text:
\section{Introduction}
\label{sec:introduction}
During the past decade, there has been a significant development on high-dimensional data analysis with applications in many fields. In this paper we shall consider simultaneous inference for mean vectors of high-dimensional stationary processes, so that one can perform family-wise multiple testing or construct simultaneous confidence intervals, an important problem in the analysis of spatial-temporal processes. To fix the idea, let $X_i$ be a stationary process in $\mathbb R^p$ with mean $\mu = (\mu_1, \ldots, \mu_p)^\top$ and finite second moment in the sense that $\E (X_i^\top X_i) < \infty$. In the scalar case in which $p = 1$ or when $p$ is fixed, under suitable weak dependence conditions, we can have the central limit theorem (CLT)
\begin{eqnarray}
\label{eq:15140307}
{1\over \sqrt n} \sum_{i=1}^n (X_i-\mu) \Rightarrow N(0, \Sigma), \mbox{ where }
  \Sigma = \sum_{k=-\infty}^\infty \E ( (X_0-\mu) (X_k-\mu)^\top).
\end{eqnarray}
See, for example, \citet{rosenblatt1956central}, \citet{ibragimov1971independent}, \citet{Wu2005}, \citet{dedecker2007weak}
and \citet{bradley2007introduction} among others. In the high dimension case in which $p$ can also diverge to infinity, \citet{portnoy1986central} showed that the central limit theorem can fail for i.i.d. random vectors if $\sqrt n = o(p)$. In this paper we shall consider an alternative form: Gaussian approximation for the largest entry of the sample mean vector  $\bar X_n = n^{-1} \sum_{i=1}^n X_i$. For a vector $v = (v_1, \ldots, v_p)^\top$, let $|v|_\infty = \max_{j\le p} |v_j|$. Specifically, our primary goal is to establish the Gaussian Approximation (GA) in $\mathbb R^p$
\begin{eqnarray}\label{eq:1507172206}
 \sup_{u \ge 0} |\P( \sqrt n |\bar X_n-\mu|_\infty \ge u) - \P(|Z|_\infty \ge u)| \to 0,
\end{eqnarray}
where both $n, p \to \infty$. Here the Gaussian vector $Z = (Z_1, \ldots, Z_p)^\top \sim N(0, \Sigma)$.   \citet{chernozhukov2013} studied the Gaussian approximation for independent random vectors. There has been limited research on high-dimensional inference under dependence. The associated statistical inference becomes considerably more challenging since the autocovariances with all lags should be considered. \citet{zhang2014} extended the Gaussian approximation in \citet{chernozhukov2013} to very weakly dependent random vectors which satisfy a uniform geometric moment contraction condition. The latter condition is also adopted in \citet{csw2015} for self-normalized sums. \citet{chernozhukov2013testing} did a similar extension to strong mixing random vectors. Here we shall establish (\ref{eq:J021209}) for a wide class of high-dimensional stationary process under suitable conditions on the magnitudes of $p$, $n$, and the mild dependence conditions on the process $(X_i)$.

In Section \ref{sec:high-dimentional time series} we shall introduce the framework of high-dimensional time series and some concepts about functional and predictive dependence measures that are useful for establishing an asymptotic theory.  The main result for Gaussian approximation of the normalized mean vector and the choice of the normalization matrix is established in Section \ref{sec:Gaussian approximation}. Depending on the moment and the dependence conditions, both high dimension and ultra high dimension cases are discussed.

To perform statistical inference based on (\ref{eq:J021209}), one needs to estimate the long-run covariance matrix $\Sigma$. The latter problem has been extensively studied in the scalar case; see \citet{politis1999subsampling}, \citet{buhlmann2002}, \citet{lahiri2003resampling}, \citet{Alexopoulos2004}, among others. In Section \ref{sec:estimation of long-run variance} we study the batched-mean estimate of long-run covariance matrices and derive a large deviation result about quadratic forms of stationary processes. The latter tail probabilities inequalities allow dependent and/or non-sub-Gaussian processes under mild conditions, which is expected to be useful in other high-dimensional inference problems for dependent vectors. The consistency of the batched-mean estimate ensures the validity of the normalized Gaussian multiplier bootstrap method.

We provide in Section \ref{sec:probability inequalities} some sharp inequalities for tail probabilities for dependent processes in both polynomial tail and exponential tail cases. Part of the proof are relegated to Section \ref{sec:proof}.

We now introduce some notation. For a random variable $X$ and $q > 0$, we write $X \in \mathcal{L}^q$ if $\|X\|_{q} := (\E |X|^q)^{1/q} < \infty$, and for a vector $v = (v_1, \ldots, v_p)^\top$, let the norm-$s$ length $|v|_s = ( \sum_{j=1}^p |v_j|^s)^{1/s}$, $s \ge 1$. Write the $p \times p$ identity matrix as $\text{Id}_p$. For two real numbers, set $x \vee y =\max(x, y)$ and $x \wedge y =\min(x, y)$. For two sequences of positive numbers $(a_n)$ and $(b_n)$, we write $a_n \asymp b_n$ (resp. $a_n \lesssim b_n$ or $a_n \ll b_n$) if there exists some constant $C > 0$ such that $C^{-1} \leq a_n/b_n \leq C$ (resp. $a_n/b_n \leq C$ or $a_n/b_n \to 0$) for all large $n$. We use $C, C_1, C_2, \cdots$ to denote positive constants whose values may differ from place to place. A constant with a symbolic subscript is used to emphasize the dependence of the value on the subscript. Throughout the paper, we assume $p=p_n \rightarrow \infty$ as $n\rightarrow \infty$.

\section{High-dimensional Time Series}
\label{sec:high-dimentional time series}
Let $\eps_i, i \in \mathbb{Z}$, be i.i.d. random variables and $\calF^{i}=(\ldots, \eps_{i-1}, \eps_i)$; let $({X}_i)$ be a stationary process taking values in $\mathbb{R}^p$ that assumes the form
\begin{equation}
\label{highdimensionrepresentation}
X_i = (X_{i1}, X_{i2}, \ldots, X_{ip})^\top=G(\calF^{i}),
\end{equation}
where $G(\cdot)=(g_1(\cdot), \ldots, g_p(\cdot))^\top$ is an $\mathbb{R}^p$-valued measurable function such that $X_i$ is well-defined. In the scalar case with $p = 1$, (\ref{highdimensionrepresentation}) allows a very general class of stationary processes (cf. \citet{wiener1958nonlinear}, \citet{rosenblatt1971markov}, \citet{priestley1988non}, \citet{tong1990non}, \citet{Wu2005}, \citet{tsay2005analysis}, \citet{wu2011asymptotic}). It includes linear processes as well as a large class of nonlinear time series models. Within this framework, $(\eps_i)$ can be viewed as independent inputs of a physical system and all the dependences among the outputs $(X_i)$ result from the underlying data-generating mechanism $G(\cdot)$. The function $g_j(\cdot)$, $1 \leq j \leq p$, is the $j$-th coordinate projection of $G(\cdot)$. Unless otherwise specified, assume throughout the paper that $\E X_i=0$ and $\max_{j \le p} \| X_{ij} \|_q < \infty $ for some $q \geq 2$. Let $\Gamma(k) = (\gamma_{ij}(k))_{i, j=1}^p = \E (X_i X_{i+k}^\top)$ be the autocovariance matrix and recall the long-run covariance matrix
\begin{eqnarray}
\label{eq:J13426p}
\Sigma=(\sigma_{i j})_{i, j=1}^p=\sum_{k=-\infty}^\infty \Gamma(k)
\end{eqnarray}
if it exists. Note that $\sigma_{jj}=\sum_{k=-\infty}^\infty \gamma_{jj}(k)$, $1 \leq j \leq p$, is the long-run variance of the component process $X_{\cdot j} = (X_{i j})_{i \in \mathbb{Z}}$. For the latter process, following \citet{Wu2005} we define respectively the functional dependence and the predictive dependence measure
\begin{eqnarray}\label{eq:1507140845}
\delta_{i, q, j} &=& \| X_{ij}-X_{ij, \{0\}} \|_q=\| X_{ij}-g_j(\calF^{i, \{0\}}) \|_q, \cr
\theta_{i, q, j}&=&\| \E(X_{ij}|\calF^{0})-\E(X_{ij}|\calF^{-1})\|_q = \| \mathcal{P}^0 X_{ij} \|_q,\cr
\theta'_{i, q, j}&=&\|\E(X_{ij}|\calF_{0}^i)-\E(X_{ij}|\calF_{1}^i)\|_q=\|\mathcal{P}_0 X_{ij}\|_q,
\end{eqnarray}
where $\mathcal{F}^{i, \{j\}}=(\ldots, \varepsilon_{j-1}, \varepsilon_{j}', \varepsilon_{j+1}, \ldots, \varepsilon_{i})$ is a coupled version of $\mathcal{F}^{i}$ with $\varepsilon_{j}$ in $\mathcal{F}^i$ replaced by $\varepsilon_{j}'$, and $\varepsilon_k, \varepsilon_l'$, $k, l \in \mathbb{Z}$, are i.i.d. random variables, $\calF_i^j=(\eps_i, \eps_{i+1}, \ldots, \eps_j)$ and $\calF_{i}=(\eps_i, \eps_{i+1}, \ldots)$. Note that $\mathcal{F}^{i, \{j\}}= \mathcal{F}^{i}$ if $j > i$. To account for the dependence in the process $X_{\cdot j}$, we define the dependence adjusted norm
\begin{eqnarray}
\label{eq:2015081932}
\|X_{\cdot j} \|_{q, \alpha} = \sup_{m \geq 0} (m+1)^\alpha \Delta_{m, q, j},\,  \alpha \geq 0,
 \mbox{ where }  \Delta_{m, q, j}= \sum_{i=m}^\infty \delta_{i, q, j}, \,  m \geq 0.
\end{eqnarray}
Due to the dependence, it may happen that $\| X_{i j} \|_q < \infty$ while $\|X_{\cdot j} \|_{q, \alpha} = \infty$. Elementary calculations show that, if $X_{i j}, i \in \mathbb{Z}$, are i.i.d., then $\| X_{i j} \|_q \le \|X_{\cdot j} \|_{q, 0} \le 2 \| X_{i j} \|_q$, suggesting that the dependence adjusted norm is equivalent to the classical $L^q$ norm.

To account for high-dimensionality, we define
\begin{eqnarray*}
\Psi_{q, \alpha}= \max_{1 \leq j \leq p}\|X_{\cdot j}\|_{q, \alpha}
 \mbox{ and }
\Upsilon_{q, \alpha}=\left(\sum_{j=1}^p \|X_{\cdot j}\|_{q, \alpha}^q \right)^{1/q},
\end{eqnarray*}
which can be interpreted as the uniform and the overall dependence adjusted norms of $(X_i)_{i \in \mathbb{Z}}$, respectively.  The form (\ref{highdimensionrepresentation}) and its associated dependence measures provide a convenient framework for studying high-dimensional time series. \citet{chen2013} and \citet{zhang2014} considered some special cases:  the former paper requires that  $\max_{1 \leq j \leq p}\|X_{\cdot j}\|_{q, \alpha} \leq C$ while the latter imposes the stronger geometric moment contraction condition $\max_{1 \leq j \leq p}\Delta_{m, q, j} \le C \rho^m$ with $\rho \in (0, 1)$, and in both cases the constant $C$ does not depend on $p$. Those assumptions can be fairly restrictive. In this paper $\Psi_{q, \alpha}$ can be unbounded in $p$. Additionally, we define the $\mathcal{L}^{\infty}$ functional dependence measure and its corresponding dependence adjusted norm for the $p$-dimensional stationary process $(X_i)$
\begin{eqnarray*}
&&\omega_{i, q} = \| |X_i - X_{i, \{0\}}|_{\infty} \|_q;\\
&& \| |X_{\cdot}|_{\infty} \|_{q, \alpha}=\sup_{m \geq 0} (m+1)^\alpha \Omega_{m,q}, \alpha\geq 0, \text{ where } \Omega_{m,q}=\sum_{i=m}^\infty \omega_{i,q}, m \geq 0.
\end{eqnarray*}
Clearly, we have $\Psi_{q, \alpha} \leq \||X_{\cdot}|_\infty\|_{q, \alpha} \leq \Upsilon_{q, \alpha}$.

\section{Gaussian Approximations}
\label{sec:Gaussian approximation}

In this section we shall present main results on Gaussian approximations. Theorem \ref{th:1507140259} concerns the finite polynomial moment case with both weaker and stronger temporal dependence. Consequently the dimension $p$ allowed can be at most a power of $n$. If the underlying process has finite dependence-adjusted sub-exponential norms, Theorem \ref{th:1507140325} asserts that an ultra-high dimension $p$ can be allowed. Theorem \ref{maintheorem} in Section \ref{sec:1507151006} provides a convergence rate of the Gaussian approximation.

Recall (\ref{eq:J13426p}) for the long-run covariance matrix $\Sigma$. Let $\Sigma_0 = {\rm diag}(\Sigma)$ be the diagonal matrix of $\Sigma$, and $D_0 = {\rm diag}(\sigma_{11}^{1/2}, \ldots, \sigma_{pp}^{1/2})$. Assume $\mu = 0$. We consider the following normalized version of (\ref{eq:1507172206}):
\begin{eqnarray}\label{eq:J021209}
 \sup_{u \ge 0} |\P( \sqrt n |D_0^{-1} \bar X_n|_\infty \ge u) - \P(|D_0^{-1} Z|_\infty \ge u)| \to 0,
\end{eqnarray}

\begin{assumption}
\label{assumption:1507172202}
There exists a constant $c > 0$ such that $\min_{1 \leq j \leq p} \sigma_{jj} \ge c$.
\end{assumption}

To state Theorem \ref{th:1507140259}, we need to define the following quantities:  $\Theta_{q, \alpha}=\Upsilon_{q, \alpha} \wedge (\||X_{\cdot}|_\infty\|_{q, \alpha} \log p)$, $L_1 = (n^{1/q-1/2} (\log p)^{1/2} \Theta_{q, \alpha})^{1/(\alpha - 1/2 + 1/q)}$, $L_2 = (\Psi_{2, \alpha} \Psi_{2, 0} (\log p)^2 )^{1/\alpha}$, $W_1 = (\Psi_{3, 0}^6 + \Psi_{4, 0}^4) (\log (p n) )^7$, $W_2 = \Psi_{2, \alpha}^2 (\log (p n) )^4$, $W_3 = (n^{-\alpha} (\log (pn))^{3/2}  \Theta_{q, \alpha})^{1/(1/2-\alpha - 1/q)}$, $N_1 = (n / \log p)^{q/2} / \Theta_{q, \alpha}^q$, $N_2 = n (\log p)^{-2} \Psi_{2, \alpha}^{-2}$, $N_3 = (n^{1/2} (\log p)^{-1/2}  \Theta_{q, \alpha}^{-1})^{1/(1/2-\alpha)}$.

