), there exists an absolute constant $A$ such that

\begin{align}
\e e^{\la W_i} &= 1 + 2 \la^2 x^2 \e X_i^2 - \la x^2 \e Y_i^2 + \f43 \la^3 x^3 \e X_i^3 - 2 \la^2 x^3 \e X_iY_i^2 + O_1 R_{x,i}, \label{0-mo} \\
&= \exp \left\{ 2 \la^2 x^2 \e X_i^2 - \la x^2 \e Y_i^2 + \f 4 3 \la^3 x^3 \e X_i^3 - 2 \la^2 x^3 \e X_iY_i^2 + O_2 R_{x,i}        \right\},  \nn
\end{align}

\begin{equation}
\e W_ie^{ \la W_i} = 4 \la x^2 \e X_i^2 - x^2 \e Y_i^2 + 4\la^2 x^3 \e X_i^3 - 4 \la x^3\e X_iY_i^2 + O_3 R_{x,i}, \label{1-mo}
\end{equation}

\begin{equation}
\e W_i^2 e^{\la W_i} = 4x^2 \e X_i^2 + 8 \la x^3 \e X_i^3 - 4 x^3 \e X_iY_i^2 + O_4 R_{x,i},  \label{2-mo}
\end{equation}

\begin{equation}
\e |W_i|^3 e^{\la W_i} = O_5 x^3\( \e |X_i|^3 + \e |Y_i|^3 \) + O_6 R_{x,i}, \label{3-mo}
\end{equation}

where $ | O_i  | \leq A $ for $ i = 1, \ldots, 6 $.

The next lemma is Lemma 2.1 in [31] with some modifications.

We have for $x$ satisfying (<ref>) that

\begin{align*}
(1 + x) ^4 (\e X_i^2)^2 &  \le 2 \de_{x,i}, \\
(1 + x)^5 \e X_i^2   \e |X_i|^3 & \le 2 \de_{x,i}, \\
( 1 + x )^6 (\e |X_i|^3)^2 & \le \de_{x,i},
\end{align*}

and similar results hold for $Y_i$. In addition, if $x$ also satisfies (<ref>), then

\begin{equation*}
(1  + x )^4 L_{3,n}^2  \le 2\de_x.
\end{equation*}

By Lemmas <ref> and <ref>, it is readily seen that under (<ref>),

\begin{align*}
\e \WT{W_i} &= x^2 \( 4\la \e X_i^2- \e Y_i^2 \) + x^3 \( 4\la^2 \e X_i^3 - 4\la \e X_iY_i^2 \) + O_1 R_{x,i},  \\
\var \WT{W_i} &= 4x^2 \e X_i^2+ x^3 \( 8\la \e X_i^3 - 4\e X_iY_i^2 \)+ O_2 R_{x,i},
\end{align*}

\begin{equation*}
\e \big| \WT{W_i} \big|^3 = O_3 x^3 \( \e \left|X_i\right|^3 + \e \left|Y_i\right|^3 \) + O_4 R_{x,i}.
\end{equation*}

Put $ m_n = \sum_{i=1}^n \e \WT{W_i}$, $ \si_n^2 = \sum_{i=1}^n Var \WT{W_i} $, $ v_n = \sum_{i=1}^n \e \big| \WT{W_i} \big|^3 $. Consequently, we obtain

\begin{align}
m_n &= \( 4\la - 1 \) x^2 + x^3 \( 4\la^2 \sum_{i=1}^n \e X_i^3 - 4\la \sum_{i=1}^n \e X_iY_i^2 \) + O_1 R_x, \label{expectation} \\
\si_n^2 &= 4x^2 + x^3 \( 8\la \sum_{i=1}^n \e X_i^3 - 4\sum_{i=1}^n \e X_iY_i^2 \) + O_2 R_x, \label{2nd-moment}
\end{align}

\begin{equation}
v_n = O_3 x^3 L_{3,n} + O_4 R_x. \label{3rd-moment}
\end{equation}

Define $ m \(\la\) = \sum_{i=1}^n \log{ \e e^{\la W_i} } $, therefore, $ m_n = m' (\la) $ and $\si_n^2 = m''( \la )$. Before proving Proposition <ref>, let us present the following lemma that will be applied in the proof.

For $ x $ satisfying (<ref>) and (<ref>), if $|\de \(x\)| < \f12 x^2 $ and $|\de_0 \(x\)|<\f12 x^2 $, then the equation

\begin{equation*}
m' \(\la\) = x^2 + \de \(x\),
\end{equation*}

has a unique solution $\la_\de$. In addition, $ \la_{\de} $ satisfies $ \f14 < \la_\de < \f34 $ and

\begin{equation}
\left| \la_\de - \( \f12 + \f {\de(x)}{4x^2} \) + x \( \la_\de^2 \sum_{i=1}^{n} \e X_i^3 - \la_\de \sum_{i=1}^{n} \e X_iY_i^2 \) \right| \le A x^{-2} R_x,
\label{pro-solu}
\end{equation}

\begin{equation}
\left| \la_\de - \la_{\de_0} - \frac{ \delta\left(x\right) -
\de_0 (x) }{ 4x^2 } \right| \le A \{ x^{-2} R_x + \left| \de (x) - \de_0 (x) \right| x^{-1} L_{3,n} \}.
\label{e6}
\end{equation}

