Various studies in the literature have proposed invariance principles (functional central limit theorems) for partial sum processes of sequences of dependent random variables such as [111, 113, 112]. Moreover, more general dependence structures are given by [28, 29]. Following the aforementioned literature and a natural extension when considering econometric models with near-epoch dependence, is to prove the weak convergence of partial sum processes of NED sequences to the $J_1$ topology. Notice that although $X_t$  may not be mixing, if it depends almost entirely on the near epoch of $\left\{ \boldsymbol{V}_t  \right\}$ it will often have properties permitting the application of limit theorems, of which the mixingale property is the most important. Furthermore, note that near-epoch dependence is not an alternative to a mixing assumption, is a property of the mapping from $\left\{ \boldsymbol{V}_t \right\}$ to $\left\{ X_t \right\}$, not of the random variables themselves. In particular, the NED property is important when $\left\{ \boldsymbol{V}_t  \right\}$ is a mixing process, because then $\left\{ X_t \right\}$ inherits certain useful characteristics. Notice that $\mathbb{E} \left( X_t | \mathcal{F}_{t-m}^{t+m}     \right)$ is a finite-lag, then the corresponding information set is considered to be a measurable function of a mixing process and hence is also mixing. Thus, NED implies that $\left\{ X_t \right\}$ is approximately mixing in the sense of being well approximated by a mixing process. Therefore, a NED function of a mixing process can be mixingale subject to suitable restrictions on the distribution moments. From the point of view of applications, NED captures the characteristics of a stable dynamic econometric model in which a dependent variable $X_t$ depends mainly on the recent histories of a collection of explanatory variables or shock processes $\boldsymbol{V}_t$, which might be assumed to be mixing.

If $\left\{ X_j \right\}_{ j \in \mathbb{Z}}$ is a stationary $m-$ dependent Gaussian sequence with $\mathbb{E} X_0 = \mu$, then there exists $\left\{ a_k \right\}_{k=0}^m$ such that

\begin{align}
\left\{ X_j \right\}_{ j \in \mathbb{Z}} = \left\{ \mu + \sum_{k=0}^{\infty} \alpha_j \eta_{j - k }  \right\}_{ j \in \mathbb{Z}}
\end{align}

where $\left\{ \eta_{j} \right\}_{ j \in \mathbb{Z}}$ are i.i.d standard normal random variables.

Let $R(j) = \text{Cov} \left( X_0, X_j \right)$, for $j \in \mathbb{R}$. $R(j)$ is a positive define function on $\mathbb{Z}$ and $R(j) = 0$ for $|j| > m$. The Riesz factorization lemma implies that there exists $\left\{ a_k \right\}_{ k = 0}^{\infty}$ such that

\begin{align}
\sum_{ j = -  m }^m R(j) e^{ij \theta} = \bigg|  \sum_{j=0}^m a_j e^{ij \theta} \bigg|^2.
\end{align}

By expanding the square and grouping the coefficients of $e^{ij \theta}$, one sees that

\begin{align}
\label{covariance}
R(j) = \sum_{ k = 0 }^{m - j} a_k a_{ k + j}, \ \ \text{for} \ \ 0 \leq j \leq m.
\end{align}

The RHS of (<ref>) is the covariance function of the finite moving average sequence $\left\{ \sum_{k=0}^m a_k \eta_{j - k }  \right\}_{ j \in \mathbb{Z} }$, then the proposition is verified.

Although all $m-$dependent stationary Gaussian sequences are finite moving average sequences, it is next shown that there are $m-$dependent stationary infinitely divisible sequences that do not have the same distribution as any finite moving average of $\textit{i.i.d}$ infinitely divisible random variables.

§.§ Approximations of NED Processes

Understanding the underline mechanism for handling NED processes and their corresponding approximations is important especially when econometric identification is concerned with spatial equilibrium dynamics based on cointegrating relations. In other words, the existence of such phenomenan can be proved by rejecting the null hypothesis of cointegration with no NED versus cointegration under NED conditions. For example, rejecting the null hypothesis of no presence of spatial generated cointegration dynamics with NED conditions, implies that economic agents decide to cointegrate only under NED and spatial separability. Next, we follow the derivations in the framework of [100].