\begin{thm}
\label{th:1507140259}
Let Assumption \ref{assumption:1507172202} be satisfied. (i) Assume that $\Theta_{q, \alpha} < \infty$ holds with some $q \geq 4$ and $\alpha > 1/2 - 1/q$ (the weaker dependence case),
\begin{eqnarray}
\label{eq:1507142040}
\Theta_{q, \alpha} n^{1/q-1/2} (\log (p n) )^{3/2} \to 0
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:1507142041}
\max(L_1, L_2) \max(W_1, W_2) = o(1) \min(N_1, N_2).
\end{eqnarray}
Then the Gaussian Approximation (\ref{eq:J021209}) holds. (ii) Assume $0 < \alpha < 1/2 - 1/q$ (the stronger dependence case). Then (\ref{eq:J021209}) holds if $\Theta_{q, \alpha} (\log p)^{1/2} = o(n^{\alpha})$ and
\begin{eqnarray}
\label{eq:1508010523}
L_2 \max(W_1, W_2, W_3) = o(1) \min(N_2, N_3).
\end{eqnarray}
\end{thm}

\begin{rmk} (Optimality of our result on the allowed dimension $p$)
Assume $\alpha > 1/2 - 1/q$. In the special case with $\Psi_{q, \alpha} \asymp 1$ and $\Theta_{q, \alpha} \asymp p^{1/q}$, (\ref{eq:1507142040}) becomes
\begin{eqnarray}
\label{eq:1507142042}
p (\log (p n) )^{3q/2} = o(n^{q/2-1}),
\end{eqnarray}
which by elementary manipulations implies (\ref{eq:1507142041}), and hence the GA (\ref{eq:J021209}). It turns out that condition (\ref{eq:1507142042}), or equivalently $p (\log p)^{3q/2} = o(n^{q/2-1})$, is optimal up to a multiplicative logarithmic term. Consider the special case in which $X_{i j}$, $i, j \in \mathbb{Z},$ are i.i.d. symmetric random variables with $\E (X_{i j}^2) = 1$ and the tail probability $\P( X_{i j} \ge u) = u^{-p} \ell(u)$, $u \ge u_0$, where $\ell(u) = (\log u)^{-2}$. By \citet{nagaev1979large}, we have the expansion: for $y \ge \sqrt n$,
\begin{eqnarray}
\label{eq:J011029}
\P(X_{1 1} + \ldots + X_{n 1} \ge y) \sim n y^{-q} \ell(y) + 1 - \Phi(y/\sqrt n).
\end{eqnarray}
Let $M_n = X_{1 1} + \ldots + X_{n 1}$, $Z = (Z_1, \ldots, Z_p)^\top \sim N(0, \text{Id}_p)$ and assume
\begin{eqnarray}
\label{eq:1507142204}
n^{q/2-1} = o(p (\log n)^{-2} (\log p)^{-q/2}).
\end{eqnarray}
Then the Gaussian approximation (\ref{eq:J021209}) {\it does not hold}. To see this, let $u = (2 \log p)^{1/2}$. Then $p \P (|Z_1| \ge u) \to 0$, and, by (\ref{eq:J011029}) and (\ref{eq:1507142204}), $p \P( M_n \ge \sqrt n u) \to \infty$. Hence $\P^p( |M_n| \le \sqrt n u) \to 0$ and $\P^p(|Z_1| \le u) \to 1$, implying that
\begin{eqnarray*}
|\P( \sqrt n |\bar X_n|_\infty \le u) - \P(|Z|_\infty \le u)|
 &=&  |\P^p( |M_n| \le \sqrt n u) - \P^p(|Z_1| \le u)| \cr
 &=& | [1-2 \P( M_n \ge \sqrt n u)]^p - \P^p(|Z_1| \le u)|
 \to 1.
\end{eqnarray*}
Note that (\ref{eq:1507142204}) is equivalent to $n^{q/2-1} = o(p (\log p)^{-2-q/2})$, suggesting that (\ref{eq:1507142042}) is optimal up to a logarithmic term. \qed
\end{rmk}

Now suppose there exist $0 \le \kappa_1 \le \kappa_2$ such that $\Psi_{q, \alpha} \asymp p^{\kappa_1}$ and $\Theta_{q, \alpha} \asymp p^{\kappa_2}$, and $p^\tau \asymp n$. Elementary but tedious calculations show that, in the weaker dependence case $\alpha > 1/2 - 1/q$, if
\begin{eqnarray}
\label{eq:1507150901}
\tau > \max\left\{\frac{\kappa_2}{1/2-1/q}, \frac{2\kappa_1}{\alpha}+8\kappa_1, \frac{2}{q}\left(\frac{2\kappa_1}{\alpha}+8\kappa_1\right)+2\kappa_2\right\},
\end{eqnarray}
then conditions in (i) of Theorem \ref{th:1507140259} are satisfied, while for the stronger dependence case with $0 < \alpha < 1/2 - 1/q$, a larger sample size $n$ is required:
\begin{eqnarray}
\label{eq:1507150902}
\tau > \max\left\{\frac{\kappa_2}{\alpha}, \frac{2\kappa_1}{\alpha}+8\kappa_1,(1-2\alpha)\left(\frac{2\kappa_1}{\alpha}+8\kappa_1\right)+2\kappa_2 \right\}
\end{eqnarray}
The lower bounds in (\ref{eq:1507150901}) and (\ref{eq:1507150902}) are both non-decreasing of $\kappa_1, \kappa_2$ and non-increasing in $q, \alpha$.

Under (\ref{eq:1507142042}), the allowed dimension $p$ can only be at most a polynomial of $n$. To ensure the validity of GA in the ultra-high dimensional case with $\log p = o(n^c)$ with some $c > 0$, we need to consider the sub-exponential case in which $X_{i j}$ has finite moment with any order. For $\nu \ge 0$ and $\alpha \ge 0$, define the dependence-adjusted sub-exponential norm
\begin{eqnarray*}
\| X_{\cdot j} \|_{\psi_\nu, \alpha} = \sup_{q\ge 2}  { {\| X_{\cdot j} \|_{q, \alpha} } \over q^\nu} \mbox{ and }
 \Phi_{\psi_\nu, \alpha} = \max_{j \le p}  \| X_{\cdot j} \|_{\psi_\nu, \alpha}
\end{eqnarray*}
Let $L_3 = ( (\log p)^{1/\beta + 1/2}  \Phi_{\psi_\nu, \alpha} )^{1/\alpha}$, $N_4 = n (\log p)^{-1-2/\beta} \Phi_{\psi_\nu, 0}^{-2}$ and $W_4 = (\log (p n))^{3+2/\beta} \Phi_{\psi_\nu, 0}^2 + (\log (p n))^{4}$. Here  $\beta=2/(1+2\nu)$.

\begin{thm}
\label{th:1507140325}
Let Assumption \ref{assumption:1507172202} be satisfied. Assume that $\Phi_{\psi_\nu, \alpha}<\infty$ for some $\nu \geq 0$, $\alpha>0$ and
\begin{eqnarray}
\max(L_2, L_3) \max(W_1, W_4) = o(N_4),\quad  L_2^\alpha\max(W_1, W_4)=o(n).
\end{eqnarray}
Then the Gaussian Approximation (\ref{eq:J021209}) holds.
\end{thm}

\begin{proof}
The proof is similar to that of Theorem \ref{th:1507140259}, and thus is omitted.
\end{proof}

If $\Phi_{\psi_\nu, \alpha} \asymp 1$, then the ultra high-dimensional case with $\log p = o(n^c)$ with some $c>0$ is allowed, where specifically we can let
\begin{equation}
c=\left\{
\begin{array}{ll}
1/(8+2/\alpha+2/\beta), & 2/3 \leq \beta \leq 2\\
1/[7+(1/\beta+1/2)(1/\alpha+2)], & 1/2 \leq \beta <2/3 \\
1/[3+2/\beta+(1/\beta+1/2)(1/\alpha+2)], & 0 <\beta <1/2
\end{array}
\right..
\end{equation}

\subsection{Simultaneous Inference of Covariances}
Let $X_1, \ldots, X_n$ be i.i.d. $p$-dimensional vectors with mean $0$ and covariance matrix $\Gamma = \Gamma_0 = (\gamma_{j k})_{j, k=1}^p = \E (X_i X_i^\top)$. We can estimate $\Gamma$ by the sample covariance matrix $\hat \Gamma = (\hat \gamma_{jk})_{j, k=1}^p =  n^{-1} \sum_{i=1}^n X_i X_i^\top$.  To perform simultaneous inference on $\gamma_{j k}, 1\le j,k \le p$, one needs to derive asymptotic distribution of the maximum deviation $\max_{j, k \le p} | \hat \gamma_{jk} - \gamma_{jk}|$ or the normalized version $\max_{j, k \le p} | \hat \gamma_{jk} - \gamma_{jk}|/\tau_{j k}$; cf Equation (2) in \citet{Xiao20132899}. \citet{jiang2004} established the Gumbel convergence of the maximum deviation assuming that all entries of $X_i$ are also independent. See  \citet{Li2006} and \citet{liu2008} for some refined results. \citet{Xiao20132899} considered the extension which allows dependence among entries of $X_i$. However the latter paper requires that the vectors $X_1, \ldots, X_n$ are i.i.d. The problem of further extension to temporally dependent $X_i$ is open. In analyzing electrocorticogram data in the format of multivariate time series, \citet{PhysRevE.79.061916} proposed to use the maximum cross correlation between time series to identify edges that connect the corresponding nodes in a network, suggesting that an asymptotic theory for maximum deviations of sample covariances is needed.

Our Theorems \ref{th:1507140259} and \ref{th:1507140325} can be applied to the above problem of further extension to temporally dependent process $(X_i)$. Let $(X_i)$ be a mean zero $p$-dimensional stationary process of form (\ref{highdimensionrepresentation}).  To apply Theorems \ref{th:1507140259} and \ref{th:1507140325}, one needs to deal with the key issue of computing the functional dependence measure of the $p^2$-dimensional vector $\mathcal{X}_i = \mbox{vec}(X_i X_i^\top-\E(X_i X_i^\top))$. Interestingly, our framework allows a natural and elegant treatment. Let $a = (j, k)$, $j, k \le p$, and $\mathcal{X}_{i a} =  X_{ij}X_{ik} - \gamma_a$, where $ \gamma_a = \gamma_{j k} = \E (X_{ij}X_{ik})$. By H\"{o}lder's inequality, the functional dependence of the component process $(\mathcal{X}_{i a})_i$
\begin{eqnarray}\label{eq:1508220838}
\varphi_{i, q/2, a} &:= &\|X_{ij}X_{ik}-\E (X_{ij}X_{ik})-X_{ij, \{0\}}X_{ik, \{0\}}
 +\E(X_{ij, \{0\}}X_{ik, \{0\}})\|_{q/2} \cr
&\leq&2\|X_{ij}X_{ik}-X_{ij, \{0\}}X_{ik, \{0\}}\|_{q/2}\cr
&\leq&2\|X_{ij}(X_{ik}-X_{ik, \{0\}})\|_{q/2} +
 2\| (X_{ij} -X_{ij, \{0\}}  ) X_{ik, \{0\}}\|_{q/2}\cr
&\leq &2\|X_{ij}\|_q\delta_{i,q,k} + 2\|X_{ik}\|_q\delta_{i,q,j}.
\end{eqnarray}
Hence, we can have an upper bound of the dependence adjusted norm of $(\mathcal{X}_{i a})$
\begin{eqnarray}\label{eq:1508220839}
\|\mathcal{X}_{\cdot a}\|_{q/2,\alpha}
 &:=& \sup_{m \geq 0} (m+1)^\alpha \sum_{i=m}^\infty \varphi_{i,q/2,j,k}\cr
 &\leq &2\|X_{\cdot j}\|_{q,0}\|X_{\cdot k}\|_{q,\alpha}+2\|X_{\cdot k}\|_{q,0}\|X_{\cdot j}\|_{q,\alpha}.
\end{eqnarray}
Consequently, the uniform and the overall dependence adjusted norms of $\mathcal{X}_i$ are
\begin{eqnarray}\label{eq:1508220840}
&& \max_a \|\mathcal{X}_{\cdot a}\|_{q/2, \alpha} \leq 4 \Psi_{q,0}\Psi_{q,\alpha},\cr
&& \left(\sum_a \|\mathcal{X}_{\cdot a}\|_{q/2, \alpha}^{q/2}\right)^{2/q}
 \leq 4 \left(\sum_{j=1}^p\|X_{\cdot j}\|_{q,0}^{q/2}\right)^{2/q}
  \left(\sum_{j=1}^p\|X_{\cdot j}\|_{q,\alpha}^{q/2}\right)^{2/q}.
\end{eqnarray}
Similarly, the $\mathcal{L}^\infty$ dependence adjusted norm for the process $(\mathcal{X}_i)$ can be calculated by
\begin{equation}\label{eq:1508220841}
\||\mathcal{X}_{\cdot}|_\infty \|_{q/2, \alpha} \leq 4 \||X_{\cdot}|_\infty\|_{q, 0} \||X_{\cdot}|_\infty\|_{q, \alpha}.
\end{equation}
With (\ref{eq:1508220838})-(\ref{eq:1508220841}), conditions in Theorems \ref{th:1507140259} and \ref{th:1507140325} can be formulated accordingly, and under those conditions we can have the following Gaussian Approximation
\begin{eqnarray}\label{eq:1508220832}
 \sup_{u \ge 0} |\P( \sqrt n \max_a | \hat \gamma_a - \gamma_a| / \tau_a \ge u)
  - \P(\max_a |Z_a / \tau_a| \ge u)| \to 0,
\end{eqnarray}
where $Z = (Z_a)_a \sim N(0, \Sigma_\mathcal{X})$, $\Sigma_\mathcal{X}$ is the $p^2 \times p^2$ long-run covariance matrix of $(\mathcal{X}_i)_i$ and $(\tau_a^2)_a$ is the diagonal matrix of $\Sigma_\mathcal{X}$.

\section{Estimation of long-run covariance matrix}
\label{sec:estimation of long-run variance}

Given the realization $X_1, \ldots, X_n$, to apply the Gaussian approximation (\ref{eq:J021209}), we need to estimate the long-run covariance matrix $\Sigma$. Note that $\Sigma/(2\pi)$ is the value of the spectral density matrix of $(X_i)$ at zero frequency. In the one-dimensional case, there is a large literature concerning spectral density estimation; see for example \citet{anderson1971}, \citet{priestley1981}, \citet{rosenblatt1985}, \citet{brockwell1991}, \citet{liu2010asymptotics} among others. In the high-dimensional setting, \citet{chen2013} studied the regularized estimation of $\Gamma(0) = \E (X_0 X_0^\top)$.  Assume $\E X_i = 0$. We then consider the batched mean estimate
\begin{equation}
\label{estimate LRCM}
\hat{\Sigma}=\frac{1}{M w}\sum_{b=1}^w Y_bY_b^\top=\frac{1}{M w}\sum_{b=1}^w (\sum_{i \in L_b}X_i)(\sum_{i \in L_b}X_i)^\top.
\end{equation}
where the window $L_b = \{1+(b-1)M, \ldots, nM\}, b=1, \ldots, w$, the window size $|L_b| = M \to \infty$ and the number of blocks $w = \lfloor n/M \rfloor$. Theorems \ref{theoremforquadraticformunderpolynomial} and \ref{theoremforquadraticformunderexponential} concern the convergence of the above estimate for processes with finite polynomial and finite sub-exponentail dependence adjusted norms, respectively. The convergence rate depends in a subtle way on the temporal dependence characterized by $\alpha$ (cf. (\ref{eq:2015081932})), the uniform and the overall dependence adjusted norms $\Psi_{q, \alpha}$ and $\Upsilon_{q, \alpha}$, respectively, the same size $n$ and the dimension $p$.