Moreover, we have

\begin{equation}
m\( \la_\de \) =  \( 2\la_\de^2 - \la_\de \) x^2 + x^3\( \f43 \la_\de^3  \sum_{i=1}^{n} \e X_i^3 - 2\la_\de^2 \sum_{i=1}^{n} \e X_iY_i^2 \) + O_1 R_x ,
\label{e4}
\end{equation}

\begin{equation}
\left| m \( \la_\de \) - m \( \la_{\de_0} \) \right| \le A \( R_x + \left| \de \(x\) - \de_0 \(x\) \right| \).
\label{e5}
\end{equation}

The proof of Lemma <ref> will be shown in Section <ref>. Now we are ready to prove Proposition <ref>.
The main idea is to use the conjugate method (see (4.9) in [24]).
Assume $ \la_1 $ is the solution to $ m'(\la) = x^2 +2xc $.
Write $ \WT {S_n} = \sum_{i=1}^n \WT {W_i} $, $ U_n = \(\WT {S_n} - m_n \)/\si_n$. It is well-known that by the conjugate method

\begin{align}
\P \( 2x S_n - x^2 V_n^2 \ge x^2 + 2xc \)
=& \exp \{ m(\la_1) \} \e e^{-\la_1 \WT {S_n}} \mathbbm{1}( \WT {S_n} \ge x^2 + 2xc ) \nn \\
=& \exp \{ m(\la_1) - \la_1 m_n\} \e e^{-\la_1 \si_n U_n} \mathbbm{1} ( U_n \ge 0 ). \nn
% &\le \exp \{ m(\la_1) - \la_1 m_n\} \{ I_1 + I_2 \} ,
\end{align}

Let $Z$ be a standard normal random variable. Define

\begin{align*}
I_1  &=  \sup \limits_{ x \in \mathbbm{R} } | \P(U_n \leq x )  -  \P (Z \leq x)|, \\
I_2  & = \e e^{-\la_1 \si_n Z} \mathbbm{1} ( Z \ge 0 )  .
\end{align*}

Integration by parts gives

\begin{align*}
\big| \e e^{-\la_1 \si_n U_n} \mathbbm{1} ( U_n \ge 0 )  - \e e^{-\la_1 \si_n Z} \mathbbm{1} ( Z \ge 0 )  \big|  \leq  2 I_1.
\end{align*}

As a result,

\begin{align}
& \big|  \P \( 2x S_n - x^2 V_n^2 \ge x^2 + 2xc \)  - \exp \{ m(\la_1) - \la_1 m_n\} I_2 \big|  \label{con-re} \\
& \hspace{1cm} \leq  2 \exp \{ m(\la_1) - \la_1 m_n\}  I_1. \nn
\end{align}

Applying the Berry-Esseen theorem to $I_1$ and by the fact of (<ref>) and (<ref>), it is easy to find that for $x > 3 $ satisfying (<ref>),

\begin{equation}
I_1 \le A \f { v_n }{ \si_n^3 } \le A ( L_{3,n} + x^{-3} R_x ).
\label{I1-bound}
\end{equation}

Regarding $ I_2 $, we have

\begin{equation*}
I_2 = \frac{e^{\lambda_1^2 \sigma_n^2/2}}{\sqrt{2\pi}} \int_{ \lambda_1 \sigma_n }^{\infty} e^{ - t ^2 / 2 } dt = \f{1}{ \sqrt{2\pi} } { 1-\Phi(\la_1 \si_n) \over \varphi (\la_1 \si_n) },
\end{equation*}

where $ \phi (x) $ is the standard normal density function.
Let $ \psi (x) = \f { 1 - \Phi(x) }{ \varphi(x) } $. For $ x > 3 $,

\begin{equation}
x^{ -1 }/2 \le \psi ( x )  \le x^{-1} \quad \mb{ and } \quad |\psi ' (x)| = |x \psi (x) - 1| \le x^{-2}. \label{property1}
\end{equation}

By (<ref>) and the fact that $ 1/4 < \lambda_1 < 3/4 $,

| \la_1 \si_n - 2 \la_1 x | = \f { \la_1 |\si_n^2 - 4x^2| }{ \si_n + 2x } \le A \( x^2 L_{3n} + x^{-1} R_x \),

therefore, under (<ref>) for small constant $ c_1 $, there exists some $\theta^*$ between $2\lambda_1 x$ and $\lambda_1 \sigma_n$, such that

\begin{align}
I_2 &= \f {1}{ \sqrt{2\pi} } \{ \psi \( 2 \la_1 x\) + \psi' \( \theta^* \) \( \la_1 \si_n - 2 \la_1 x \) \} \label{I2-bound}  \\
&= \f {1}{ \sqrt{2\pi} }  \psi \( 2 \la_1 x\) \left\{ 1+O_1 ( x L_{3,n} +  x^{-2} R_x ) \right\} \nn  \\
&= e^{ 2 \la_1^2 x^2 } \[ 1-\Phi \( 2 \la_1 x \) \] \left\{ 1 + O_1 (  x L_{3,n} +  x^{-2} R_x ) \right\}.
\nn
\end{align}

Since $ 1 /4 < \lambda_1 < 3/4 $, we have for $x > 3 $

\begin{align*}
e^{ 2 \la_1^2 x^2 } \[ 1-\Phi \( 2 \la_1 x \) \]  \geq  A x^{  -1 },
\end{align*}

which yields

\begin{align*}
I_1  \leq  A  e^{ 2 \la_1^2 x^2 } \[ 1-\Phi \( 2 \la_1 x \) \]   ( x L_{3, n} + x^{ -2 } R_x ).
\end{align*}