Assume that the stationary process $Y_t$ and $\boldsymbol{X}_t$ are $\mathbb{R}^1$ and $\mathbb{R}^p-$valued random fields, respectively and

\begin{align}
Y_t &:= \Psi_Y \left(  \varepsilon_t, \varepsilon_{t-1}, \varepsilon_{t-2},... \right)
\\
\boldsymbol{X}_t &:= \Psi_Y \left(  \varepsilon_t, \varepsilon_{t-1}, \varepsilon_{t-2},... \right),
\end{align}

where $\boldsymbol{X}_t = \left( X_{t1},..., X_{td} \right)^{\prime} \in \mathbb{R}^d$.

The idea of the approximations for an NED process as proposed by [100] is to be able to express an NED process using an $\alpha-$mixing process approximation.

In particular, the $\alpha-$mixing process approximation is given by the following expression

\begin{align}
Y_t &= Y_t^{(m)} + \left( Y_t - Y_t^{(m)} \right) := Y_t^{(m)} + \delta_{Y,t}^{(m)}
\\
\boldsymbol{X}_t &= \boldsymbol{X}_t^{(m)} + \left( \boldsymbol{X}_t - \boldsymbol{X}_t^{(m)} \right)   := \boldsymbol{X}_t^{(m)} + \delta_{X,t}^{(m)}
\end{align}

\begin{align}
\mathbb{E} \left[ \delta_{Y,t}^{(m)} \right] = \mathcal{O} \left( \nu_2 (m) \right) \ \ \ \text{and} \ \ \ \mathbb{E} \left[ \delta_{X,t}^{(m)} \right] = \mathcal{O} \left( \nu_2 (m) \right), \ \ \text{as} \ \ m \to \infty.
\end{align}

where the $\alpha-$mixing coefficient satisfies the following condition

\begin{align}
\alpha_m(k) \leq 1 \ \ \ \text{for} \ \ k = 0,1,...,m \ \ \text{and} \ \ \alpha_m(k) = \alpha (k-m) \ \ \text{for} \ \ k \geq m + 1
\end{align}

Therefore, in order to be able to estimate the local linear quantile estimator under NED when regressors are integrated we need to modify the approximation based on the I(1) reprentation of the process. Furthermore, the error term of the process is asymptotically equivalent in the case we impose the assumption of a linear process representation for the innovation sequence.

Notice that the main intuition of this step is that the near-epoch dependence can be considered as a type of $m-$dependence processes which means that even if the data around the neighborhood of the block size are dependent, asymptotically as $m \to \infty$, then an asymptotic independence assumption holds which can facilitate the development of the asymptotic theory without necessitating to consider finite-sample covariance matrix corrections when constructing test statistics.

Let $m = m_T$ be a positive integer tending to $\infty$ as $T \to \infty$. Then, it holds that

\begin{align}
\mathbb{E} \left[ Y_i^{(m)} K \left(  \frac{ \boldsymbol{x} - \boldsymbol{X}_i^{(m)}}{b_T} \right) \right]
\mathbb{E} \left[ Y_i K \left(  \frac{ \boldsymbol{x} - \boldsymbol{X}_i }{b_T} \right) \right] + \mathcal{O}_p \left( b_T^{-1} L \nu^{1/2} (m) \right)
\end{align}

which in practice means that the convergence rate is a function of the block size. Thus the larger the block size the slower the convergence rate to the true parameter space. Therefore, in order to consider the efficiency bound of the approximation which depends both on the block size as well as on the $\alpha-$mixing property we take the expected value difference between the stochastic process under near-epoch dependence and the corresponding stochastic process without the near-epoch dependence as given by the following expression elememtwise:

\begin{align}
\mathcal{I}:= \mathbb{E} \left|  \left( \boldsymbol{W}_T^{(m)} \right)_i -  \left( \boldsymbol{W}_T \right)_i \right|
\mathbb{E} \left| \left( T b_T^d \right)^{-1} \sum_{j=1}^T \left\{ Z_j^{(m)} K_i \left( \frac{ \boldsymbol{X}_j^{(m)} - \boldsymbol{x} }{ b_T } \right) - Z_j K_i \left( \frac{ \boldsymbol{X}_j - \boldsymbol{x} }{ b_T } \right) \right\} \right|
\end{align}