For a random variable $X$, we define the operator $\E_0$ as $\E_0 (X) := X-\E X$.

\begin{thm}
\label{theoremforquadraticformunderpolynomial}
Assume $\Psi_{q, \alpha} < \infty$ with $q>4$ and $\alpha>0$, and $M=O(n^{\varsigma})$ for some $0< \varsigma <1$. Let $F_\alpha = w M$ (resp. $w M^{q/2-\alpha q/2}$ or $w^{q/4-\alpha q/2} M^{q/2-\alpha q/2}$) for $\alpha>1-2/q$ (resp. $1/2-2/q< \alpha <1-2/q$ or $\alpha < 1/2-2/q$). Then for $x \geq \sqrt{w}M \Psi_{q, \alpha}^2$, we have
 \begin{eqnarray}\label{eq:201508181936}
&&\P(n|\diag(\hat{\Sigma})-\E \diag(\hat{\Sigma})|_\infty\geq x)
\lesssim {{ F_\alpha \Upsilon_{q, \alpha}^q } \over {x^{q/2}}}+ p \exp\left(-\frac{C_{q, \alpha} x^2}{wM^2\Psi_{4, \alpha}^4}\right),
\cr
&& \P(n|\hat{\Sigma}-\E \hat{\Sigma}|_\infty \geq x)
\lesssim {{p F_\alpha \Upsilon_{q, \alpha}^q } \over {x^{q/2}}}
 + p^2 \exp\left(-\frac{C_{q, \alpha} x^2}{wM^2\Psi_{4, \alpha}^4}\right)
\end{eqnarray}
for all large $n$, where the constants in $\lesssim$ only depend on $\varsigma$, $\alpha$ and $q$.
\end{thm}

\begin{proof}
Fix $1 \leq j, k \leq p$; let $T=\sum_{b=1}^w Y_{bj} Y_{bk}$, where $Y_{b j} = \sum_{i \in L_b}X_{i j}$.
For $\tau \geq 0$, define $X_{ij, \tau}=\E(X_{ij}|\eps_{i-\tau}, \ldots, \eps_{i})$, $Y_{bj, \tau}=\sum_{i \in L_b}X_{ij, \tau}$ and $T_{\tau}=\sum_{b=1}^w Y_{bj,\tau}Y_{bk, \tau}$.
We will first prove for any $x>0$
\begin{equation}
\P(|\E_0(T-T_{M})| \geq x)
\lesssim \left\{
\begin{array}{ll}
\label{differenceforquadraticform}
x^{-q/2}wM^{q/2-\alpha q/2}\xi_{q, \alpha}^{q/2}+E_{q, \alpha}(x), & \alpha >1/2-2/q \\
x^{-q/2}w^{q/4-\alpha q/2}M^{q/2-\alpha q/2}\xi_{q, \alpha}^{q/2}+E_{q, \alpha}(x), & \alpha <1/2-2/q \\
\end{array}
\right. ,
\end{equation}
where the constants in $\lesssim$ only depend on $\varsigma$, $\alpha$ and $q$, and
\begin{eqnarray*}
&& \xi_{q, \alpha}=\|X_{\cdot j}\|_{q, 0}\|X_{\cdot k}\|_{q, \alpha}+\|X_{\cdot k}\|_{q, 0}\|X_{\cdot j}\|_{q, \alpha},\\
&& E_{q, \alpha}(x)=\exp\{-C_{q, \alpha}(wM^{2-2\alpha}\xi_{4, \alpha}^2)^{-1}x^{2}\}.
\end{eqnarray*}

Following the argument in the proof of Lemma \ref{lemma for m-approximation, polynomial tail}, let $L=\lfloor (\log w)/ (\log 2)\rfloor$, $\varpi_l=2^l$, $1 \leq l <L$, $\varpi_L=w$ and $\tau_l= M \varpi_l$ for $1 \leq l \leq L$. Let $\varpi_0 = 1$ and $\tau_0 = M$. Write
\begin{equation}
\label{T-T_{M}}
T-T_{M} = T-T_{Mw} + \sum_{l=1}^L V_{w, l}, \mbox{ where } V_{w, l}=T_{\tau_l}-T_{\tau_{l-1}}.
\end{equation}
By the argument in Lemma 9 of \citet{xiao2012covariance}, we have
\begin{eqnarray}
\| \E_0(T-T_{Mw})\|_{q/2}
&\leq & C_q M\sqrt{w}(\Delta_{0, q, j}\Delta_{Mw+1, q, k}+\Delta_{Mw+1, q, j}\Delta_{0, q, k})\nonumber \\ \label{appximationmoment}
&\leq & C_q M\sqrt{w}(Mw)^{-\alpha}\xi_{q, \alpha}
\end{eqnarray}
for some constant $C_q > 0$. By Markov's inequality, for $x > 0$,
\begin{equation}
\label{T-T_{Mw}}
\P(|\E_0(T-T_{Mw})|\geq x)\leq \frac{C_q M^{q/2-\alpha q/2}w^{q/4-\alpha q/2}\xi_{q, \alpha}^{q/2}}{x^{q/2}}.
\end{equation}
By the same argument for proving (\ref{appximationmoment}), we have
\begin{equation*}
\|\E_0(V_{w, l})\|_{q/2} \leq C_q M\sqrt{w} \tau_l^{-\alpha}\xi_{q, \alpha}.
\end{equation*}
Let $c=q/4-1-\alpha q/2$, $\lambda_l= 3 l^{-2} \pi^{-2}$ if $1 \leq l \leq L/2$ and $\lambda_l= 3 (L+1-l)^{-2} \pi^{-2}$ if $L/2 < l \leq L$. Then $\sum_{l=1}^L \lambda_l < 1$.  By the Nagaev (1979) inequality, it follows that
\begin{eqnarray}
\P(|\sum_{l=1}^L \E_0(V_{w, l})| \geq x) &\leq& \sum_{l=1}^L \P(|\E_0(V_{w, l})| \geq \lambda_lx) \nonumber \\
& \leq & \sum_{l=1}^L \frac{C_1 w \varpi_l^{-1}(M{\varpi_l}^{1/2} \tau_l^{-\alpha})^{q/2}\xi_{q, \alpha}^{q/2}}{(\lambda_l x)^{q/2}}+4 \sum_{l=1}^L \exp \left(-\frac{C_2 (\lambda_l x)^2\tau_l^{2 \alpha}}{wM^2\xi_{4, \alpha}^{2}} \right)\nonumber \\
&\leq & \frac{C_3wM^{q/2-\alpha q/2}\xi_{q, \alpha}^{q/2}}{x^{q/2}}\sum_{l=1}^L \frac{\varpi_l^{c}}{\lambda_l^{q/2}}+C_4 \sum_{l=1}^L E_{q, \alpha}(\lambda_l \varpi_l^\alpha x). \label{V_{w,l}}
\end{eqnarray}
Elementary calculations show that
\begin{equation}
\label{poly}
\sum_{l=1}^L \frac{\varpi_l^{c}}{\lambda_l^{q/2}} \leq C_5 \text{ for } c<0 \text{ and } \sum_{l=1}^L \frac{\varpi_l^{c}}{\lambda_l^{q/2}} \leq C_6 \varpi_L^c=C_6 w^c \text{ for } c>0.
\end{equation}
Furthermore, we can use (\ref{exponentialterm}) to obtain
\begin{equation}
\label{exp}
\sum_{l=1}^L E_{q, \alpha}(\lambda_l \varpi_l^\alpha x) \lesssim E_{q, \alpha} (x).
\end{equation}
Putting (\ref{T-T_{M}}), (\ref{T-T_{Mw}}), (\ref{V_{w,l}}), (\ref{poly}) and (\ref{exp}) together, we then have (\ref{differenceforquadraticform}).

Now it suffices to consider $\P(|\E_0(T_M)|\geq x)$. Observe that $(Y_{bj, M}Y_{bk, M})_{b \text{ is odd}}$ are independent and so are $(Y_{bj, M}Y_{bk, M})_{b \text{ is even}}$. By Corollary 1.7 of \citet{nagaev1979large}, for any $J > 1$,
\begin{eqnarray*}
\P(|\E_0(T_M)|\geq x) &\leq & \sum_{b=1}^w \P(|\E_0(Y_{bj, M}Y_{bk, M}
)|\geq x/(2J))+2\left(\frac{\sum_{b=1}^w \|\E_0(Y_{bj, M}Y_{bk, M})\|_{q/2}^{q/2}}{J x^{q/2}}\right)^{J}\\
&&+4\exp\left\{-\frac{C_q x^2}{\sum_{b=1}^w \|\E_0(Y_{bj, M}Y_{bk, M})\|_2^2}\right\}.
\end{eqnarray*}
Note that $\|Y_{bj, M}\|_{q} \leq C_q \sqrt M \|X_{\cdot j}\|_{q, 0}$. Hence for $1 \leq b \leq w$, $1 \leq j, k\leq p$ and $q \geq 4$,
\begin{equation*}
\|\E_0(Y_{bj, M}Y_{bk, M})\|_{q/2} \leq 2\|Y_{bj, M}Y_{bk, M}\|_{q/2} \leq 2\|Y_{bj, M}\|_{q}\|Y_{bk, M}\|_{q}\leq C_q M \|X_{\cdot j}\|_{q, 0}\|X_{\cdot k}\|_{q, 0}.
\end{equation*}
Since
\begin{equation*}
\E |Y_{bj, M}Y_{bk, M}| \leq \|Y_{bj, M}\|_2 \|Y_{bk, M}\|_2  \leq M \|X_{\cdot j}\|_{2, 0}\|X_{\cdot k}\|_{2, 0}
 \le {x \over \sqrt w},
\end{equation*}
we have
\begin{eqnarray*}
\P(|\E_0(T_M)|\geq x)& \leq& \sum_{b=1}^w \P(|Y_{bj, M}Y_{bk, M}|\geq x/(4J)) \\
&&+ 2\left(\frac{wM^{q/2}\|X_{\cdot j}\|_{q,0}^{q/2}\|X_{\cdot k}\|_{q,0}^{q/2}}{J x^{q/2}}\right)^{J}
 +4\exp\left(-\frac{C_q x^2}{wM^2\Psi_{4,0}^4}\right).
\end{eqnarray*}
Recall that $M=O(n^\varsigma)$ with $0 < \varsigma < 1$. Let $J = 1+ (2q-2) (q-4)^{-1} (1- \varsigma)^{-1}$. Since  $x \geq \sqrt{w}M \|X_{\cdot j}\|_{q,0}\|X_{\cdot k}\|_{q,0}$, elementary calculations show that for sufficiently large $n$ the second term in the above expression is no greater than $C_J wM\|X_{\cdot j}\|_{q,0}^{q/2}\|X_{\cdot k}\|_{q,0}^{q/2}/x^{q/2}$. As for the first term, we have
\begin{equation*}
\P(|Y_{bj, M}Y_{bk, M}| \geq x/(4J))  \leq  \P(|Y_{bj, M}| \geq \sqrt{x/(4J)})+\P(|Y_{bk, M}| \geq \sqrt{x/(4J)}).
\end{equation*}
By Lemma \ref{theorem for the sum polynomial}, for $\alpha>1/2-1/q$ and $\alpha <1/2-1/q$, respectively, we have
\begin{equation*}
\P(|Y_{bj, M}| \geq \sqrt{x}) \leq \left\{
\begin{array}{ll}
C_{q, \alpha}x^{-q/2}M\|X_{\cdot j}\|_{q, \alpha}^q+ C_{q, \alpha} \exp\left(-\frac{C_{q, \alpha}x}{M \|X_{\cdot j}\|_{2, \alpha}^2} \right), \\
C_{q, \alpha}x^{-q/2}M^{q/2-\alpha q}\|X_{\cdot j}\|_{q, \alpha}^q+ C_{q, \alpha} \exp\left(-\frac{C_{q, \alpha}x}{M \|X_{\cdot j}\|_{2, \alpha}^2} \right). \\
\end{array}
\right.
\end{equation*}
A similar inequality holds for $\P(|Y_{bk, M}| \geq \sqrt{x})$. Let $\phi_{q, \alpha}=\|X_{\cdot j}\|_{q, \alpha}^q+\|X_{\cdot k}\|_{q, \alpha}^q$. Hence, it follows that  for $\alpha>1/2-1/q$ and $\alpha <1/2-1/q$ respectively,
\begin{equation}
\P(|\E_0(T_M)|\geq x) \leq \left\{
\begin{array}{ll}
\label{maintermforquadraticform}
C_{q, \alpha}x^{-q/2} wM \phi_{q, \alpha} + C_{q, \alpha} \exp\left(-\frac{C_{q, \alpha}x^2}{wM^2 \Psi_{4, \alpha}^4} \right), \\
C_{q, \alpha}x^{-q/2}w M^{q/2-\alpha q}\phi_{q, \alpha}+ C_{q, \alpha} \exp\left(-\frac{C_{q, \alpha}x^2}{wM^2 \Psi_{4, \alpha}^4} \right). \\
\end{array}
\right.
\end{equation}
Combining (\ref{differenceforquadraticform}) and (\ref{maintermforquadraticform}), and noticing that $\xi_{q, \alpha}^{q/2} \leq C_q\phi_{q, \alpha}$, it follows that
\begin{eqnarray*}
 \P(|\E_0(T)|\geq x) \le C_{q, \alpha} x^{-q/2} F_\alpha \phi_{q, \alpha}
  +C_{q, \alpha} \exp\left(-\frac{C_{q, \alpha} x^2}{wM^2\Psi_{4, \alpha}^4}\right).
\end{eqnarray*}
which implies (\ref{eq:201508181936}) by the Bonferroni inequality by summing over $j$ and $k$.
\end{proof}

Under stronger moment conditions, we can have an exponential inequality.

\begin{thm}
\label{theoremforquadraticformunderexponential}
Assume $\Phi_{\psi_\nu, 0} < \infty$ for some $\nu \geq 0$. Then for all $x >0$, we have
\begin{eqnarray}
&& \P(n|\diag(\hat{\Sigma})-\E \diag(\hat{\Sigma})|_\infty \geq x) \lesssim   p \exp\left(-\frac{x^\gamma}{4e\gamma (\sqrt{w}M\Phi_{\psi_\nu, 0}^2)^\gamma }\right), \label{exponentialforquadraticform1}\\
&&\P(n|\hat{\Sigma}-\E \hat{\Sigma}|_\infty \geq x) \lesssim  p^2 \exp\left(-\frac{x^\gamma}{4e\gamma (\sqrt{w}M\Phi_{\psi_\nu, 0}^2)^\gamma }\right), \label{exponentialforquadraticform2}
\end{eqnarray}
where $\gamma=1/(1+2 \nu)$ and the constants in $\lesssim$ only depend on $\nu$.
\end{thm}
\begin{proof}
Let $T=\sum_{b=1}^w Y_{bj} Y_{bk}$. By the Burkholder inequality, we have
\begin{eqnarray}\label{eq:1508182136}
\| \E_0 T \|_{q/2}^2 \le (q/2-1) \sum_{l=-\infty}^{w M} \|{\cal P}^l T \|_{q/2}^2
 \le (q/2-1) \sum_{l=-\infty}^{w M} \left( \sum_{b=1}^w  \|{\cal P}^l Y_{bj} Y_{bk} \|_{q/2} \right)^2
\end{eqnarray}
By Theorem 3 in \citet{wu2011asymptotic}, $\| Y_{bj}  \|_q \le (q-1)^{1/2} \sqrt M \|X_{\cdot j}\|_{q, 0}$. Since $\| Y_{bk} - Y_{bk, \{l\}} \|_{q}  \le \sum_{h=1+(b-1)M}^{b M} \delta_{h-l, q, k}$, we have
\begin{eqnarray*}
\sum_{b=1}^w  \|{\cal P}^l Y_{bj} Y_{bk} \|_{q/2}
 &\le& \sum_{b=1}^w   \| Y_{bj} Y_{bk} - Y_{bj, \{l\} } Y_{bk, \{l\}} \|_{q/2}  \cr
  &\le& \sum_{b=1}^w  ( \| Y_{bj} \|_q \| Y_{bk} - Y_{bk, \{l\}} \|_{q}  +
 \| Y_{bj} - Y_{bj, \{l\} } \|_q  \|Y_{bk, \{l\}}\|_q ) \cr
  &\le&  (q-1)^{1/2} \sqrt M \left(\|X_{\cdot j}\|_{q, 0} \sum_{h=1}^{w M} \delta_{h-l, q, k}
  + \|X_{\cdot k}\|_{q, 0} \sum_{h=1}^{w M} \delta_{h-l, q, j} \right),
\end{eqnarray*}
which by (\ref{eq:1508182136}) implies that
\begin{eqnarray}\label{eq:1508182144}
\| \E_0 T \|_{q/2}^2 \le (q/2-1) \sum_{l=-\infty}^{w M} \|{\cal P}^l T \|_{q/2}^2
 \le (q-2)(q-1) w M^2 \|X_{\cdot j}\|_{q, 0}^2 \|X_{\cdot k}\|_{q,0}^2.
 \end{eqnarray}
Let $R_{j k} = \E_0 T /(\sqrt w M)$. Similarly as the argument for proving Lemma \ref{m-approximationtheorem2},
if $\gamma h \geq 2$, it follows that $\|R_{jk}\|_{\gamma h} \leq (2\gamma h-1)(2\gamma h)^{2 \nu}\|X_{\cdot j}\|_{\psi_\nu, 0}\|X_{\cdot k}\|_{\psi_\nu, 0}$. Let $\tau_0=(2e\gamma \|X_{\cdot j}\|_{\psi_\nu, 0}^\gamma\|X_{\cdot k}\|_{\psi_\nu, 0}^\gamma)^{-1}$. Notice that $-2\nu=1-1/\gamma$. Then
\begin{eqnarray*}
\frac{t^h\|R_{jk}^\gamma\|^h_h}{h!} &\leq& \frac{t^h(2\gamma h-1)^{\gamma h}(2\gamma h)^{2 \nu \gamma h}\|X_{\cdot j}\|_{\psi_\nu, 0}^{\gamma h}\|X_{\cdot k}\|_{\psi_\nu, 0}^{\gamma h}}{C_1(h/e)^h a_h^{-1}}\\
&\leq & \frac{a_h t^h (2\gamma h -1)^{\gamma h}}{C_1 \tau_0^h(2\gamma h)^{\gamma h}} \leq \frac{a_h t^h}{C_1 \sqrt{e} \tau_0^h}.
\end{eqnarray*}
If $\gamma h <2$, then $\|R_{jk}\|_{\gamma h} \leq \|R_{jk}\|_2 \leq \sqrt{6}\cdot 4^{2 \nu} \|X_{\cdot j}\|_{\psi_\nu, 0}\|X_{\cdot k}\|_{\psi_\nu, 0}$. So we have
\begin{eqnarray*}
\E [\exp(t R_{jk}^\gamma)]&\leq & 1+ \sum_{1 \leq h <2/\gamma}\frac{t^h(\sqrt{6}\cdot 4^{2 \nu} \|X_{\cdot j}\|_{\psi_\nu, 0}\|X_{\cdot k}\|_{\psi_\nu, 0})^{\gamma h}}{h!}+\sum_{h \geq 2/\gamma}\frac{a_h t^h}{C_1 \sqrt{e} \tau_0^h}\\
&\leq & 1+C_\gamma \sum_{h=1}^\infty a_h\frac{t^h}{\tau_0^h} \leq 1+ C_\gamma \frac{t/\tau_0}{(1-t/\tau_0)^{1/2}}.
\end{eqnarray*}
By choosing $t=\tau_0/2$, and applying the Markov inequality and the Bonferroni inequality, (\ref{exponentialforquadraticform1}) and (\ref{exponentialforquadraticform2}) are obtained.
\end{proof}