Consequently, by (<ref>)–(<ref>) and $ m_n = m'(\la_1) = x^2 + 2 x c  $, we arrive at

\begin{align}
\P (2x &S_n - x^2V_n^2 \ge x^2 + 2xc )   \label{mid-re} \\
&= \exp \big\{ m( \la_1 ) + ( 2\la_1^2 - \la_1 ) x^2 - 2 \la_1 xc  \big\} \nn \\
& \quad \quad \times \[ 1 - \Phi (2\la_1 x) \]\left\{ 1 + O_1 x L_{3,n} +O_2 x^{-2} R_x \right\} \nn \\
&= \exp \big\{ m( \la_1 ) + \f12 (x+c)^2  -\la_1 ( x^2 + 2xc ) \big\} \nn \\
& \quad \quad \times \[ 1 - \Phi (x+c) \]  e^{O_1 R_x} \{ 1 + O_2 x L_{3,n} \}, \nn
\end{align}

and the last equality holds because

\begin{equation*}
|  \la_1  -  \gamma | \le A_1 x L_{3,n} + A_2 x^{-2} R_x.
\end{equation*}

By applying (<ref>) and (<ref>) in Lemma <ref>, where $ \gamma = (1 + c /x ) /2  $.  Noticing that $ x^4 L_{3,n}^2 \le 2\de_x < 2R_x $ by Lemma <ref>, we obtain for $x$ satisfying (<ref>) and (<ref>) that

\begin{equation*}
(2\lambda_1^2-\lambda_1)x^2+\frac{1}{2}(x+c)^2-\lambda_1(x^2+2xc)= 2x^2 (\lambda_1 -\gamma)^2 \leq O_1 R_x.
\end{equation*}

Thus (<ref>) implies

\begin{align}
& \; m( \la_1 ) + \f12 (x+c)^2  -\la_1 ( x^2 + 2xc ) \label{transform} \\
=\; &  \f43 \ga^3 x^3 \sum_{i=1}^n \e X_i^3 - 2 \ga^2 x^3 \sum_{i=1}^n \e X_i Y_i^2 + O_1 R_x. \nn
\end{align}

Finally, we arrive at the desired result (<ref>) by substituting (<ref>) into (<ref>). The proof is completed.

§.§ Proof of Proposition <ref>

The proof again relies on the conjugate method and a randomized concentration inequality for independent random variables. Recalling the definitions of $\la_1,  \widetilde{S_n}  $ and $ U_n  $ at the very beginning of Section <ref>, we have by the conjugate method

\begin{align}
&\P \( 2x S_n - x^2 V_n^2 \ge x^2 + 2xc - x \De_{n} \) \label{P_1+P_2}\\
&\hspace{1cm}  = \exp \{ m(\la_1) \} \e e^{-\la_1 \WT {S_n}} \mathbbm{1}(  \WT {S_n} \ge x^2 + 2xc - x \WT\De_{n} ) \nn \\
&\hspace{1cm}  = \exp \{ m(\la_1) - \la_1 m_n\} \e e^{-\la_1 \si_n U_n} \mathbbm{1} ( U_n \ge -\f {x \WT\De_{n}}{\si_n}) \nn \\
& \hspace{1cm} :=  P_1 + P_2 , \nn
\end{align}

\begin{align*}
P_1 &= \exp \{ m(\la_1) - \la_1 m_n\} \e e^{- \la_1 \si_n U_n } \mathbbm{1} ( U_n > 0 ),\\
P_2 &= \exp \{ m(\la_1) - \la_1 m_n\} \e e^{- \la_1 \si_n U_n } \mathbbm{1} ( -\f {x \WT\De_{n}}{\si_n} \le U_n \le 0 ),
\end{align*}

and $\WT\De_{n} =  \min \{ 2 x ( \sum_{i = 1}^n \eta_i^2 - 1 )^2 , x^{ -1 }  (2 \lor 72 R_x) \} $. We already established in (<ref>) that

\begin{align}
P_1 & = \exp \{ m(\la_1) + \( 2\la_1^2 - \la_1\) x^2 - 2 \la_1 xc \} \label{P1} \\
& \hspace{2cm} \times [ 1 - \Phi ( 2 \la_1 x  ) ] ( 1 + O_1 x L_{3,n} + O_2 x^{-2} R_x ).  \nn
\end{align}

As for $ P_2 $, because $ x \WT\De_{n} \le 2 + 72 R_x $,

\begin{equation}
P_2 \le e^{ 2 + 72 R_x  } \exp \{ m\( \la_1 \) - \la_1 m_n \} \P \Big( -\f {x \WT\De_{n}}{\si_n} \le U_n \le 0 \Big). \label{P2}
\end{equation}

As $ U_n  $ is a normalized sum of independent random variables, we apply the randomized concentration inequality by [29] (Theorem 4.1 therein) to estimate the concentration probability involved in $ P_2 $. As $  \Delta_n \leq 2 x ( V_n^2 -1  )^2 $, it is sufficient to bound the concentration probability as if $ \Delta_n = 2 x ( V_n^2 -1  )^2  $ and $ \widetilde{\Delta}_n = 2 x ( \sum_{i = 1}^n \eta_i^2 - 1 )^2 $. Denote $ Z_i = Y_i^2 - \e Y_i^2 $ and define $\De_{n}^{(i)}$ as $ \De_{n}^{(i)}= 2x(\sum_{ j \ne i } Z_j)^2  $. $\WT\De_{n}^{(i)}$ is defined in the same manner as $\De_{n}^{(i)}$ but with $ \{Y_i \}_{ 1 \leq i \leq n  }$ replaced by $\{ \eta_i  \}_{1 \leq i \leq n}$. We obtain

\begin{equation}
\P \Big( -\f {x \WT\De_{n}}{\si_n} \le U_n \le 0 \Big) \le 17 \f {v_n}{ \si_n^3 } + \f {5x}{ \si_n }  \e \big| \WT\De_{n} \big| + \f {2 x}{ \si_n^2}  \sum_{i=1}^n \e \big| \WT {W_i} \{ \WT\De_{n} - \WT\De_{n}^{(i)} \} \big|. \label{BE}
\end{equation}