\begin{align*}
\mathcal{I}
\left( T b_T^d \right)^{-1} \sum_{j=1}^T \mathbb{E} \left| Z_j^{(m)} K_i \left( \frac{ \boldsymbol{X}_j^{(m)} - \boldsymbol{x} }{ b_T } \right) - Z_j K_i \left( \frac{ \boldsymbol{X}_j - \boldsymbol{x} }{ b_T } \right) \right|
\\
b_T^{-d} \mathbb{E} \left| Z_j^{(m)} - Z_j \right| K_i \left( \frac{ \boldsymbol{X}_j^{(m)} - \boldsymbol{x} }{ b_T } \right)
b_T^{-d} \mathbb{E} \bigg| Z_j \bigg| \mathbf{1}\left\{  |Z_j | \leq L \right\}
+ \mathbf{1}\left\{  |Z_j | \geq L \right\} \bigg| K_i \left( \frac{ \boldsymbol{X}^{(m)}_j - \boldsymbol{x} }{ b_T } \right) -  K_i \left( \frac{ \boldsymbol{X}_j - \boldsymbol{x} }{ b_T } \right) \bigg|
\\
\mathcal{O}_p \left( b_T^{-d} \sqrt{ \nu_2(m)} \right) +  \mathcal{O}_p \left( b_T^{- (d+1) } \sqrt{ \nu_2(m)} \right) +  \mathcal{O}_p \left( b_T^{-d} L^{ -( \delta + 1 )}  \right)
\end{align*}

In other words, we need to identify which of the three terms of the above expansion stochastically dominates the other. Then, if we find that the particular term is bounded in probability we have proved the desired result. The next step is to employ the $\alpha-$mixing approximation to the above expression. In other words, the fundamental idea to prove the asymptotic normality of $\boldsymbol{W}_T (\boldsymbol{x})$ is to divide the expression for $\boldsymbol{W}_T (\boldsymbol{x})$ into two parts such that $m = m_T \to \infty$ which implies that we can write the following expression withour worrying that the block size can affect the asymptotic approximation result.

\begin{align}
\boldsymbol{W}_T (\boldsymbol{x}) = \boldsymbol{W}_T^{(m)} (\boldsymbol{x}) + \big[ \boldsymbol{W}_T (\boldsymbol{x}) - \boldsymbol{W}_T^{(m)} (\boldsymbol{x})  \big]
\end{align}

Therefore, considering the main result of the lemma that we need to prove, we have that

\begin{align}
\left( T b_T^{d} \right)^{1/2} \mathbb{E} \big[ \boldsymbol{W}_T (\boldsymbol{x}) - \boldsymbol{W}_T^{(m)} (\boldsymbol{x}) \big] \to 0.
\end{align}

which implies that

\begin{align}
\left( T b_T^{d} \right)^{1/2} \left( \boldsymbol{c}^{\prime} \frac{ \boldsymbol{W}_T (\boldsymbol{x}) - \mathbb{E} \big[ \boldsymbol{W}_T^{(m)} (\boldsymbol{x})  \big]  }{ \sigma } \right)
\end{align}

is asymptotically standard normal as $T \to \infty$. Recalling that

\begin{align}
\eta_j^{(m)} ( \boldsymbol{x} ) := Z_j^{(m)} K_c \left( \boldsymbol{x} - \boldsymbol{X}_j^{(m)} \right) \ \ \ \text{and} \ \ \  \Delta_j^{(m)} ( \boldsymbol{x} )  := \eta_j^{(m)} ( \boldsymbol{x} )  - \mathbb{E} \left[ \eta_j^{(m)} ( \boldsymbol{x} ) \right]
\end{align}

Using the Bahadur representation for the  conditional quantile functional form as in [126], the proof is similar to the arguments for the conditional mean functional form case as in [100]. Suppose that

\begin{align}
W_n :=
\begin{bmatrix}
\\
\end{bmatrix},
\ \ \ ( W_n )_j :=
\left( n h_n^p  \right)^{-1} \sum_{i=1}^n \psi_{\uptau} \left(  Y_i^* \right) \left( \frac{X_i - x}{ h_n } \right)_j K \left( \frac{X_i - x}{ h_n } \right), \ \ \ \text{for} \ \ j = 0,...,p,
\end{align}