\begin{rmk}
An alternative estimate of $\Sigma$, which also works with unknown mean $\E X_i$, is
\begin{equation}
\label{long run covariance estimate minus mean}
\tilde{\Sigma}=\frac{1}{w M}\sum_{b=1}^w (\sum_{i \in L_b}X_i-M \bar{X})(\sum_{i \in L_b}X_i-M\bar{X})^\top,
\end{equation}
where $\bar{X} = (w M)^{-1}\sum_{i=1}^{w M} X_i$, $w = \lfloor n/M \rfloor$. Then $|\tilde{\Sigma} - \hat{\Sigma}|_\infty = M |\bar X|_\infty^2$. Applying Lemma \ref{theorem for the sum polynomial} to $\sum_{i=1}^{w M} X_{i j}$, one can conclude that Theorems \ref{theoremforquadraticformunderpolynomial} and \ref{theoremforquadraticformunderexponential} still hold for $\tilde{\Sigma}$ with $\E\hat{\Sigma}$ therein replaced by $\Sigma_M:=\sum_{i=-M}^M (1-|i|/M) \Gamma_i$ (which equals to $\E\hat{\Sigma}$ if $\E X_i = 0$).
\end{rmk}

\begin{cor}
\label{corollary for Sigma}
(i) Under conditions in Theorem \ref{theoremforquadraticformunderpolynomial},  we have $| \tilde{\Sigma} - \Sigma|_\infty = O_\P(r_n)$, where
 \begin{equation*}
 r_n =
n^{-1} \max\{p^{2/q} F^{2/q}_\alpha \Upsilon_{q, \alpha}^2, \, \sqrt w M \Psi_{4, \alpha}^2 \sqrt{\log p}, \,
 \sqrt w M \Psi_{q, \alpha}^2 \}  + \Psi_{2,0}\Psi_{2, \alpha} v(M),
 \end{equation*}
where $v(M)=1/M$ if $\alpha>1$, $v(M)=\log M/M$ if $\alpha=1$ and $v(M)=1/M^\alpha$ if $0<\alpha <1$. (ii) Under conditions in Theorem \ref{theoremforquadraticformunderexponential}, we have $| \tilde{\Sigma} - \Sigma|_\infty = O_\P(r_n)$ with $r_n =  n^{-1} \sqrt{w}M \Phi_{\psi_\nu,0}^2 (\log p)^{1/\gamma} + \Psi_{2,0}\Psi_{2, \alpha} v(M)$.
\end{cor}

The above Corollary easily follows from Theorems \ref{theoremforquadraticformunderpolynomial} and \ref{theoremforquadraticformunderexponential} since the bias $|\Sigma_M-\Sigma|_{\infty} \lesssim \Psi_{2,0}\Psi_{2, \alpha} v(M)$; see the proof of Lemma \ref{lemmafortwogaussian}.

For the estimate $\tilde{\Sigma}$ in (\ref{long run covariance estimate minus mean}), let $\tilde{D}_0=[\diag(\tilde{\Sigma})]^{1/2}$. Let $\tilde{Z} = \tilde{\Sigma}^{1/2} \eta$, where $\eta \sim N(0, \text{Id}_p)$ is independent of $(X_i)_i$. Then conditioning on $(X_i)_i$, $\tilde{Z} \sim N(0, \tilde{\Sigma})$. Let $0 < \theta < 1$; let $\tilde \chi_\theta$ be the conditional $\theta$-quantile of $|\tilde{D}_0^{-1}\tilde{Z}|_\infty$ given $(X_i)_{i=1}^n$. We can use $\tilde \chi_\theta$ to estimate the $\theta$-quantile of $|D_0^{-1}(\bar{X}_n-\mu)|_\infty$, thus constructing simultaneous confidence intervals for $\mu = (\mu_1, \ldots, \mu_p)^\top$ as $\hat \mu_j \pm \tilde \chi_\theta \tilde \sigma_{j j}^{1/2}$, $1 \le j \le p$. Assume that $r_n = o(1/\log^2 p)$. Then $\pi(|\tilde{\Sigma}-\Sigma|_\infty)=o(1)$, and by Lemma 3.1 in \citet{chernozhukov2013}, the latter simultaneous confidence intervals have the asymptotically correct coverage probability $\theta$. Note that $\tilde \chi_\theta$ can be obtained by sample quantile estimates from extensive simulations of $\tilde{Z} = \tilde{\Sigma}^{1/2} \eta$.

\section{Tail probability inequalities under dependence}
\label{sec:probability inequalities}
Tail probability inequalities play an important role in simultaneous inference. Here we shall provide some Nagaev-type tail probability inequalities. They are of independent interest. Let $\eps_i, \eps'_j, i, j, \in \mathbb{Z}$, be i.i.d. random variables. We start with the one-dimensional stationary process $(e_i)_{i=-\infty}^\infty$ of the form
\begin{equation}
\label{representation}
e_i=g(\ldots, \eps_{i-1}, \eps_i),
\end{equation}
where $g$ is a measurable function such that $e_i$ is well-defined. Recall $\calF_i^j=(\eps_i, \eps_{i+1}, \ldots, \eps_j)$, $\calF^j=(\ldots, \eps_{j-1}, \eps_j)$ and $\calF_{i}=(\eps_i, \eps_{i+1}, \ldots)$. Let the projection operators $\mathcal{P}^0 \cdot = \E( \cdot |\calF^{0}) - \E( \cdot | \calF^{-1})$, $\mathcal{P}_0 \cdot = \E( \cdot |\calF_{0}^i) - \E( \cdot |\calF_{1}^i)$. As in (\ref{eq:1507140845}), define respectively the functional and the predictive dependence measures
\begin{eqnarray*}
 \delta_{i, q}=\| e_i-g(\calF^{i, \{0\}}) \|_q, \,\, \theta_{i, q} =\|\mathcal{P}^0 e_i\|_q, \,\, \mbox{ and }
 \theta_{i, q}' =\|\mathcal{P}_0 e_i\|_{q},
\end{eqnarray*}
where $\calF^{i, \{0\}} = ( \ldots, \eps_{-1}, \eps_0', \eps_{1}, \ldots, \eps_i)$. Let $\delta_{i, q}= 0$ if $i < 0$; let $\Delta_{m, q} = \sum_{i=m}^\infty \delta_{i, q}$, $m \ge 0$, be the tail dependence measures, and the dependence adjusted norm
\begin{equation}
\label{dependenceadjustednorm}
\| e_\cdot \|_{q, \alpha}:= \sup_{m \geq 0} (m+1)^{\alpha} \Delta_{m, q}, \text{ for }\alpha \geq 0.
\end{equation}
Here $\delta_{i, q}$ measures the dependence of $e_i$ on $\eps_0$ and $\Delta_{m, q}$ measures the cumulative impact of $\eps_0$ on $(e_i)_{i \geq m}$. The projections $(\mathcal{P}_{-i}\cdot)_{i \in \mathbb{Z}}$ and $(\mathcal{P}^{i}\cdot)_{i \in \mathbb{Z}}$ induces martingale differences with respect to $(\calF_{-i})$ and $(\calF^i)$, respectively. Both predictive dependence measures provide an evaluation to the effect on the prediction of $e_i$ when part of the previous inputs is concealed, and they satisfy $\theta_{i, q} \leq \delta_{i, q}$ and $\theta'_{i, q} \leq \delta_{i, q}$ in view of Jensen's inequality.

\subsection{Inequalities with Finite Polynomial Moments}
For $m\geq 0$, the $m$-dependence approximation of $e_i$ is denoted by $e_{i, m}$ where
\begin{equation*}
e_{i, m}=\E(e_i|\eps_{i-m},\eps_{i-m+1}, \ldots, \eps_i).
\end{equation*}
Let $S_n=\sum_{i=1}^n e_i$, $S_{n, m}=\sum_{i=1}^n e_{i, m}$.
With the dependence adjusted norm (\ref{dependenceadjustednorm}), we are able to provide tail probability inequalities for error bounds when approximating $(e_i)$ by the $m$-dependent process $(e_{i, m})$. In lemmas below the constant $C_{q, \alpha}$ only depends on $q$ and $\alpha$ and its values may change from line to line.

\begin{lem}
\label{m-approximationtheorem1}
Assume $\|e_\cdot\|_{q, \alpha} < \infty$, where $q >2$ and $\alpha>0$. (i) If $\alpha>1/2-1/q$, then
\begin{equation}
\label{alpha>}
\P(|S_n-S_{n, m}| \geq x) \leq \frac{C_{q, \alpha} n m^{q/2-1-\alpha q}\|e_\cdot\|_{q, \alpha}^q}{x^q}+ C_{q,\alpha} \exp\lp - \frac{C_{q, \alpha} x^2 m^{2 \alpha}}{n\|e_\cdot\|_{2, \alpha}^2}\rp
\end{equation}
holds for all $x>0$ and $1 \leq m \leq n$.
(ii) If $0 < \alpha < 1/2-1/q$, we have
\begin{equation}
\label{alpha<}
\P(|S_n-S_{n, m}| \geq x) \leq \frac{C_{q,\alpha} n^{q/2-\alpha q} \|e_\cdot\|_{q, \alpha}^q}{x^q}+ C_{q, \alpha} \exp\lp - \frac{C_{q, \alpha} x^2 m^{2 \alpha}}{n\|e_\cdot\|_{2, \alpha}^2}\rp.
\end{equation}
\end{lem}

\begin{proof}[Proof of Lemma \ref{m-approximationtheorem1}]
It is a special case of Lemma \ref{lemma for m-approximation, polynomial tail} for $p=1$.
\end{proof}

\begin{lem}[cf. Theorem 2 of \citet{wu2014}]
\label{theorem for the sum polynomial}
Assume that $\|e_\cdot\|_{q, \alpha} < \infty$, where $q >2$ and $\alpha>0$. (i) If $\alpha>1/2-1/q$, then there exists some constant $C_{q, \alpha}$ depending on $q$ and $\alpha$ only such that,  for $x>0$,
\begin{equation}
\label{eq:1508220935}
\P(|S_n| \geq x) \leq \frac{C_{q, \alpha} n \|e_\cdot\|_{q, \alpha}^q}{x^q}+ C_{q,\alpha} \exp\lp - \frac{C_{q, \alpha} x^2}{n\|e_\cdot\|_{2, \alpha}^2}\rp.
\end{equation}
(ii) If $0 < \alpha < 1/2-1/q$, we have the following inequality,
\begin{equation}
\label{eq:1508220936}
\P(|S_n| \geq x) \leq \frac{C_{q,\alpha} n^{q/2-\alpha q} \|e_\cdot\|_{q, \alpha}^q}{x^q}+ C_{q, \alpha} \exp\lp - \frac{C_{q, \alpha} x^2}{n\|e_\cdot\|_{2, \alpha}^2}\rp.
\end{equation}
\end{lem}

\begin{rmk}
By Markov's inequality and Lemma 1 of \citet{liu2010asymptotics}, one obtains
\begin{equation}
\P(|S_n-S_{n, m}|\geq x) \leq \frac{\|S_n-S_{n,m}\|_q^q}{x^q}
 \le C_q \frac{n^{q/2}m^{-\alpha q}\|e_\cdot\|_{q, \alpha}^q}{x^q}.
\end{equation}
In comparison, the polynomial tail bounds in (\ref{eq:1508220935}) and (\ref{eq:1508220936}) are sharper.
\end{rmk}