Furthermore, it follows from the distribution of $ (\xi_i, \eta_i ) $ defined in (<ref>) that

\begin{align}
\e \big| \WT\De_{n} \big| &= \exp \{ - m (\la_1) \} \e \left\{ |\De_{n}| e^{ \la_1 \sum_{i=1}^n W_i }  \right\} \label{esti-1} \\
&   \le 2 x \exp \{ - m (\la_1) \} \e \left\{  \( V_n^2 -1 \)^2 e^{ \la_1 \sum_{i=1}^n W_i }  \right\} , \nn
\end{align}

\begin{align}
& \e \big| \WT {W_i} \{ \WT\De_{n} - \WT\De_{n}^{(i)} \} \big| \label{esti-2}  	 \\
& = \exp \{ - m (\la_1) \} \e \left\{ \big| W_i \{ \De_{n} -  \De_{n}^{(i)}  \}  \big| e^{ \la_1 \sum_{i=1}^n W_i } \right\}  \nn \\
& \leq  2 x \exp \{ - m (\la_1) \} \e \Big \{ |W_i| \Big| ( V_n^2 - 1 )^2 - \Big(\sum_{ j \ne i } Z_j\Big)^2 \Big| e^{ \la_1 \sum_{i=1}^n W_i }  \Big \}   \nn \\
& =  2 x \exp \{ - m (\la_1) \} \e \Big\{ |W_i| \Big| Z_i^2 + 2 Z_i \sum_{j \ne i} Z_j \Big| e^{ \la_1 \sum_{i=1}^n W_i }  \Big\}. \nn
\end{align}

The lemma below presents the bounds of (<ref>) and (<ref>) and the proof will be presented in Section <ref>.

For $ x $ satisfying (<ref>) and (<ref>) and $ 1 /4 < \la_1 < 3 / 4  $, there exists an absolute constant $A$ such that for $ x > 3 $,

\begin{align}
\e \left\{ \( V_n^2 -1 \)^2 e^{ \la_1 \sum_{i=1}^n W_i } \right\} &\le A_1 \exp \{ m( \la_1 ) \} x^{-2} R_x e^{ A_2 R_x },  \label{error1} \\
\sum_{ i =1 }^n \e \Big\{ \big|W_i\big| \Big| Z_i^2 + 2 Z_i \sum_{j \ne i} Z_j \Big| e^{ \la_1 \sum_{i=1}^n W_i }  \Big\} &\le A_1 \exp \{ m( \la_1 ) \} x^{-2} R_x e^{ A_2 R_x }. \label{error2}
\end{align}

In general, assume $0 < r < r_0 < 1  $ for a  constant $r_0$. For a number $    \omega >   r_0 $, there exist constants $A_1 $ and $ A_2$ depending on $r_0   $ and $ \omega $ such that

\begin{align}
& \e \Big\{ \big( V_n^2 -1 \big)^2  e^{ \sum_{i=1}^n (2rxX_i- \omega r x^2Y_i^2) } \Big\}         \label{error1generalversion}  \\
&   \le \f{A_1 R_x}{x^2} \exp\Big\{  (2r^2- \omega r )x^2-2 \omega r^2 x^3\sum_{ i =1 }^n  \e X_iY_i^2  +   \frac{4}{3}r^3x^3\sum_{ i =1 }^n \e X_i^3+ A_2 R_{x}   \Big\}  .  \nn
\end{align}

Consequently, by assembling (<ref>)–(<ref>) and (<ref>)–(<ref>), it holds that

\begin{align}
P_2 &\le A \exp \{ A R_x \}  \exp \{ m( \la_1 ) - \la_1 m_n \}  ( L_{3,n} +  x^{-1} R_x )  .  \label{Porp2p2}
\end{align}

Moreover, by observing that $ m_n = m' (\la_1 ) = x^2 + 2 x c  $ and $  1 - \Phi ( 2 \la_1 x ) \geq  C x^{ -1 } e^{ - 2 \la_1^2 x^2   } $ for $ x > 2 $ and $ 1/ 4 <  \la_1 < 3 / 4 $, we obtain

\begin{align}
P_2 & \leq  A \exp \{ A R_x \} \exp \{ m( \la_1 ) + (2 \la_1^2 - \la_1) x^2 - 2 \la_1 xc \}   \\
& \quad \quad \times \[ 1 - \Phi( 2 \la_1 x ) \] \{ A_4 x L_{3,n} + A_5 R_x \},  \nn
\end{align}

which combined with (<ref>) yields

\begin{align*}
&\P ( 2x S_n - x^2 V_n^2 \ge x^2 + 2xc - x \De_{n} )  \\
&\le \exp \{ m( \la_1 ) + (2 \la_1^2 - \la_1) x^2 - 2 \la_1 xc \}   \[ 1 - \Phi( 2 \la_1 x ) \] e^{A R_x} (1 + A  x L_{3,n} ) \\
&= \exp \big\{ m( \la_1 ) + \f12 (x+c)^2  -\la_1 ( x^2 + 2xc ) \big\} \\
& \hspace{2cm} \times \[ 1 - \Phi (x+c) \]  e^{A_3 R_x}  ( 1 + A_4  x L_{3,n} ),
\end{align*}

where the last equality is derived by a similar procedure to (<ref>). Finally, we arrive at the desired result (<ref>) by using (<ref>). The proof is completed.