Furthermore, we employ the Cramer-Wold device. For all $c := \left( c_0, c_1^{\prime} \right)^{\prime} \in \mathbb{R}^{ p+1}$, let

\begin{align}
A_n := \left( n h_n^p  \right)^{1/2} c^{\prime} W_n = \phi_{\uptau} (x) \frac{1}{ \sqrt{n h_n^p } } \sum_{i=1}^n \psi_{\uptau} \left(  Y_i^* \right) K \left( \frac{X_i - x}{ h_n } \right)
\end{align}

A second-order Taylor expansion of $g(u)$ about x gives

\begin{align}
\mathbb{E} \left[ Y_{in} | X_{in} = u \right] = g(x) + g^{\prime}(x)^{\top} (u-x) + \frac{1}{2} ( u- x )^{\prime} g^{\prime \prime} \big( x + \theta (u-x) \big) (u-x)
\end{align}

for some scalar $| \theta | < 1$, where $u$ is an open neighborhood of $x$.

Therefore, using the properties of the kernel $K_c(.)$ we get the following expression:

\begin{align*}
\mathbb{E} &\left[ \eta_j - \eta_j^{(m)} \right]^2
\leq \mathbb{E} \left[ v_j - v_j^{(m)} \right]^2
\\
\mathbb{E} \left\{ \left[ \psi_{\uptau} (Y_i^{*}) - \psi_{\uptau}(Y_j^{*(m)}) \right] K_n \left(  \frac{ X_j^{(m)} - x }{ h_n } \right)  + \psi_{\uptau}(Y_j^{*}) \left[ K_n \left( \frac{ X_j - x }{ h_n } \right) - K_n \left(  \frac{ X_j^{(m)} - x }{ h_n } \right) \right]   \right\}^2
\\
2 \left\{  \mathbb{E} \left[ \psi_{\uptau} (Y_i^{*}) - \psi_{\uptau}(Y_j^{*(m)}) \right]^2 K_n^2 \left(  \frac{ X_j^{(m)} - x }{ h_n } \right)  + \mathbb{E}\ \left[ \psi_{\uptau}(Y_j^{*}) \right] \left[ K_n \left( \frac{ X_j - x }{ h_n } \right) - K_n \left(  \frac{ X_j^{(m)} - x }{ h_n } \right) \right]^2 \right\}
\end{align*}

Therefore, it can be proved that

\begin{align}
\nu_{n23} := ( n h_n^p )^{-1} \sum_{ 1 \leq i \leq j \leq n  } \mathbb{E} \left[ \eta_j - \eta_j^{(m)} \right]^2 \to 0.
\end{align}

Next we focus on the prove of Theorem 3.2 in [126] as given below.

We only give the proof of the main argument used in the proof of Theorem 3.2 as below:

\begin{align*}
\widehat{q}_{\uptau}(x)  - q_{\uptau}(x)
\left( n h_n^p \right)^{-1} \phi_{\uptau} \sum_{i=1}^n \psi_{\uptau} (Y_i^{*}) K_i + o_p \left( \frac{1}{H_n} \right)
\\
\left( n h_n^p \right)^{-1} \phi_{\uptau} \sum_{i=1}^n  \big\{ \psi_{\uptau}(Y_i^{*}) - \mathbb{E}\left[ \psi_{\uptau}(Y_i^{*}) \right] K_i \big\} + \left( n h_n^p \right)^{-1} \phi_{\uptau} \sum_{i=1}^n \mathbb{E}\left[ \psi_{\uptau}(Y_i^{*}) \right] K_i + o_p \left( \frac{1}{H_n} \right).
\end{align*}

Furthermore, we can that $c = ( 1, 0)$ therefore we obtain the following analytical expression

\begin{align}
\mathbb{E} \left[ Q_{n1}^2 \right] = \left( n h_n^p \right)^{-1} \psi_{\uptau}^2 \mathbb{E} \big[ c^{\prime} V_n(0) - c^{\prime} \mathbb{E} V_n (0) \big]^2 = \mathcal{O}_p \left(  \left( nh_n^p \right)^{-1} \right) = \mathcal{O}_p \left( H_n^{-2} \right).
\end{align}

In other words, the key step of this proof is to consider the local Bahadur representation.