\subsection{Inequalities with Finite Exponential Moments}
If $e_i$ satisfies stronger moment condition than the existence of finite $q$-th moment, we can have an exponential inequality. We shall assume $\|e_\cdot\|_{q, \alpha} < \infty$ for all $q >0$ and some $\alpha\geq 0$ and we further assume for some $\nu \geq 0$, the dependence adjusted sub-exponential norm
\begin{equation}
\label{assumption}
\|e_\cdot\|_{\psi_\nu, \alpha}:=\sup\limits_{q \geq 2} q ^{-\nu} \|e_\cdot\|_{q, \alpha} < \infty.
\end{equation}
By this definition, if $e_i$ are i.i.d., $\|e_\cdot\|_{\psi_\nu, \alpha}$ reduces to the sub-Gaussian norm ($\nu=1$) or sub-exponential norm ($\nu=1/2$) of the random variable by the equivalence of $\|e_\cdot\|_{q, \alpha}$ and $\|e_i\|_q$. The parameter $\nu$ measures how fast $\|e_\cdot\|_{q, \alpha}$ increases with $q$.
\begin{lem}
\label{m-approximationtheorem2}
Assume (\ref{assumption}). Let $J_n=(S_n-S_{n, m})/\sqrt{n}$ and $\beta=2/(1+2\nu)$.
Then
\begin{equation*}
h(t):= \sup_{n \in \mathbb{N}} \E[\exp(tJ_n^{\beta})] \leq 1+C_{\beta}(1-t/t_0)^{-1/2} t/t_0
\end{equation*}
holds for $0 \leq t < t_0$ with $t_0=m^{\alpha\beta}/(e\beta\|e_\cdot\|_{\psi_\nu, \alpha}^\beta)$. Consequently, letting $t=t_0/2$, for $x>0$,
\begin{equation}
\label{conclusion1}
\Prob(|J_n| \geq x) \leq \exp(-t x^\beta)h(t) \leq C_\beta\exp\left(-\frac{x^\beta m^{\alpha \beta}}{2e\beta \|e_\cdot\|^\beta_{\psi_\nu, \alpha}}\right).
\end{equation}
\end{lem}

\begin{lem}[cf. Theorem 3 of \citet{wu2014}]
\label{theorem for the sum exponential}
Assume (\ref{assumption}) holds for $\alpha=0$. Let $\beta=2/(1+2\nu)$. Then for $x>0$,
\begin{equation}
\Prob(|S_n/\sqrt{n}| \geq x) \leq C_\beta \exp\left(-\frac{x^\beta}{2e\beta \|e_\cdot\|^\beta_{\psi_\nu, 0}}\right).
\end{equation}
\end{lem}

\begin{proof}[Proof of Lemma \ref{m-approximationtheorem2}]
Let $Q_{n, l}=\sum_{i=1}^n\mathcal{P}_{i-l}X_i$, $l \geq 0$. Then $Q_{n, l}$ is a backward martingale. By Burkholder's inequality, we have
\begin{equation*}
\|Q_{n, l}\|_q^2 \leq (q-1) \sum_{i=1}^n \|\mathcal{P}_{i-l}X_i\|_q^2 =(q-1)n (\theta_{l, q}')^2.
\end{equation*}
By $\theta_{l, q}' \leq \delta_{l, q}$, we have $\|J_n\|_q \leq (q-1)^{1/2}\Delta_{m+1, q}$ in view of $\sqrt{n}J_n= \sum_{l=m+1}^\infty Q_{n, l}$. Write the negative binomial expansion $(1-s)^{-1/2}=1+\sum_{k=1}^\infty a_k s^k$ with $a_k=(2k)!/(2^{2k}(k!)^2)$ for $|s|<1$. By Stirling's formula, we have $a_k \sim (k\pi)^{-1/2}$ as $k \rightarrow \infty$. Hence, there exists absolute constants $C_1, C_2>0$ such that for all $k \geq 1$,
\begin{equation}
C_1 (k/e)^ka_k^{-1} \leq k! \leq C_2 (k/e)^k a_k^{-1}.
\end{equation}
Under condition (\ref{assumption}), if $k\beta \geq 2$, then $\|e_\cdot\|_{\beta k, \alpha} \leq \|e_\cdot\|_{\psi_\nu, \alpha}(\beta k)^\nu$ and hence
\begin{eqnarray*}
\frac{t^k\|J_n^\beta\|^k_k}{k!} \leq \frac{t^k(\beta k-1)^{\beta k/2}\Delta_{m+1,\beta k}^{\beta k}}{C_1(k/e)^k a_k^{-1}}
\leq \frac{a_k t^k (\beta k -1)^{\beta k/2}}{C_1 t_0^k(\beta k)^{\beta k/2}} \leq \frac{a_k t^k}{C_1 \sqrt{e} t_0^k}.
\end{eqnarray*}
If $k\beta <2$, then $\|J_n\|_{\beta k} \leq \|J_n\|_2 \leq 2^\nu m^{-\alpha}\|e_\cdot\|_{\psi_\nu, \alpha}$. In $e^y= \sum_{k=0}^\infty y^k/k!$, let $y = t J_n^\beta$, then
\begin{eqnarray*}
h(t) &\leq & 1+ \sum_{1 \leq k <2/\beta}\frac{t^k(2^\nu m^{-\alpha} \|e_\cdot\|_{\psi_\nu, \alpha})^{\beta k}}{k!}+\sum_{k \geq 2/\beta}\frac{a_k t^k}{C_1 \sqrt{e} t_0^k}\\
&\leq & 1+C_\beta \sum_{k=1}^\infty a_k\frac{t^k}{t_0^k} \leq 1+ C_\beta \frac{t/t_0}{(1-t/t_0)^{1/2}},
\end{eqnarray*}
where $C_\beta>0$ only depends on $\beta$. So (\ref{conclusion1}) follows by Markov's inequality.
\end{proof}

\subsection{Inequalities for High-dimensional Time Series with Finite Polynomial Moments }
In this section we shall derive powerful tail probability inequalities for high-dimensional stationary vectors; cf Lemmas \ref{lemma for m-approximation, polynomial tail} and \ref{lemma for the sum, polynomial tail}. The proofs require Theorem 4.1 of \citet{pinelis1994optimum}, a deep Rosenthal-Burkholder type bound on moments of Banach-spaced martingales. Lemma \ref{lemma for high-dim Bulkholder} follows from Theorem 4.1 of \citet{pinelis1994optimum}. Lemma \ref{lemma for independent high-dim Nagaev} is a Fuk-Magaev type inequality for the sum of independent random vectors. For a $p$-dimensional vector $v = (v_1, \ldots, v_p)$ recall the $s$-length $|v|_s = (\sum_{j=1}^p |v_j|^s)^{1/s}$, $s \ge 1$.

\begin{lem}
\label{lemma for high-dim Bulkholder}
Let $D_i$, $1 \leq i \leq n$, be $p$-dimensional martingale difference vectors with respect to the $\sigma$-field $\mathcal{G}_i$. Let $s>1$ and $q\geq 2$. Then
\begin{equation*}
\||D_1+\ldots+D_n|_{s}\|_{q} \le c \left\{ q \| \sup_{i} |D_i|_s \|_q +\sqrt{q(s-1)}\left\|\left[\sum_{i=1}^n \E (|D_i|_s^2|\mathcal{G}_{i-1})\right]^{1/2}\right\|_q \right\},
\end{equation*}
where $c$ is an absolute constant.
\end{lem}

\begin{lem}
\label{lemma for independent high-dim Nagaev}
Assume $s>1$. Let $X_1, \ldots, X_n$ be $p$-dimensional independent random vectors with mean zero such that for some $q>2$, $\||X_i|_s\|_q < \infty$, $1 \leq i \leq n$. Let $T_n=\sum_{i=1}^n X_i$ and $\sigma_i=(\|X_{i1}\|_2, \ldots, \|X_{ip}\|_2)^\top$. Then for any $y>0$,
\begin{equation}
\P \left(|T_n|_s\geq 2 \E |T_n|_s+y\right) \leq C_q y^{-q}\sum_{i=1}^n \E |X_i|_s^q+\exp\left(-\frac{y^2}{3\sum_{i=1}^n |\sigma_i|_s^2}\right),
\end{equation}
where $C_q$ is a positive constant only depending on $q$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma for independent high-dim Nagaev}]
For $s>1$, we apply Theorem 3.1 of \citet{einmahl2008characterization} with the Banach space $(\mathbb{R}^p, |\cdot|_s)$ and $\eta=\delta=1$. The unit ball of the dual of $(\mathbb{R}^p, |\cdot|_s)$ is the set of linear functions $\{u=(u_1, \ldots, u_p)^\top \mapsto \lambda^\top u: \lambda \in \mathbb{R}^p, |\lambda|_a \leq 1\}$ where $1/a+1/s=1$. By Minkowski's and H\"{o}lder's inequalities, we have
\begin{equation*}
\|\lambda^\top X_i\|_2 \leq \sum_{j=1}^p |\lambda_j| \cdot \|X_{ij}\|_2 \leq |\lambda|_a |\sigma_i|_s.
\end{equation*}
Hence, the $\Lambda_n$ therein is bounded by $\sum_{i=1}^n |\sigma_i|_s^2$.
\end{proof}

Let $X_i$ be a mean zero $p$-dimensional stationary process, and $T_n = \sum_{i=1}^n X_i$, $T_{n,m}=\sum_{i=1}^n X_{i,m}$ where $X_{i,m}= \E(X_i| \eps_{i-m}, \ldots, \eps_{i})$. We are interested in bounding the tail probabilities of $\P (|T_n-T_{n,m}|_\infty \geq x)$ and $\P (|T_n|_\infty \geq x)$ for large $x$. Wrtie $\ell = \ell(p) = 1\vee \log p$.

\begin{lem}
\label{lemma for m-approximation, polynomial tail}
Assume $\| |X_\cdot|_\infty \|_{q, \alpha} < \infty$, where $q>2$ and $\alpha \geq 0$. Also assume $\Psi_{2, \alpha} < \infty$. (i) If $\alpha>1/2-1/q$, then for $x \gtrsim [\sqrt{n \ell}\Psi_{2, \alpha}+n^{1/q} \ell \| |X_{\cdot}|_\infty \|_{q, \alpha}]m^{-\alpha}$,
\begin{equation}
\label{alpha>}
\P(|T_n-T_{n, m}|_\infty \geq x) \leq \frac{C_{q, \alpha} n m^{q/2-1-\alpha q} \ell^{q/2}\||X_\cdot|_\infty\|_{q, \alpha}^q}{x^q}+ C_{q,\alpha} \exp\lp - \frac{C_{q, \alpha} x^2 m^{2 \alpha}}{n \Psi_{2, \alpha}^2}\rp
\end{equation}
holds for all $1 \leq m \leq n$.
(ii) If $0 < \alpha < 1/2-1/q$, the inequality is
\begin{equation}
\label{alpha<}
\P(|T_n-T_{n, m}|_\infty \geq x) \leq \frac{C_{q,\alpha} n^{q/2-\alpha q} \ell^{q/2} \| |X_\cdot|_\infty \|_{q, \alpha}^q}{x^q}+ C_{q, \alpha} \exp\lp - \frac{C_{q, \alpha} x^2 m^{2 \alpha}}{n \Psi_{2, \alpha}^2}\rp.
\end{equation}
\end{lem}

\begin{proof}[Proof of Lemma \ref{lemma for m-approximation, polynomial tail}]
Let $s = \ell = 1\vee \log p$. Then $\P (|T_n-T_{n,m}|_\infty \geq x)$ is equivalent to $\P (|T_n-T_{n,m}|_s \geq x)$, since for any vector $v=(v_1, \ldots, v_p)^\top$, $|v|_\infty \leq |v|_s \leq p^{1/s} |v|_\infty$.
Let $L=\lfloor (\log n - \log m)/ (\log 2)\rfloor$, $\varpi_l=2^l$ if $1 \leq l <L$, $\varpi_L=\lfloor n/m \rfloor$ and $\tau_l=m \cdot \varpi_l$ for $1 \leq l < L$, $\tau_0=m$, $\tau_L=n$. Define $M_{n, l}=T_{n, \tau_l}-T_{n, \tau_{l-1}}$ for $1 \leq l \leq L$ and write
\begin{equation}
\label{sum}
T_n-T_{n, m}= T_n-T_{n, n}+\sum\limits_{l=1}^L M_{n, l}.
\end{equation}
Notice that $T_n-T_{n, n}=\sum\limits_{j=n}^\infty T_{n, j+1}-T_{n, j}$. By Lemma \ref{lemma for high-dim Bulkholder},
\begin{equation*}
 \| |T_n-T_{n, n}|_s \|_q \leq \sum\limits_{j=n}^\infty \| |T_{n, j+1}-T_{n, j}|_s \|_q \leq \sum\limits_{j=n}^\infty C_q (ns)^{1/2}\omega_{j+1, q}=C_q (ns)^{1/2}\Omega_{n+1, q},
\end{equation*}
where $C_q$ is a constant only depending on $q$. By Markov's inequality, we have
\begin{equation}
\label{secondterm}
\Prob(|T_n-T_{n, n}|_s \geq x) \leq \frac{\| |T_n-T_{n, n}|_s \|_q ^q}{x^q} \leq \frac{C_q (ns)^{q/2}\Omega_{n+1, q}^q}{x^q}.
\end{equation}
For each $1 \leq l \leq L$, define
\begin{eqnarray*}
&&Y_{i,l}=\sum_{k=(i-1)\tau_l+1}^{(i\tau_l)\wedge n}\lp X_{k,\tau_l}-X_{k,\tau_{l-1}}\rp, \quad \text{for } 1 \leq i \leq \lfloor n/\tau_l\rfloor;\\
&&R_{n, l}^e= \sum_{i \text{ is even}} Y_{i,l} \text{ and } R_{n, l}^o= \sum_{i \text{ is odd}} Y_{i,l}.
\end{eqnarray*}
Let $c=q/2-1-\alpha q$; let $\lambda_1, \lambda_2, \cdots, \lambda_L$ be a positive sequence such that $\sum_{l=1}^L \lambda_l \leq 1$, specifically, $\lambda_l=l^{-2}/(\pi^2/3)$ if $1 \leq l \leq L/2$ and $\lambda_l=(L+1-l)^{-2}/(\pi^2/3)$ if $L/2 < l \leq L$.
Since $Y_{i,l}$ and $Y_{i',l}$ are independent for $|i-i'|>1$, by Lemma \ref{lemma for independent high-dim Nagaev}, for any $x>0$,
\begin{eqnarray*}
\P( |R_{n,l}^e|_s-2 \E | R_{n,l}^e|_s\geq \lambda_lx) \leq  \frac{C_q \sum\limits_{i \text{ is even}}\E| Y_{i,l}|_s^{q}}{\lp \lambda_l x \rp^{q}}+ \exp\lp -\frac{\lp \lambda_l x \rp^2}{3 \sum\limits_{i \text{ is even}} |\sigma_{Y_i, l}|_s^2} \rp,
\end{eqnarray*}
where $\sigma_{Y_i, l}=(\|Y_{i1, l}\|_2, \ldots, \|Y_{ip,l}\|_2)^\top$.
By Lemma \ref{lemma for high-dim Bulkholder}, $\| |Y_{i, l}|_s \|_{q} \leq C_q (\tau_l s)^{1/2} \tilde{\omega}_{l, q}$ where $\tilde{\omega}_{l, q} =\sum_{k=\tau_{l-1}+1}^{\tau_l} \omega_{k, q} \leq \tau_{l-1}^{-\alpha} \||X_\cdot|_\infty \|_{q, \alpha}$. For $1 \leq j \leq p$, by the Bulkholder inequality, $\|Y_{ij,l}\|_2 \leq \sqrt{\tau_l} \tilde{\delta}_{l,2,j}$ where $\tilde{\delta}_{l,2,j}= \sum_{k=\tau
_{l-1}+1}^{\tau_l} \delta_{k, 2, j} \leq \tau_{l-1}^{-\alpha} \|X_{\cdot j}\|_{2, \alpha}$, which implies $|\sigma_{Y_i, l}|_s \lesssim \tau^{1/2}\tau_{l-1}^{-\alpha} \Psi_{2, \alpha}$. So we obtain
\begin{equation}
\P( | R_{n,l}^e|_s -2\E | R_{n,l}^e|_s \geq \lambda_l x) \leq  \frac{C_1 n s^{q/2}}{x^{q}}\cdot \frac{\tau_{l}^{q/2-1}\tilde{\omega}_{l, q}^{q}}{\lambda_l^{q}}+ \exp\lp -\frac{C_2\lp\lambda_l x \rp^2\tau_{l-1}^{2\alpha}}{n \Psi_{2, \alpha}^2} \rp. \label{R}
\end{equation}
By Lemma 8 of \citet{chernozhukov2014comparison}, for $s = \log p \vee 1$,
\begin{equation}
\E | R_{n,l}^e|_s \lesssim \sqrt{ns} \tau_{l-1}^{-\alpha} \Psi_{2, \alpha} + n^{1/q}s \tilde{\omega}_{l,q} \lesssim  [\sqrt{ns}\Psi_{2, \alpha}+n^{1/q} s \| |X_{\cdot}|_\infty\|_{q, \alpha}]m^{-\alpha} \varpi_l^{-\alpha}.
\end{equation}
Notice that $\min_{l \geq 0} \lambda_l \varpi_l^{\alpha} >0$. Hence, $\E | R_{n,l}^e|_s \lesssim \lambda_l x$ and (\ref{R}) implies
\begin{equation*}
\P( | R_{n,l}^e|_s \geq \lambda_l x) \leq  \frac{C_1 n s^{q/2}}{x^{q}}\cdot \frac{\tau_{l}^{q/2-1}\tilde{\omega}_{l, q}^{q}}{\lambda_l^{q}}+ \exp\lp -\frac{C_2\lp\lambda_l x \rp^2\tau_{l-1}^{2\alpha}}{n \Psi_{2, \alpha}^2} \rp.
\end{equation*}