§.§ Proof of Proposition <ref>

We use the truncation technique to estimate the error term. Let $ B = 2500 \lor  200c_0 $. Here the constants $2500$ and $200$ are set large enough for simplicity of proof, because we are not pursuing the best possible constants. The integrating region can be partitioned into three parts as follows,

\begin{equation}
\P\Big( S_n > x V_n + c, | V_n^2 - 1 | >   x^{-1} ( 1 \lor 6 R_x^{1/2})  \Big) \le \sum_{i=1}^3 \P \( ( S_n, V_n^2 ) \in \Omega_i \) ,
\end{equation}

where $ \{ \Omega_i, i=1,2,3 \} $ are given by

\begin{align*}
\Omega_1 &= \big\{ (u,v) \in (R,R^+): ~u > x \sqrt{v} + c,~  1 +  x^{-1} ( 1 \lor 6 R_x^{1/2}) < v \le B \big\}, \\
\Omega_2 &= \big\{ (u,v) \in (R,R^+):~ u > x \sqrt{v} + c,~v < 1 - x^{-1} ( 1 \lor 6 R_x^{1/2}) \big\}, \\
\Omega_3 &= \big\{ (u,v) \in (R,R^+): ~ u > x \sqrt{v} + c, ~ v > B  \big\}.
\end{align*}

For the first part, we choose

\begin{equation*}
r_1= x + c ,\quad   t_1 =  \f{2}{5}x(x + c).
\end{equation*}

By Chebyshev's inequality we obtain

\begin{align}
&\P \big( \big( S_n , V_n^2 \big) \in \Omega_1 \big)  \label{er-1} \\
&\le x^2 \exp \Big\{ - \inf_{ (u,v) \in \Omega_1 } ( r_1 u - t_1 v ) \Big\}\e \left\{ \( V_n^2 - 1 \)^2 e^{ r_1 S_n - t_1 V_n^2 } \right\}. \nn
\end{align}

\begin{align}
& \inf_{ (u,v) \in \Omega_1 } ( r_1 u - t_1 v )  \label{inf} \\
&= r_1 \Big( c + x \sqrt { 1 +   x^{-1} ( 1 \lor 6 R_x^{1/2}) } \Big) - t_1 \big( 1 + x^{-1} ( 1 \lor 6 R_x^{1/2})  \big) . \nn
\end{align}

It follows from applying (<ref>) in Lemma <ref> with $ r  = r_1 / (2 x ) $ and $ t =t_1 / x^2$ (It is easy to verify the conditions for (<ref>) are satisfied when $|c| \leq x / 5 $) that

\begin{align}
&\e \{ \( V_n^2 - 1 \)^2 e^{ r_1 S_n - t_1 V_n^2 } \}  \label{er-11}	 \\
& \leq  \frac{A_1 R_x } { x^2 } \exp \Big\{  \f { r_1^2 }{2} - t_1 + \f {r_1^3}{6} \sum_{i=1}^n \e X_i^3 - r_1 t_1 \sum_{i=1}^n \e [X_i Y_i^2] + A_2 R_x \Big \}. \nn
\end{align}

By plugging (<ref>), (<ref>) and the value of $ r_1,~t_1 $ into (<ref>), we arrive at

\begin{align}
& \P ( ( S_n, V_n^2  ) \in \Omega_1 )  \label{Omega1} \\
& \leq A_1 R_x \Psi_x^*  \exp \Big\{  - \frac { ( x + c )^2 } { 2 } + \f{2}{5} \gamma^2 x^3 \sum_{i = 1}^n \e [X_i Y_i^2] \nn \\
& \quad   -  2 x \gamma ( 1 \lor 6 R_x^{1/2} ) \Big[\big( \sqrt{ 1 + x^{-1} ( 1 \lor 6 R_x^{1/2} ) } + 1 \big)^{-1} - \f{2}{5}   \Big] + A_2 R_x \Big\} , \nn
\end{align}

where $ \gamma = ( x + c ) / (2 x ) $. For $ x > 3  $ satisfying (<ref>) with a small constant $ c_1 \leq \frac {1} {324}$, it holds that $ x^{-1 } (1 \lor 6 R_x^{1/2} ) \leq 1/3 $. Note also that $ x^3  |\e [ X_i Y_i^2 ] | \leq \frac 23  x^3 L_{3, n} \leq x R_x^{1/2} $ by Lemma <ref>. Hence

\begin{align*}
& \P ( ( S_n, V_n^2  ) \in \Omega_1  ) \\
& \leq A_3 x  R_x \Psi_x^* \big[ 1 - \Phi (x + c ) \big] \exp \Big\{ 0.4 \gamma^2 x R_x^{1/2}  -  2 x \gamma  ( 1 \lor 6 R_x^{1/2} ) \cdot 0.06  + A_2 R_x \Big\}.
\end{align*}