Define with

\begin{align}
\eta_i^{(m)}(x)
&:= \psi_{\uptau} \left( Y_i^{*(m)} \right) K_c \left( \frac{ X_i^{(m)} - x}{h_n} \right)
\\
\zeta_{ni}^{(m)}(x)&:= h_n^{-p/2} \left( \eta_i^{(m)}(x) - \mathbb{E} \left[ \eta_i^{(m)}(x) \right] \right),
\end{align}

and let $S_n^{(m)} := \sum_{i=1}^n \zeta_{ni}^{(m)}$. Then, it holds that

\begin{align}
n^{-1/2} S_n^{(m)} = \left( n h_n^p \right)^{1/2} c^{\prime} \left( W_n^{(m)}(x) - \mathbb{E} \left[ W_n^{(m)}(x) \right] \right) = A_n^{(m)} - \mathbb{E} \left[ A_n^{(m)} \right]
\end{align}

Therefore, by decomposing $n^{-1/2} S_n^{(m)}$ into smaller pieces involving "large" and "small" blocks. More specifically, consider

\begin{align}
U^{(m)} (1,n,x,k) := \sum_{j = k(p^{*} + q^{*}) + p^{*} + 1}^{(k+1)(p^{*} + q^{*})} \zeta_{nj}^{(m)}(x)
\end{align}

The general approach consists in showing that, as $n \to \infty$,

\begin{align}
Q_1^{(m)} &:= \left| \mathbb{E} \mathsf{exp} \left[ i u Y^{(m)}(n,x,1) \right] - \prod_{j=0}^{r-1} \mathbb{E} \left[ \mathsf{exp} \left\{ i u Y^{(m)}(n,x,1) \right\} \right] \right| \to 0
\\
Q_2^{(m)} &:= n^{-1} \mathbb{E} \left[ Y^{(m)} (n,x,2) \right]^2 \to 0,
\end{align}

Assume that the probability space $\left( \Omega_n, \mathcal{F}_n, P_n \right)$ supports square integrable random variables $S_{n,1}, S_{n,2},..., S_{n,k_n}$, and that $S_{n,t}$ are adapted to $\sigma-$algebras $\mathcal{F}_{m,t}, 1 \leq t \leq k_n$, where

\begin{align}
\mathcal{F}_{n,1} \subset \mathcal{F}_{n,2} \subset ... \subset \mathcal{F}_{n,k_n} \subset \mathcal{F}_n.
\end{align}

Let $X_{n,t} = S_{n,t} - S_{n,t-1}, S_{n,0} = 0$ and $U_{n,t}^2 \sum_{s=1}^t X_{n,s}^2$.

A family of random variables $\left\{ X_i, i \in I \right\}$ is said to be uniformly integrable if the following holds $\displaystyle
\underset{ a \to \infty }{ \mathsf{lim} } \ \underset{ i \in I }{ \mathsf{sup} } \int_{| X_i| \leq a } | X_i | dP = 0$.

Then, it suffices to show that for all $\delta > 0$, $\sum_{t=2}^T \mathbb{E} \left[ Y_{Tt}^2 \mathbf{1} \left\{ | Y_{Tt} | > \delta \right\} | \Omega_{T,t-1} \right]  \overset{p}{\to} 0$,such that

\begin{align*}
\sum_{t=2}^T \mathbb{E} \left[ Y_{Tt} | \mathcal{F}_{T,t-1} \right]
\sum_{t=2}^{P(T)} Y_{Tt} + \sum_{t=P(T) + 1 }^T \mathbb{E} \left[ Y_{Tt} | \mathcal{F}_{T,t-1} \right]
\sum_{t=P(T) + 1 }^T Y_{Tt} \overset{p}{\to} 0,
\end{align*}

Therefore, it can be proved that

\begin{align}
\underset{ \delta \to 0 }{ \mathsf{lim} }  \underset{ T \to \infty } \ { \mathsf{lim \ sup} } \mathbb{P} \ \left(  \frac{ \tilde{\sigma}_T }{ \sigma_T } > \delta \right) = 1.
\end{align}

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