A similar inequality holds for $R_{n, l}^o$. Therefore,
\begin{eqnarray}
\Prob (|\sum\limits_{l=1}^L M_{n, l}|_s  \geq x ) \nonumber
&\leq& \sum_{l=1}^L \P\lp | M_{n,l}|_s \geq \lambda_l x \rp\\ \nonumber
&\leq& \sum_{l=1}^L \P\lp \lal  R_{n,l}^e \ral_s \geq \lambda_l x/2 \rp+\sum_{l=1}^L \P \lp \lal R_{n,l}^o \ral_s \geq \lambda_l x/2 \rp\\ \nonumber
&\leq& \sum_{l=1}^L \frac{C_1 n s^{q/2}}{x^{q}}\cdot \frac{\tau_{l}^{q/2-1}\tilde{\omega}_{l, q}^{q}}{\lambda_l^{q}}+ 2\sum_{l=1}^L\exp\lp -\frac{C_2\lp\lambda_l x \rp^2\tau_{l-1}^{2\alpha}}{n \Psi_{2, \alpha}^2} \rp \\ \label{thirdterm}
&\leq & \frac{C_3 n m^{c}s^{q/2}\| |X_\cdot|_\infty \|_{q, \alpha}^q}{x^q} \sum_{l=1}^L \frac{\varpi_l^c}{\lambda_l^q}+C_4 \sum_{l=1}^L \exp\left(-\frac{C_5 x^2 m^{2 \alpha} \lambda_l^2 \varpi_l^{2\alpha}}{n \Psi_{2, \alpha}^2}\right).
\end{eqnarray}
By the definition of $\varpi_l$ and $\lambda_l$ and by some elementary calculation, there exists some constant $C_6>1$ such that for all $t \geq 1$,
\begin{equation}
\label{exponentialterm}
\sum_{l=1}^L \exp(-C_5 t \lambda_l^2 \varpi_l^{2\alpha}) \leq C_6 \exp(-C_5 t \mu),
\end{equation}
where $\mu=\min_{l \geq 1} \lambda_l^2 \varpi_l^{2 \alpha} >0$. If $c>0$, it can be obtained that $\sum_{l=1}^L \varpi_l^c/ \lambda_l^q \leq C_7 \varpi_L^c \leq C_7 n^c/m^c$. If $c <0$, then $\sum_{l=1}^L \varpi_l^c/ \lambda_l^q \leq C_8$. Hence, combining (\ref{sum}), (\ref{secondterm}), (\ref{thirdterm}), (\ref{exponentialterm}), Lemma   \ref{lemma for m-approximation, polynomial tail} follows.
\end{proof}

\begin{lem}
\label{lemma for the sum, polynomial tail}
Assume $\| |X_\cdot|_\infty \|_{q, \alpha} < \infty$, where $q>2$ and $\alpha \geq 0$. Also assume $\Psi_{2, \alpha} < \infty$. (i) If $\alpha>1/2-1/q$, then for $x \gtrsim \sqrt{n \ell}\Psi_{2, \alpha}+n^{1/q} \ell \| |X_{\cdot}|_\infty \|_{q, \alpha}$,
\begin{equation}
\label{alpha> for the sum}
\P(|T_n|_\infty \geq x) \leq \frac{C_{q, \alpha} n \ell^{q/2} \| |X_\cdot|_\infty \|_{q, \alpha}^q}{x^q}+ C_{q,\alpha} \exp\lp - \frac{C_{q, \alpha} x^2}{n \Psi_{2, \alpha}^2}\rp.
\end{equation}
(ii) If $0 < \alpha < 1/2-1/q$, we have the following inequality,
\begin{equation}
\label{alpha< for the sum}
\P(|T_n|_\infty \geq x) \leq \frac{C_{q,\alpha} n^{q/2-\alpha q} \ell^{q/2} \| |X_\cdot|_\infty \|_{q, \alpha}^q}{x^q}+ C_{q, \alpha} \exp\lp - \frac{C_{q, \alpha} x^2}{n \Psi_{2, \alpha}^2}\rp.
\end{equation}
\end{lem}

\begin{proof}[Proof of Lemma \ref{lemma for the sum, polynomial tail}]
The proof is similar to that of Lemma \ref{lemma for m-approximation, polynomial tail}, and thus is omitted.
\end{proof}

\section{Proofs}
\label{sec:proof}

\subsection{Proof of Theorems \ref{th:1507140259} and \ref{th:1507140325}}
\label{sec:1507151006}

We shall apply the $m$-dependence approximation approach. For $m \ge 0$, define
\begin{equation}
X_{i, m}=(X_{i1,m}, \ldots, X_{ip, m})^\top=\E(X_i|\eps_{i-m},\eps_{i-m+1}, \ldots, \eps_i).
\end{equation}
Write $T_{X}=\sum_{i=1}^n X_i$ and $T_{X, m}=\sum_{i=1}^n X_{i, m}$.
For simplicity, suppose $n=(M+m)w$, where $M \gg m$ and $M, m, w \rightarrow \infty$ (to be determined) as $n \to \infty$. We apply the block technique and split the interval $[1, n]$ into alternating large blocks $L_b= [ (b-1)(M+m)+1,  bM+(b-1)m ]$ and small blocks $S_b =[ bM+(b-1)m +1, b(M+m) ]$, $1 \leq b \leq w$. Let
\begin{eqnarray*}
Y_b = \sum_{i \in L_b}X_i, \,\, Y_{b, m}=\sum_{i \in L_b} X_{i, m}, \,\,
 T_{Y}=\sum_{b=1}^w Y_{b}, \,\, T_{Y, m}=\sum_{b=1}^w Y_{b, m}.
\end{eqnarray*}
Let $Z_{b}$, $1 \leq b \leq w$, be i.i.d. $N(0, M B)$ and $Z_{b, m}$ be i.i.d. $N(0, M \tilde B)$, where the covariance matrices $B$ and $\tilde B$ are respectively given by
\begin{eqnarray}
\label{eq:1507172300}
B = (b_{ij})_{i,j=1}^p = \mbox{Cov}(Y_b/\sqrt{M}) \mbox{ and }
\tilde{B}=(\tilde{b}_{ij})_{i,j=1}^p = \mbox{Cov}(Y_{b, m}/\sqrt{M}).
\end{eqnarray}
Write $T_{Z, m}=\sum_{b=1}^w Z_{b, m}$ and let $Z \sim N(0,\Sigma)$.

\begin{lem}
\label{lemmafordifference}
(i) Assume $\Theta_{q, \alpha}< \infty$ for some $q>2$ and $\alpha>0$. Then there exists some constant $C_{q, \alpha}$ such that for $y > 0$
\begin{equation}\label{eq:1508122226}
\P(|T_X-T_{Y, m}|_\infty \geq y) \lesssim f^*_1(y)+f^*_2(y)=: f^*(y)
\end{equation}
where the constant in $ \lesssim$ only depends on $q$ and $\alpha$,
\begin{equation}
f^*_1(y)=\left\{
\begin{array}{ll}
  y^{-q}n m^{q/2-1-\alpha q}\Theta_{q, \alpha}^q+ p\exp\lp - \frac{C_{q, \alpha}y^2 m^{2 \alpha}}{n\Psi_{2, \alpha}^2}\rp, & \alpha>1/2-1/q \\
  y^{-q}n^{q/2-\alpha q} \Theta_{q, \alpha}^q+  p \exp\lp - \frac{C_{q, \alpha}y^2 m^{2 \alpha}}{n\Psi_{2, \alpha}^2}\rp, & \alpha<1/2-1/q
\end{array}
\right.
\end{equation}
and
\begin{equation}
f^*_2(y)=\left\{
\begin{array}{ll}
  y^{-q}wm \Theta_{q, \alpha}^q+  p\exp\lp - \frac{C_{q, \alpha}y^2 }{mw\Psi_{2, \alpha}^2}\rp, & \alpha>1/2-1/q \\
  y^{-q}(wm)^{q/2-\alpha q} \Theta_{q, \alpha}^q+ p\exp\lp - \frac{C_{q, \alpha} y^2 }{wm\Psi_{2, \alpha}^2}\rp, & \alpha<1/2-1/q
\end{array}
\right. .
\end{equation}
(ii) Assume $\Phi_{\psi_\nu, \alpha}<\infty$ for some $\nu\geq 0$ and $\alpha>0$. Let $\beta=2/(1+2\nu)$. Then there exists a constant $C_\beta >0$ such that for $y>0$,
\begin{equation}\label{eq:1508122228}
\P(|T_X-T_{Y, m}|_\infty \geq y) \lesssim f^\diamond_1(y)+ f^\diamond_2(y)=: f^\diamond(y),
\end{equation}
where the constant in $ \lesssim$ only depends on $\beta$ and $\alpha$,
\begin{equation*}
f^\diamond_1(y)=   p\exp\left\{-C_\beta\left(\frac{ y m^{\alpha}}{\sqrt{n}\Phi_{\psi_\nu, \alpha}}\right)^\beta\right\} \,\,
 \text{and} \,\, f^\diamond_2(y)= p\exp\left\{-C_\beta\left(\frac{ y }{\sqrt{mw} \Phi_{\psi_\nu, 0}}\right)^\beta\right\}.
\end{equation*}
\end{lem}
\begin{proof}
Let $P_1 = \P(|T_X-T_{X, m}|_\infty \geq y/2)$ and $P_2 = \P(|T_{X, m}-T_{Y, m}|_\infty \geq y/2)$. Lemmas \ref{m-approximationtheorem1} and \ref{lemma for m-approximation, polynomial tail} imply that $P_1 \leq f^*_1(y)$. Write $T_{X, m}-T_{Y, m} = \sum_{b=1}^w \sum_{i \in S_b} X_{i, m}$. By Lemmas \ref{theorem for the sum polynomial} and \ref{lemma for the sum, polynomial tail}, we also have $P_2 \leq f^*_2(y)$. Hence both cases with $\alpha>1/2-1/q$ and $\alpha<1/2-1/q$ of Lemma \ref{lemmafordifference}(i) follow in view of $\P(|T_X-T_{Y, m}|_\infty \geq y) \leq P_1+P_2$.

The exponential moment case (ii) similarly follows from $P_1 \leq f^\diamond_1(y)$ and $P_2 \leq f^\diamond_2(y)$.
\end{proof}

\begin{lem}
\label{mainlemma}
Let $D=(d_{i j})_{i, j=1}^p$ be a diagonal matrix. Assume that there exist constants $c>0, c_2>c_1>0$ such that $c<\min_{1 \leq j \leq p} d_{jj}$ and $c_1 \leq \tilde{b}_{jj}/d_{jj} \leq c_2$ for all $1 \leq j \leq p$. Assume $\Psi_{q, 0}< \infty$ for some $q \geq 4$. Then for all $\lambda \in (0,1)$,
\begin{eqnarray*}
&&\sup\limits_{t \in \mathbb{R}} \left| \P(|D^{-1/2}T_{Y, m}/\sqrt{n}|_\infty \leq t)-\P(|D^{-1/2}T_{Z, m}/\sqrt{n}|_\infty \leq t)  \right| \\
&\lesssim &  w^{-1/8}(\Psi_{3, 0}^{3/4}\vee\Psi_{4, 0}^{1/2})(\log(pw/\lambda))^{7/8} +w^{-1/2}(\log(pw/\lambda))^{3/2} u_m(\lambda)+\lambda\\
&=:& h(\lambda, u_m(\lambda)),
\end{eqnarray*}
where the constant in $\lesssim$ depends on $c, c_1, c_2$, and $q$ and $\alpha$ for (i), and $\beta$ for (ii) below, and $u_m(\lambda) \le u^*_m(\lambda)$ in (i), and $u_m(\lambda) \le u^\diamond_m(\lambda)$ in (ii).

(i) Assume $\Theta_{q, \alpha}< \infty$ for some $q \geq 4$ and $\alpha>0$, then
\begin{equation}
u^*_m(\lambda) = \left\{
\begin{array}{ll}
 \max\{\Theta_{q, \alpha}(\lambda^{-1}w)^{1/q}M^{1/q-1/2}, \Psi_{2, \alpha}\sqrt{\log(pw/\lambda)}\}, & \alpha>1/2-1/q \\
 \max\{\Theta_{q, \alpha}(\lambda^{-1}w)^{1/q}M^{-\alpha}, \Psi_{2, \alpha}\sqrt{\log(pw/\lambda)}\}, & \alpha<1/2-1/q.
\end{array} \right.
\end{equation}

(ii) Assume $\Phi_{\psi_\nu, 0}<\infty$ for some $\nu\geq 0$. Then
\begin{equation}
u^\diamond_m(\lambda) = \max\{\Phi_{\psi_\nu, 0}(\log (pw/\lambda))^{1/\beta}, \sqrt{\log (pw/\lambda)}\}.
\end{equation}
\end{lem}

\begin{proof}
For $1<l\leq q$, define $R_l=\max_{1 \leq j \leq p} \| M^{-1/2} Y_{bj, m}\|_l$.
Since $X_{ij, m}=\sum_{k=0}^m \mathcal{P}_{i-k}X_{ij}$, by Burkholder's inequality (\citet{burkholder1973}),
\begin{equation*}
\|\sum_{i=1}^M \mathcal{P}_{i-k}X_{ij}\|_l^2 \leq C_l \sum_{i=1}^M \|\mathcal{P}_{i-k}X_{ij}\|_l^2 \leq C_l M (\theta'_{k, l, j})^2,
\end{equation*}
then we have
\begin{equation}
\label{moment for the sum}
\|\sum_{i=1}^M X_{ij, m}\|_l \leq C_l \sum_{k=0}^m \|\sum_{i=1}^M \mathcal{P}_{i-k}X_{ij}\|_l \leq C_l M^{1/2} \Delta_{0, l, j},
\end{equation}
which implies $R_l \leq C_l \Psi_{l, 0}$.
For $0<\lambda<1$ and the diagonal matrix $D=(d_{ij})_{i,j=1}^p$, define $u_{Y, m}(\lambda)$ as the infimum over all numbers $u>0$ such that
\begin{equation*}
\P(|M^{-1/2} d_{jj}^{-1/2}Y_{bj, m}| \leq u, 1 \leq b \leq w, 1\leq j \leq p) \geq 1-\lambda.
\end{equation*}
Also define $u_{Z, m}(\lambda)$ by the corresponding quantity for the analogue Gaussian case, namely with $Y_{b, m}$ replaced by $Z_{b, m}$ in the above definition. Let $u_{m}(\lambda):= u_{Y, m}(\lambda) \vee u_{Z, m}(\lambda)$.
By Theorem 2.2 of \citet{chernozhukov2013}, for all $\lambda \in (0,1)$,
\begin{eqnarray*}
&&\sup\limits_{t \in \mathbb{R}} \left| \P(|D^{-1/2}T_{Y, m}/\sqrt{n}|_\infty \leq t)-\P(|D^{-1/2}T_{Z, m}/\sqrt{n}|_\infty \leq t)  \right| \\
&\lesssim & w^{-1/8}(R_3^{3/4}\vee R_4^{1/2})(\log(pw/\lambda))^{7/8}+w^{-1/2}(\log(pw/\lambda))^{3/2} u_m(\lambda)+\lambda,
\end{eqnarray*}
Now we shall find a bound on the function $u_{m}(\lambda)$.
(i) By Lemmas \ref{theorem for the sum polynomial} and \ref{lemma for the sum, polynomial tail}, we have
\begin{eqnarray*}
&&\P(|M^{-1/2} d_{jj}^{-1/2} Y_{bj, m}| > u \text{ for some } b, j) \leq \P(|M^{-1/2}Y_{b, m}|_\infty > c^{1/2}u) \\
&&\leq \left\{
\begin{array}{ll}
C_{q, \alpha} u^{-q}wM^{1-q/2} \Theta_{q, \alpha}^q+ C_{q,\alpha} pw \exp\lp -\frac{C_{q, \alpha}u^2}{\Psi_{2,\alpha}^2}\rp, & \alpha>1/2-1/q \\
C_{q,\alpha} u^{-q}w M^{-\alpha q}\Theta_{q, \alpha}^q + C_{q, \alpha} pw \exp\lp - \frac{C_{q, \alpha} u^2 }{\Psi_{2,\alpha}^2}\rp, & \alpha<1/2-1/q
\end{array}
\right. .
\end{eqnarray*}
This implies $u_{Y, m}(\lambda) \leq C_{q,\alpha} \max\{\Theta_{q, \alpha}(\lambda^{-1}w)^{1/q}M^{1/q-1/2}, \Psi_{2, \alpha}\sqrt{\log(pw/\lambda)}\}$ if $\alpha>1/2-1/q$ and $u_{Y, m}(\lambda) \leq C_{q, \alpha} \max\{\Theta_{q, \alpha}(\lambda^{-1}w)^{1/q}M^{-\alpha}, \Psi_{2, \alpha}\sqrt{\log(pw/\lambda)}\}$ if $\alpha<1/2-1/q$. For $u_{Z, m}(\lambda)$, since  $M^{-1/2} Z_{bj, m} \sim N(0, \tilde{b}_{jj})$, we have $\E(\exp\{M^{-1}Z^2_{bj, m}/(4\tilde{b}_{jj})\})\leq C$. Hence
\begin{eqnarray}
\P(|M^{-1/2}d_{jj}^{-1/2} Z_{bj, m}| > u \text{ for some } b, j) &\leq& \sum_{b=1}^w \sum_{j=1}^p\P(|M^{-1/2}Z_{bj, m}| > d_{jj}^{1/2}u) \nonumber \\ \label{gaussianboundforu}
&\leq & Cpw \exp(-d_{jj}u^2/(4\tilde{b}_{jj})).
\end{eqnarray}
With the assumption $c_1 \leq \tilde{b}_{jj}/d_{jj} \leq c_2$,  $u_{Z, m}(\lambda) \leq C \sqrt{\log(pw/\lambda)}$.\\
(ii)  By Bonferroni inequality and Lemma \ref{theorem for the sum exponential},
\begin{equation}
\label{exponentialboundforu}
\P(|M^{-1/2}d_{jj}^{-1/2} Y_{bj, m}| > u \text{ for some } b, j)
\leq C_\beta pw \exp\left\{-C_\beta\frac{u^\beta}{\Phi^\beta_{\psi_\nu, 0}}\right\},
\end{equation}
where $\beta=2/(1+2\nu)$ and $C_\beta$ is a constant that depends on $\beta$ only. Combining (\ref{gaussianboundforu}) and (\ref{exponentialboundforu}), it follows that $u_{m}(\lambda)\leq C_{\beta}\max\{\Phi_{\psi_\nu, 0}(\log (pw/\lambda))^{1/\beta}, \sqrt{\log (pw/\lambda)}\}$.
\end{proof}