Furthermore, observe that $ 1 \lor 6 R_x^{1/2}  \geq 1/2 + 3 R_x^{1/2}$ and $ 2 /5 \leq  \gamma \leq 3/5 $, therefore

\begin{align}
& \P ( ( S_n , V_n^2 ) \in \Omega_1 ) \label{om1} \\
& \leq A_3 x R_x \Psi_x^* \big[ 1 - \Phi (x + c) \big] \exp \{  - 0. 024x  -  0.048 x R_x^{1/2} + A_2 R_x \} \nn \\
& \leq A_4 R_x \Psi_x^*  \big[ 1 - \Phi (x + c) \big] e^{ A_2 R_x }. \nn
\end{align}

As for the second error term, by choosing $ r_2 =  x + c $ and $ t_2 = 2 x (x + c) $, we have under (<ref>) that

\begin{equation}
\inf_{ (u,v) \in \Omega_2 } ( r_2 u - t_2 v ) = r_2 \Big( c + x \sqrt { 1 - x^{-1} (1  \lor   R_x^{1/2} ) } \Big) - t_2 \Big( 1 -  x^{-1} (1  \lor   R_x^{1/2} )  \Big). \label{inf2}
\end{equation}

In the same manner as the proof above, combining (<ref>), (<ref>) and (<ref>) with $r_1,t_1$ replaced by $r_2,t_2$, we obtain

\begin{align}
&\P \( \( S_n , V_n^2 \) \in \Omega_2 \) \label{Omega2} \\
&\qquad \le A_1 x R_x \[ 1 - \Phi (x + c ) \] \Psi_x^*  \nn \\
&\hspace{1.5cm} \times \exp \Big\{ - \f x2 \lor 3xR_x^{1/2}   - 6 \ga^2  x^3 \sum_{i=1}^n \e X_i Y_i^2 + A_2 R_x \Big\} \nn \\
&\qquad \le A_1 R_x [ 1 - \Phi(x + c) ] \Psi_x^* e^{ A_2 R_x }. \nn
\end{align}

Next we deal with $ \P \( \( S_n , V_n^2 \) \in \Omega_3 \) $. Recall the notations

\begin{gather*}
\hatX_i = X_i \mathbbm{1} ( | (1 + x ) X_i | \le 1 ) , \quad \hatS_n = \sum_{i=1}^n \hatX_i , \\
\hatY_i = Y_i \mathbbm{1} ( | (1 + x) Y_i | \le 1 )  ,  \quad  {\hatV_n}^2 = \sum_{i=1}^n \hatY_i^2.
\end{gather*}

We also denote

\begin{gather*}
\bar{X}_i = X_i \mathbbm{1} ( | (1 + x ) X_i | >  1 ) , \quad \bar{S}_n = \sum_{i=1}^n \bar{X}_i , \\
\bar{Y}_i = Y_i \mathbbm{1} ( | (1 + x) Y_i | >  1 )  ,  \quad  {\bar{V}_n}^2 = \sum_{i=1}^n \bar{Y}_i^2.
\end{gather*}

It is evident that

\begin{align*}
& \P \( \( S_n , V_n^2 \) \in \Omega_3 \)   \\
\le& \P \( \hatS_n > \f{  x V_n + c  }{10}, V_n > \sqrt{ B } \) +
\P \( \bar{S}_n > \f{ 9\( x V_n + c  \)  }{10}, V_n > \sqrt{ B } \) \\
\le& K_1 + K_2 + K_3,
\end{align*}

\begin{align*}
K_1 &= \P \( \hatS_n > \f{\sqrt{B}}{10} x + \f {c}{10},\quad \hatV_n^2 > \f B2 \), \\
K_2 &= \P \( \hatS_n > \f{\sqrt{B}}{10} x + \f {c}{10},\quad \bar{V}_n^2 > \f B2 \), \\
K_3 &= \P \( \bar{S}_n > \f{ 9\( x V_n + c \) }{10}, V_n > \sqrt{ B } \).
\end{align*}

By Chebyshev's inequality and recalling that  $ B = \max \{ 2500, 200 c_0 \} $ we have

\begin{equation}
K_1 \le \f{1}{ \( \f B2  - 1 \)^2 } \exp \left\{  - \f 32 x^2 - \f{1}{20} x c \right\} \e \[ \(\hatV_n^2 - 1 \)^2 e^{ \f {x \hatS_n}{2} } \]. \label{K1}
\end{equation}

Let $ \hat{Z}_i = \hatY_i^2 - \e \hatY_i^2 $, then it holds that

\begin{align}
&  \e \[ \(\hatV_n^2 - 1 \)^2 e^{ \f {x \hatS_n}{2} } \]  \label{err}  \\
&  \hspace{0.5cm}= \e \[ \(\sum_{ i =1 }^n \hat{Z}_i - \sum_{ i =1 }^n \e \bar{Y}_i^2 \)^2 e^{ \f {x \hatS_n}{2} } \] \nn \\
&  \hspace{0.5cm}\le  2 \e \[ \(\sum_{ i =1 }^n \hat{Z}_i \)^2 e^{ \f {x \hatS_n}{2} } \] +2 \e \[ \(\sum_{ i =1 }^n \e \bar{Y}_i^2 \)^2 e^{ \f {x \hatS_n}{2} } \]  \nn  \\
&  \hspace{0.5cm} \le  2 \e \[ \(\sum_{ i =1 }^n \hat{Z}_i \)^2 e^{ \f {x \hatS_n}{2} } \] +2 \e \[ x^{-4}\(\sum_{ i =1 }^n \e x^3 \bar{Y}_i^3 \)^2 e^{ \f {x \hatS_n}{2} } \]  \nn  \\
&   \hspace{0.5cm}\le 2 \sum_{i=1}^n \f { \e \hat{Z}_i^2 e^{\f{ x \hatX_i }{2}}} { \e e^{\f{ x \hatX_i }{2}} }
\prod_{ j=1 }^n \e e^{\f{ x \hatX_j }{2}} + 2 x^{-4} R_x^2
\prod_{i=1}^{n} \e e^{\f{ x \hatX_j }{2}}  \nn   \\
& \hspace{1cm} + 4 \sum_{ i \ne j }
\f { \e \hat{Z}_i e^{ \f{ x \hatX_i }{2} } } { \e e^{\f{ x \hatX_i }{2}} }
\f { \e \hat{Z}_j e^{ \f{ x \hatX_j }{2} } } { \e e^{\f{ x \hatX_j }{2}} } \prod_{ j=1 }^n \e e^{ \f{ x \hatX_j }{2} }. \nn
\end{align}