Now we consider the comparison between $Z$ and $T_{Z, m}$. Let $\pi(x)= x^{1/3}(1 \vee \log(p/x))^{2/3}$ for $x>0$.
\begin{lem}
\label{lemmafortwogaussian}
Assume $\Psi_{2, \alpha}<\infty$ for some $\alpha>0$. Let $D=(d_{ij})_{i, j=1}^p$ be a diagonal matrix such that there exist some constants $0< C_1< C_2$ such that $C_1 \leq \sigma_{jj}/d_{jj} \leq C_2$ for all $1 \leq j \leq p$. Then we have
\begin{eqnarray*}
\label{Tz,m-Tz}
&&\sup\limits_{t \in \mathbb{R}} \left| \P(|D^{-1/2}T_{Z, m}/\sqrt{n}|_\infty \leq t)-\P(|D^{-1/2}Z|_\infty \leq t)  \right| \\
&\lesssim & \pi(\max_{1\leq j \leq p}d^{-1}_{jj}\Psi_{2, \alpha} \Psi_{2, 0} (m^{-\alpha}+v(M))+wm/n),
\end{eqnarray*}
where $v(M)$ is the same as defined in Corollary \ref{corollary for Sigma}.
\end{lem}
\begin{proof}
By the definition of $T_{Z, m}$ and $Z$ and (\ref{eq:1507172300}),
\begin{eqnarray*}
&&\Sigma^{Z, m}:=\Cov(D^{-1/2}T_{Z, m}/\sqrt{n})= \frac{Mw}{n}D^{-1/2}\tilde{B}D^{-1/2}, \\ &&\Sigma^{Z}:=\Cov(D^{-1/2}Z)= D^{-1/2}\Sigma D^{-1/2}.
\end{eqnarray*}
Let $S_{M j} = \sum_{i=1}^M X_{ij}$ and $S_{M j, m} = \sum_{i=1}^M X_{ij, m}$. By the moment inequality in \citet{Wu2005}, $ \| S_{M j} \|_2 \leq M^{1/2} \Delta_{0, 2, j}$, $\| S_{M j, m}\|_2 \leq M^{1/2} \Delta_{0, 2, j}$ and  $\|S_{M j} - S_{M j, m}  \|_2 \leq M^{1/2} \Delta_{m+1, 2, j}$. Note that $b_{jk}=M^{-1}\E(S_{M j} S_{M k})$ and $\tilde{b}_{jk}=M^{-1}\E(S_{M j, m}  S_{M k, m})$. Then
\begin{eqnarray*}
|b_{jk}-\tilde{b}_{jk}|&=& \frac{1}{M} |\E (S_{M j} S_{M k}  - S_{M j, m}  S_{M k, m})| \\
&\leq & \frac{1}{M}\left(\|S_{M j}  \|_2\cdot\|S_{M k}  - S_{M k, m}\|_2+\|S_{M k, m} \|_2\cdot\| S_{M j} - S_{M j, m}  \|_2\right) \cr
&\leq & 2\Psi_{2, \alpha}\Psi_{2, 0} m^{-\alpha}.
\end{eqnarray*}
Recall that $\sigma_{jk}=\sum_{l=-\infty}^\infty \gamma_{jk}(l)$ and
\begin{equation*}
b_{jk}=M^{-1}\E(S_{Mj} S_{Mk})=M^{-1}\sum_{l=-M}^M (M-|l|)\gamma_{jk}(l).
\end{equation*}
It follows that
\begin{equation*}
\sigma_{jk}-b_{jk}=\sum_{|l|> M} \gamma_{jk}(l) +M^{-1}\sum_{l=-M}^M |l| \gamma_{jk}(l).
\end{equation*}
By $X_{ij}=\sum_{h=0}^\infty \mathcal{P}^{i-h}X_{ij}$, we have
\begin{equation*}
|\gamma_{jk}(l)|=|\sum_{h=0}^\infty \E[(\mathcal{P}^{-h}X_{0j})(\mathcal{P}^{-h}X_{lk})]| \leq \sum_{h=0}^\infty |\E[(\mathcal{P}^{-h}X_{0j})(\mathcal{P}^{-h}X_{lk})]| \leq \sum_{h=0}^\infty \delta_{h, 2, j}\delta_{h+l, 2, k}.
\end{equation*}
Hence, it can be obtained that
\begin{equation*}
\left|\sum_{|l|>M} \gamma_{jk}(l)\right| \leq 2 \sum_{l=M+1}^\infty |\gamma_{jk}(l)| \leq 2\sum_{l=M+1}^\infty \sum_{h=0}^\infty \delta_{h, 2, j}\delta_{h+l, 2, k} \leq 2 \Delta_{0, 2, j}\Delta_{M+1, 2, k},
\end{equation*}
and
\begin{equation*}
\left|\frac{1}{M}\sum_{l=-M}^M |l| \gamma_{jk}(l)\right| \leq \frac{2}{M} \sum_{l=1}^M \sum_{\iota=k}^M \sum_{h=0}^\infty \delta_{h, 2, j}\delta_{h+\iota, 2, k} \leq \frac{2}{M} \Delta_{0, 2, j} \sum_{l=1}^M \Delta_{l, 2, k}.
\end{equation*}
Since $\Delta_{0, 2, j} \leq \Psi_{2,0}$ and $\Delta_{m, 2, j} \leq  \Psi_{2, \alpha}m^{-\alpha}$, $\max_{1 \leq j,k \leq p}|b_{jk}-\sigma_{jk}|\leq \Psi_{2,\alpha}\Psi_{2,0}v(M)$. Hence,
\begin{eqnarray*}
|\Sigma^{Z, m}-\Sigma^{Z}|_\infty &\leq & \max_{1\leq j \leq p}d^{-1}_{jj}(|\tilde{B}-B|_\infty+ |B-\Sigma|_{\infty})+(1-Mw/n)|D^{-1/2}\Sigma D^{-1/2}|_\infty\\
&\leq &\max_{1\leq j \leq p}d^{-1}_{jj}\Psi_{2, \alpha} \Psi_{2, 0} (m^{-\alpha}+v(M))+ C_2 wm/n.
\end{eqnarray*}
By Theorem 2 of \citet{chernozhukov2014comparison}, the result follows.
\end{proof}

\begin{thm}
\label{maintheorem}
 Let $\Sigma_0$ be the diagonal matrix of the long run covariance matrix $\Sigma$ and $D_0=\Sigma_0^{1/2}$. Let Assumption \ref{assumption:1507172202} be satisfied. (i) Assume that $\Theta_{q, \alpha} < \infty$ holds with some $q \geq 4$ and $\alpha>0$. Then for every $\lambda \in (0, 1)$ and $\eta>0$,
\begin{eqnarray}
\rho_n&:=&\sup\limits_{t \in \mathbb{R}} \left| \P(|D_0^{-1}T_{X}/\sqrt{n}|_\infty \leq t)-\P(|D_0^{-1}Z|_\infty \leq t)  \right| \nonumber \\ \label{maintheorem1}
&\lesssim &  f^*(\sqrt{n}\eta)+ \eta \sqrt{\log p}+ h(\lambda, u^*_m(\lambda))+\pi(\Psi_{2, \alpha}\Psi_{2, 0} (m^{-\alpha}+v(M))+wm/n).
\end{eqnarray}
(ii) Assume $\Phi_{\psi_\nu, \alpha} < \infty$ for some $\nu \geq 0$ and $\alpha>0$. Then for every $\lambda \in (0, 1)$ and $\eta>0$,
\begin{eqnarray}
\rho_n&:=&\sup\limits_{t \in \mathbb{R}} \left| \P(|D_0^{-1}T_{X}/\sqrt{n}|_\infty \leq t)-\P(|D_0^{-1}Z|_\infty \leq t)  \right| \nonumber \\ \label{maintheorem2}
&\lesssim & f^\diamond(\sqrt{n}\eta)+ \eta \sqrt{\log p}+ h(\lambda, u^\diamond_m(\lambda))+\pi(\Psi_{2, \alpha}\Psi_{2, 0} (m^{-\alpha}+v(M))+wm/n).
\end{eqnarray}
\end{thm}
\begin{proof}
(i)
By Lemma \ref{mainlemma} (i) and Lemma \ref{lemmafortwogaussian}, we have for every $\lambda \in (0, 1)$,
\begin{eqnarray}
&&\sup\limits_{t \in \mathbb{R}} \left| \P(|D_0^{-1}T_{Y, m}/\sqrt{n}|_\infty \leq t)-\P(|D_0^{-1}Z|_\infty \leq t)  \right| \nonumber  \\ &\lesssim & h(\lambda, u^*_m(\lambda))+\pi(\Psi_{2, \alpha}\Psi_{2, 0} (m^{-\alpha}+v(M))+wm/n). \label{sup_part1}
\end{eqnarray}
Observe that each component of the Gaussian vector $D_0^{-1}Z$ has variance 1. By Theorem 3 of \citet{chernozhukov2014comparison}, for every $\eta>0$,
\begin{equation}
\label{anticoncentrationforSZ}
\sup\limits_{t \in \mathbb{R}} \P(\left||D_0^{-1}Z|_\infty-t \right|\leq \eta) \lesssim \eta \sqrt{\log p}.
\end{equation}
By the triangle inequality, for every $\eta>0$, we have
\begin{eqnarray*}
&&\sup\limits_{t \in \mathbb{R}} \left| \P(|D_0^{-1}T_{X}/\sqrt{n}|_\infty > t)-\P(|D_0^{-1}T_{Y, m}/\sqrt{n}|_\infty > t)  \right|\\
&\leq& \P(|D_0^{-1}(T_{X}-T_{Y, m})/\sqrt{n}|_{\infty}>\eta)+\sup\limits_{t \in \mathbb{R}} \P(\left||D_0^{-1}T_{Y, m}/\sqrt{n}|_\infty-t \right|\leq \eta),
\end{eqnarray*}
which implies Theorem \ref{maintheorem} (i) in view of Lemma \ref{lemmafordifference} (i), (\ref{sup_part1}) and (\ref{anticoncentrationforSZ}).

(ii) Inequality (\ref{maintheorem2}) can be obtained by replacing $f^*$ and $u^*_m$ with $f^\diamond$ and $u^\diamond_m$ in the above proof.
\end{proof}

\subsection{Proof of Theorem \ref{th:1507140259}}

\begin{proof}
Recall (\ref{eq:1508122226}) for $f^*(\cdot)$. By Theorem \ref{maintheorem}, for $\alpha > 1/2-1/q$, to have (\ref{eq:J021209}), we need
\begin{equation}
\label{b}  \pi(\Psi_{2, \alpha}\Psi_{2, 0} (m^{-\alpha}+v(M))+wm/n) \rightarrow 0
\end{equation}
and for some $\eta>0$ and $\lambda \in (0,1)$,
\begin{eqnarray}
\label{c} && f^*(\sqrt{n}\eta)+\eta \sqrt{\log p}\rightarrow 0,\\
\label{d} && h(\lambda, u^*_m(\lambda))\rightarrow 0.
\end{eqnarray}
Firstly, (\ref{b}) requires $m \gg L_2$, $wm \ll n(\log p)^{-2}$, $w \ll n(\log p)^{-2} (\Psi_{2, \alpha}\Psi_{2,0})^{-1}$ if $\alpha>1$ and $w \ll n/L_2$ if $0<\alpha <1$. Moreover, (\ref{c}) requires $m \gg \max(L_1, (\Psi_{2, \alpha} \log p)^{1/\alpha})$ and $w m \ll \min (N_1, N_2)$. And (\ref{d}) needs (\ref{eq:1507142040}) and $w \gg \max(W_1, W_2)$. We also need $M\asymp n/w \gg m$. Notice that $(\Psi_{2, \alpha} \log p)^{1/\alpha} \lesssim L_2$, $N_2 \lesssim n(\log p)^{-2}$ and $N_2 \leq n(\log p)^{-2}(\Psi_{2, \alpha}\Psi_{2,0})^{-1}$. If
\begin{equation}
\label{e}
\max(L_1, L_2) \max(W_1, W_2) = o(1) \min(n, N_1, N_2),
\end{equation}
then we can always choose $m$ and $w$ such that (\ref{eq:J021209}) holds. Observe that $N_2 \lesssim n$, then (\ref{e}) is reduced to (\ref{eq:1507142041}).

For $0< \alpha < 1/2-1/q$, the function $f^*$ in (\ref{c}) is replaced by $f^\diamond$ (cf. (\ref{eq:1508122228})), which implies $\Theta_{q, \alpha} (\log p)^{1/2} = o(n^{\alpha})$, $m \gg (\Psi_{2, \alpha} \log p)^{1/\alpha} $ and $wm \ll \min(N_2, N_3)$. And $u^*_m$ in (\ref{d}) is replaced by $u_m^\diamond$, implying $w \gg \max(W_1, W_2, W_3)$. By the similar argument, if (\ref{eq:1508010523}) is further assumed, then (\ref{eq:J021209}) also holds for the case $0<\alpha < 1/2-1/q$.
\end{proof}

\begin{rmk}
In the proof of Theorem \ref{th:1507140259}, we exclude the case $\alpha=1$ when $\alpha>1/2-1/q$. If $\alpha=1$, we need to impose the additional assumption
\begin{equation}
\label{add}
\max(W_1, W_2)=o(n/(L_2\log n))
\end{equation}
to ensure (\ref{b}). The above condition is very mild since (\ref{eq:1507142041}) implies $\max(W_1, W_2) = o(n/L_2)$. If $\log n \lesssim (\log p)^2 \Psi_{2, \alpha}^2$, which trivially holds in the high-dimensional case $p \asymp n^\kappa$ with some $\kappa > 0$, we have $N_2= O(n/\log n)$ and hence (\ref{eq:1507142041}) implies (\ref{add}). Similarly, it is further assumed $\max(W_1, W_4)=o(n/(L_2\log n))$ in Theorem \ref{th:1507140325} if $\alpha=1$.
\end{rmk}

\bibliographystyle{plainnat

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Boil pasta, as per package directions; then, strain and set aside.
Combine the remaining ingredients in a saucepan until heated through, reducing sauce with continued heat to thicken. If using thickener, use some of the heated sauce and add to the powdered thickener in a separate bowl, to dissolve and to temper the thickener to the desired consistency; then add to sauce and stir. Pour over pasta. Season to taste.
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Exploiting Grids intuitively requires developers to alter their applications, which calls for expertise on Grid programming. Gridification tools address this problem by semi-automatically making user applications to be Grid-aware. However, most of these tools produce monolithic Grid applications in which common tuning mechanisms (e.g. parallelism) are not applicable, and do not reuse existing Grid middleware services. We propose BYG (BYtecode Gridifier), a gridification tool that relies on novel bytecode rewriting techniques to parallelize and easily execute existing applications via Grid middlewares. Experiments performed by using several computing intensive applications on a cluster and a simulated wide-area Grid suggest that our techniques are effective while staying competitive compared to programmatically using such services for gridifying applications.


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720 Fusion Fresh
Steamboat Colorado Asian Cuisine
Asian Food Blog
Know the Different Types of Soy Sauces: Let's Explore
Posted on April 18, 2018 April 18, 2018 720fusionfreshownerPosted in Asian Restaurant Steamboat Colorado, Chinese Food Steamboat, Japanese Food Steamboat Colorado
What do wine and soy sauce have in common? The more they age the more flavorful and complex the taste. Soy sauce is a 2,500-year-old condiment, and over the years there has been a lot of variation. Let's explore!
Soy sauce is a liquid made from soybeans, wheat, water, and salt. Generally speaking, it falls into two groups: naturally brewed or fermented, and chemically produced. Naturally brewed soy sauce is fermented for months or longer. The best soy sauce is actually two years or older. Chemical or non-brewed soy sauce is produced quickly within days, a mixture of hydrolyzed soy protein and flavorings such as corn syrup and caramel. Naturally brewed soy sauce has an aroma and complexity of flavor. Industrial soy sauce is one-note and super salty.
Among brewed soy sauces, there's another split: the geographic style. While soy sauce styles vary among cuisines including Korean, Indonesian, Thai, Filipino, and more, the types you're most likely to find at US grocery stores are either Japanese or Chinese in origin and/or style.
Japanese soy sauce, or shoyu, is brewed with roasted wheat. Chinese soy sauce, which traditionally left out the wheat, is nowadays brewed with wheat flour. In addition, Chinese sauce sometimes contains added sugar, according to Roberts.
This difference in ingredients, as well as brewing time, gives Japanese soy sauce a slightly sweeter, rounder flavor and Chinese soy sauce a denser, saltier taste. Chinese sauce also tends to be much thicker and darker than the Japanese style. With Chinese soy sauce, you'll see it coats the bottle.
There's a third distinction: light or dark. The difference is largely due to the length of aging. Both Japanese and Chinese light soy sauces are thinner and lighter in color (relatively speaking) but more intense in flavor than their dark counterparts. Dark Chinese soy sauce, which is aged the longest, also tends to be sweeter due to the addition of molasses or another sweetener.\
And here's another wrinkle: Dark Japanese soy sauce is what's most available and used, even if the bottle isn't labeled as such, which it usually isn't.
In Chinese cooking, it's the opposite: light soy sauce is more common. Bottles will indicate light—sometimes labeled "thin" or "superior"—or dark. The latter is reserved for stews or as a flavor booster or finisher in sauces,
Tamari is a Japanese soy sauce brewed with soybeans and no wheat—but sometimes there is a trace amount. So, not all tamari is gluten-free. Check the label to make sure.
And not every bottle labeled "gluten-free" is considered tamari. For instance, Kikkoman's gluten-free soy sauce is made with soybeans and rice instead of wheat.
The key words to look for on the bottle are "brewed" or "traditionally brewed." That and a scan of the ingredients list will tell you what you're getting. The fewer ingredients, the better, Young and Roberts said.
Beyond that, it's up to your palate, as brands vary widely in sodium levels and nuances of flavor.
BOTTLE SIZE AND STORAGE
Soy sauce will last a long time unopened. The best-by date on the bottle should be a good year to two years out. But oxygen is its enemy, so you're better off choosing a smaller bottle.
And once opened, keep it in a cool, dry place. It's not necessary to refrigerate soy sauce, but doing so will extend its flavor and freshness.
Asian Inspired Food, Soy Sauces, Steamboat Springs Colorado Chinese Food
5 of the Most Popular and Delicious Vegetables in Asian Cuisine
The Mysterious Origins of the Chinese Fortune Cookie
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Monday 11:00 am – 10:00 pm
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Exploring Ramen Noodles Popularity Across the World
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2019 Reviews: Best Asian Fusion Restaurant in Steamboat Springs, Colorado
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☞ 720 Lincoln Ave, Steamboat Springs, CO,80487
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We are closed on May 22 (Monday) and May 23(Tuesday).


FREE INTRODUCTORY CLASS!!! We offer anyone who has not tried out our IKI/Krav Maga system a chance to try out what we have to offer by attending one class for free! Please call or email to arrange an appointment (click "contact" at top of this page or use email address on home page). We try to schedule introductory classes for Wednesday evenings or Saturday mornings. It is helpful if you can call or email to let us know you will be there but on those days it is not technically required. If your schedule does not permit you to try out on these days, let us know and we will do what we can to accommodate your needs on other days.
Monday and Wednesday evenings from 6:30 PM to 8:00 PM; Saturdays: From 10:15 AM to 1:00 PM. All classes are open to all students, all ranks.
We typically include conditioning and endurance training drills in our classes, an integral part of the IKI approach to self defense.
Dues are $99 per month. You can attend as many of our classes as you like.
Family rates are available. Essentially first family member pays full price others are half of regular price.
Law Enforcement Personnel, EMT/EMS and Fire Services Personnel rates are $60 per month.
At this time the minimum age for students is 14 years. Please contact us for details.
Georgia Mountain Krav Maga's Curriculum and requirements follow the recommended programs established by Israeli Krav International.
LAW ENFORCEMENT TRAINING: Portions of our training program have been accepted (POST CERTIFIED) by the Georgia Peace Officers Standards and Training Council.
Since its inception, GMKM has been taught in what some call a "seminar style" as opposed to a rigid, structured sequential approach. The approach definitely works and it works well. For lack of a better term (until now) I have called this approach "organized chaos".
One reason this approach works so well is that the IKI Krav system is based primarily on principle, concept and logic instead of memorizing techniques. As a result, the techniques ultimately become instinctive, customized for each individual.
One of our newest students, Dr. Kathye Light, is a PHD and a physical therapist at Brenau University here in Georgia. As she was watching the approach at our recent seminar and then in our subsequent classes she told me that our method was actually "Psychomotor Training" (motor learning) or "Variable Random Practice".
While we do have ranks and belt colors up through black belt ranks, acquiring rank should not be the student's main focus. Rank is awarded based on performance, attitude and understanding of the self defense programs we teach.
By design our belt tests are not easy. We do not and never will "sell" rank or certificates. GMKM strives for quality performance. We do recognize that everyone has different capabilities, talents and limitations. All of this is taken into account when we test students. The testing at GMKM is very demanding. So is survival on the street!
Learning is the most important factor. If the student is willing to work hard and learn what is offered, rank will come. In most cases the principles involved in our techniques are more important than the techniques themselves. However the techniques must still be learned, and learned correctly.
All rank promotions will be approved by Moshe Katz and certificates of rank will be issued and signed by Mr. Katz, founder and head instructor for Israeli Krav International.
A handbook outlining all of the basic rank requirements through black belt, a good bit of history and other pertinent information is available to all students.
Our training allows successful application of self defense techniques regardless of terrain, clothing, age or injury. We include training in mental discipline and "will to survive" or "will to succeed" as part of our curriculum. Students are trained to help each other achieve their goals and to develop the mentality needed for self defense.
Environmental awareness is an integral part of our training approach. This training enhances the students in innumerable ways beyond the application of Krav Maga and self defense techniques.
Our training involves extremely close quarters situations, gun threats, knife attacks and knife threats, random attacks with and without weapons, terrorist attacks and hostage survival among others.
We also train in third party defense so that we may be able to help someone else survive.


We are fully conversant with all the latest tyre pressure monitoring systems.
We have the latest equipment to not only identify, but also rectify a problem should you have an issue with any of your valve sensors. And all at the fraction of the cost at a main dealer.


The Woodrow Wilson New Mexico MBA Fellowship in Education Leadership recruits and prepares experienced educators, who will complete an executive-style MBA program that blends an education-based business curriculum with clinical experience in schools, corporations, and nonprofits.
The WW MBA Fellowship in Education Leadership at NMSU is PED approved as an administrator preparation program and comprises 12 courses taken over two years. All classes will be delivered in a blended format with content online, as well as face-to-face.
Each Fellow receives a stipend toward tuition and expenses during the MBA program.
The curriculum provides specialized preparation in areas such as leadership, quality management, talent management, data analysis, and organizational change—all seen through an education lens.
Fellows commit to serve for three years in approved school or district leadership positions within their states. They gain the knowledge, skills, expertise, and tools to progress into district and/or state leadership positions or go on to lead other education entities. They also receive three years of executive coaching once they enter their leadership positions.
Questions regarding the NMSU MBA in Education Leadership should be addressed to the Program Director, Dr. Steven Elias, at selias@nmsu.edu or 575-646-7642.
To nominate a candidate and learn more, visit woodrow.org/fellowships/ww-ed-mba.
This entry was posted in News and tagged Fellowships, Graduate Programs, Leadership, MBA Program, Online Programs, Steve Elias, Woodrow Wilson MBA Fellowship in Education Leadership Program, Woodrow Wilson MBA Fellowship Program. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Both comments and trackbacks are currently closed.


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text:
Don't Let's Go to the Dogs Tonight: An African Childhood
by Fuller, Alexandra
Publisher: Random House Trade Paperbacks
NEW YORK TIMES BESTSELLER • A worthy heir to Isak Dinesen and Beryl Markham, Alexandra Fuller shares visceral memories of her childhood in Africa, and... Read more
of her headstrong, unforgettable mother. "This is not a book you read just once, but a tale of terrible beauty to get lost in over and over."—Newsweek "By turns mischievous and openhearted, earthy and soaring . . . hair-raising, horrific, and thrilling."—The New Yorker Though it is a diary of an unruly life in an often inhospitable place, Don't Let's Go to the Dogs Tonight is suffused with Fuller's endearing ability to find laughter, even when there is little to celebrate. Fuller's debut is unsentimental and unflinching but always captivating. In wry and sometimes hilarious prose, she stares down disaster and looks back with rage and love at the life of an extraordinary family in an extraordinary time. From 1972 to 1990, Alexandra Fuller—known to friends and family as Bobo—grew up on several farms in southern and central Africa. Her father joined up on the side of the white government in the Rhodesian civil war, and was often away fighting against the powerful black guerilla factions. Her mother, in turn, flung herself at their African life and its rugged farm work with the same passion and maniacal energy she brought to everything else. Though she loved her children, she was no hand-holder and had little tolerance for neediness. She nurtured her daughters in other ways: She taught them, by example, to be resilient and self-sufficient, to have strong wills and strong opinions, and to embrace life wholeheartedly, despite and because of difficult circumstances. And she instilled in Bobo, particularly, a love of reading and of storytelling that proved to be her salvation. Alexandra Fuller writes poignantly about a girl becoming a woman and a writer against a backdrop of unrest, not just in her country but in her home. But Don't Let's Go to the Dogs Tonight is more than a survivor's story. It is the story of one woman's unbreakable bond with a continent and the people who inhabit it, a portrait lovingly realized and deeply felt.Praise for Don't Let's Go to the Dogs Tonight"Riveting . . . [full of] humor and compassion."—O: The Oprah Magazine "The incredible story of an incredible childhood."—The Providence Journal
Speak Up: An Illustrated Guide to Public Speaking
Edition: Second Edition
Publisher: Bedford/St. Martin's
Critical Writing
Edition: Third Edition
Type: Plastic Comb
The World Must Know: The History of the Holocaust as Told...
Edition: Revised
Publisher: Johns Hopkins University Press


Consistently a leader in sales quality, revenue generation and customer satisfaction, as a representative and a manager. Recognized multiple times as winner or runner-up Sales Rep of the Year and Manager of the Year.
Currently lead local marketing team in multi-channel brand imaging and sales campaigns. Oversight of customer retentio initiatives at the branch level.


text:
ISL 2017: Steve Coppell hoping to repeat Chennaiyin's heroics against Bengaluru FC
Ramachandran Sriram
Bengaluru v Jamshedpur
Jamshedpur FC coach Steve Coppell is taking inspiration from Chennaiyin FC's victory against Bengaluru FC ahead of their clash against the Blues...
Jamshedpur take on Bengaluru in the battle of the newbies in Indian Super League (ISL) at the Kanteerava Stadium, Bengaluru on Thursday. Placed sixth in the ISL points table with six points from five matches, The Men of Steel will be eyeing nothing less than a win from this crucial away fixture.
Bengaluru FC fumbled for the second time this season when they suffered a 1-2 defeat at the hands of Chennaiyin FC in their last game. Jamshedpur boss Steve Coppell is taking inspiration from Chennaiyin's triumph at Blues' fortress.
Coppell said, "In a similar way to Chennaiyin I suppose. They are a good team. The coach made five or six changes in the last game I think. He got confidence in the depth and quality of his squad. Bengaluru FC probably have the deepest squad in the ISL. It's a great challenge for us. Chennaiyin came and won and I hope we can also do the same."
Jamshedpur are a great defensive unit which has conceded just one goal so far in the competition. But their attack has been equally poor. On the team's style, Coppell opined, "From day one I think we concentrated on laying a solid foundation."
"We worked hard on our defence. We are not really prolific in front of the goal but we need to improve. In training, we take every opportunity to improve in that department. I am confident things will change as the games progress."
On Anas' injury status, the Jamshedpur boss said, "Anas was our first pick in the draft. We had a thought process for selecting him. A strong Indian player in the central defence means you can play an international player up the pitch."
"So not having Anas meant we had to play an extra foreign defender which affected the firepower upfront. We are hopeful that he will recover soon and we can bring him back to the squad. His injury has definitely affected the team's performance."
Agent Ronaldo! How CR7 helped sign De Ligt for Juve
ISL 2017 Preview: Bengaluru FC v Jamshedpur FC Roca: We have to go all out from the beginning


text:
using System.IO;
using System.IO.Compression;
using System.Reflection;

namespace DotSpatial.Projections
{
    internal static class DeflateStreamReader
    {
        /// <summary>
        /// Decodedes embedded resource and returns it as Stream
        /// </summary>
        /// <param name="resourceName">Resource name</param>
        /// <returns>Stream</returns>
        public static Stream DecodeEmbeddedResource(string resourceName)
        {
            using (var s = Assembly.GetCallingAssembly().GetManifestResourceStream(resourceName))
            {
                var msUncompressed = new MemoryStream();
                using (var ds = new DeflateStream(s, CompressionMode.Decompress, true))
                {
                    var buffer = new byte[4096];
                    int numRead;
                    while ((numRead = ds.Read(buffer, 0, buffer.Length)) != 0)
                    {
                        msUncompressed.Write(buffer, 0, numRead);
                    }
                }
                msUncompressed.Seek(0, SeekOrigin.Begin);
                return msUncompressed;
            }
        }
    }
}

redpajama_set_name: RedPajamaGithub