Furthermore, by Taylor expansion and the same argument used in Lemma <ref>, we can obtain

\begin{align}
\e e^{\f{ x \hatX_j }{2}} &=
\exp \Big\{\f18 x^2 \e X_i^2 + \f{1}{48} x^3 \e X_i^3	+ O_1 \de_{x,i} \Big\} ,  \label{0-m}   \\
\e \hat{Z}_i e^{\f{ x \hatX_j }{2}} &= O \( x \( \e |X_i|^3 + \e |Y_i|^3 \) \)
+ O \( x^{-2} \de_{x,i} \), \label{1-m}  \\
\e \hat{Z}_i^2 e^{\f{ x \hatX_j }{2}} &= O \( x^{-4} \de_{x,i} \), \quad {\rm and}\quad \e \bar{Y}_i^2 e^{\f{ x \hatX_j }{2}} = O \( x^{-2} \de_{x,i} \). \label{2-m}
\end{align}

By plugging (<ref>)$-$(<ref>) into (<ref>), and the fact that $ x^4 L_{3,n}^2 \le 4\de_x  $, it is readily seen that

\begin{equation}
\e \[ \(\hatV_n^2 - 1 \)^2 e^{ \f {x \hatS_n}{2} } \] \le \frac{A  R_x}{x^2} \exp \left\{\f18 x^2 + \f{1}{48} x^3 \sum_{i=1}^n \e X_i^3	+ A_1 R_x \right\}. \label{err-bd}
\end{equation}

Substituting (<ref>) into (<ref>) yields for $ |c| \le x/5 $,

\begin{align}
K_1 &\le C_1 x^{-2} R_x \exp \Big\{ - \f{11}{8} x^2 - \f{xc}{20} + \f{1}{48} x^3 \sum_{i=1}^n \e X_i^3
+ A_1  R_x  \Big\} \label{bound-K1}  \\
&\le C_2 x^{-2} R_x \exp \left\{ - \f { (x + c)^2 }{2} - \f 14 x^2 + A_1 R_x \right\} \nn  \\
&\le C_3 \[ 1 - \Phi( x + c) \] R_x \exp \left\{ - \f 18 x^2 + A_1 R_x  \right\} \nn\\
&\le C_4 R_x  \[ 1 - \Phi( x + c) \] \Psi_x^* e^{ A_1 R_x} ,
\nn
\end{align}

and the last inequality holds because $ x $ satisfies (<ref>) and

\begin{align*}
\big| x^3 \sum_{i=1}^n \e X_i^3  \big| \le x^3 L_{3,n}  ,\quad \mb{and} \quad  \big| x^3 \sum_{i=1}^n \e X_i Y_i^2 \big| \le x^3 L_{3,n}.
\end{align*}

Similarly, exploiting the upper bounds (<ref>) and (<ref>), we have

\begin{equation}
K_2 < \f 2B \exp \left\{ - \f 32 x^2 - \f{xc}{20} \right\}
\e \( \bar{V}_n^2 e^{ \f{ x \hat{S}_n }{2}} \),
\end{equation}

\begin{align*}
\e \( \bar{V}_n^2 e^{ \f{ x \hat{S}_n }{2}} \) &= \sum_{i=1}^n \f{\e \Big( \bar{Y}_i^2   e^{ \f{ x \hat{X}_i }{2}} \Big) }
{ \e e^{ \f{ x \hat{X}_i }{2}} }\prod_{ j = 1 }^n \e e^{ \f{ x \hat{X}_j }{2}} \nn  \\
&\le \f{AR_x}{x^2} \exp \left\{ \f18 x^2 + \f{1}{48} x^3 \sum_{i=1}^n \e X_i^3	+ A_1 R_x \right\}.
\end{align*}

Hence, in the same manner as the proof of (<ref>), it follows that

\begin{equation}
K_2 \le A R_x \[ 1 - \Phi( x + c) \] \Psi_x^* e^{A_1 R_x}. \label{bound-K2}
\end{equation}

Finally, as for the bound for $ K_3 $, denote

\begin{equation*}
\bar{X}_i^{(1)}=\bar{X}_i\mathbbm{1}\big( 2xX_i\leq \f{X_i^2}{Y_i^2+c_0\e Y_i^2}   \big):=\bar{X}_i\mathbbm{1}(\mathcal{M}),\quad {\rm and} \quad \bar{X}_i^{(2)}=\bar{X}_i- \bar{X}_i^{(1)}.
\end{equation*}

Cauchy inequality leads to

\begin{align*}
K_3 &= \P \Big( \sum_{i=1}^n \bar{X}_i  > \f { 9 \( x V_n + c \) }{10}, \;\;  V_n > \sqrt{B} \Big) \nn  \\
&\leq \P\Big( \sum_{i=1}^n \bar{X}_i^{(1)}  > \f {  \( x V_n + c \) }{100}, \;\;  V_n > \sqrt{B} \Big) \nn  \\
&\qquad + \P\Big( \sum_{i=1}^n \bar{X}_i^{(2)}  > \f { 89 \( x V_n + c \) }{100}, \;\;  V_n > \sqrt{B} \Big) \nn  \\
&\le \P\Big( \sum_{i=1}^n 2x\bar{X}_i^{(1)}  > \f {  \( x^2 V_n + cx \) }{50}, \;\;  V_n > \sqrt{B} \Big), \nn \\
&\qquad + \P \Bigg( \sqrt{ \sum_{i=1}^n \f{ (\bar{X}_i^{(2)})^2 }{ Y_i^2 + c_0 \e Y_i^2 } } > \f{89}{100} \f{x V_n + c }{ \sqrt{ V_n^2 + c_0 } }, \;\;  V_n >  \sqrt{B}   \Bigg) \nn \\
& := K_4+K_5.
\end{align*}

Recalling that $ B = \max \{ 2500, 200 c_0 \} $. we further have for $ |c| \le x/5 $

\begin{align}
K_4 &\le \P \(  \sum_{i=1}^n 2x\bar{X}_i^{(1)} >  0.996x^2    \)  \label{bound-K3} \\
&\le  \f {C_1 x^{-2} }{ \exp \{ 0.986 x^2 \} }  \e \[ \sum_{i=1}^n 2x\bar{X}_i^{(1)}  e^{ 0.99 \sum_{i=1}^n 2x\bar{X}_i^{(1)}  } \]  \nn \\
&\le  \f { C_2x^{-2} }{ \exp \{ 0.986 x^2 \} }  \sum_{i=1}^n \e \Big[ e^{2x\bar{X}_i^{(1)} }\mathbbm{1}\big( |(1  + x)X_i|>1, \mathcal{M}  \big) \Big]  \prod_{ j \ne i } \e e^{ 2x\bar{X}_j^{(1)} } \nn\\
&\le A R_x \[ 1 - \Phi (x + c) \] \Psi_x^* \exp{\{ R_x  \} }, \nn
\end{align}

\begin{align}
K_5 &\le \P \(  \sum_{i=1}^n \f{ (\bar{X}_i^{(2)})^2 }{ Y_i^2 + c_0 \e Y_i^2 }> ( 0.884x )^2   \)  \nn \\
&\le  \f { C_1x^{-2} }{ \exp \{ 0.773 x^2 \} }  \e \[ \sum_{i=1}^n \f {(\bar{X}_i^{(2)})^2}{Y_i^2 + c_0 \e Y_i^2}  e^{ 0.99 \sum_{i=1}^n \f {(\bar{X}_i^{(2)})^2}{Y_i^2 + c_0 \e Y_i^2}  } \]  \nn \\
&\le \f { C_2 x^{-2} }{ \exp \{ 0.773 x^2 \} }  \sum_{i=1}^n \e \Big[ e^{\f {(\bar{X}_i^{(2)})^2}{Y_i^2 + c_0 \e Y_i^2}}\mathbbm{1}\big( |x'X_i|>1, \mathcal{M}^c  \big) \Big] \prod_{ j \ne i } \e e^{ \f {(\bar{X}_i^{(2)})^2}{Y_j^2 + c_0 \e Y_j^2}  } \nn\\
&\le A R_x \[ 1 - \Phi (x + c) \] \Psi_x^* \exp{\{ R_x  \} }, \nn
\end{align}

where we have used the condition $ x L_{3,n} \le c_1 $, for $c_1$ being some small enough constant, and that

\begin{align*}
\prod_{ j \ne i } \e e^{ 2x \bar{X}_i^{(1)}} &\le  \prod_{ j \ne i } \left\{ 1 + \e e^{ 2x \bar{X}_i^{(1)}} \mathbbm{1} ( |x' X_i | > 1, \mathcal{M} ) \right\}   	\\
&\le \exp \left\{ \sum_{i=1}^n \e e^{ 2x \bar{X}_i^{(1)} } \mathbbm{1} ( |x' X_i | > 1, \mathcal{M} ) \right\}   \\
&\le \exp \{ r_x \} \le \exp \{ R_x \},\\
\prod_{ j \ne i } \e e^{ \f {(\bar{X}_i^{(2)})^2}{Y_j^2 + c_0 \e Y_j^2}} &\le  \prod_{ j \ne i } \left\{ 1 + \e e^{\f {X_i^2}{Y_i^2 + c_0 \e Y_i^2} } \mathbbm{1} ( |x' X_i | > 1, \mathcal{M}^c ) \right\}   	\\
&\le \exp \left\{ \sum_{i=1}^n \e e^{\f {X_i^2}{Y_i^2 + c_0 \e Y_i^2} } \mathbbm{1} ( |x' X_i | > 1, \mathcal{M}^c ) \right\}   \\
&\le \exp \{ r_x \} \le \exp \{ R_x \}, \\
R_x\geq r_x & = \sum_{i=1}^n \e \Big[ e^{2x\bar{X}_i^{(1)} }\mathbbm{1}\big( |x'X_i|>1, \mathcal{M}  \big) \Big]\\
& \qquad + \sum_{i=1}^n \e \Big[ e^{\f {(\bar{X}_i^{(2)})^2}{Y_i^2 + c_0 \e Y_i^2}}\mathbbm{1}\big( |x'X_i|>1, \mathcal{M}^c  \big) \Big].
\end{align*}