Appendix 1: Proofs for Sect. 2
Let \(B_\mathbf{k}(\mathbf{R_k})=\bigcap _\mathbf{i\in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \) and \(\overline{B}_\mathbf{k}(\mathbf{R_k})=\bigcup _\mathbf{i\in R_k}\left\{ X_\mathbf{i}> u_\mathbf{k,i}\right\} \). For \(\mathbf{k,l\in R_n}\) such that \(\mathbf{k}\ne \mathbf{l}\) and \(u_{\mathbf{l},\mathbf{i}}\ge u_{\mathbf{k},\mathbf{i}}\), let \(m_{l_{i}}=\log l_{i}\). Note that \(k_{1}k_{2}\le l_{1}l_{2}\). Let \(\mathbf {M^{*}}=\mathbf {M^{*}}_{\mathbf {kl}}=\mathbf {R_{k}}\cap \mathbf {R_{l}}\) and \(\mathbf {M}_{\mathbf {kl}}=\{(x_{1},x_{2}): (x_{1},x_{2})\in \mathbf {N}^{2}, 0\le x_{i}\le \sharp (\prod _{i}(\mathbf {M}^{*}))+m_{l_{i}}, i=1,2\}\), where \(\sharp \) denotes cardinality. Note that \(\mathbf {M^{*}}\subset \mathbf {M_{kl}}\).
The proof of Theorem 2.1 will be given by means of several lemmas.
Lemma 4.1
Let \(\mathbf{X}\) be a nonstationary random field satisfying condition \(D^*(u_{\mathbf {n},\mathbf {i}})\) over \(\mathcal {F}\). Assume that \(\left\{ n_{1}n_{2}\max \left\{ P\left( X_{\mathbf {i}}>u_{\mathbf {n},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {n}\right\} \right\} _{\mathbf {n}\ge \mathbf {1}}\) is bounded and \(\alpha _{\mathbf {l},m_{l_1},m_{l_2}}\ll (\log l_1 \log l_2)^{-(\epsilon +1)}\). Then, for \(\mathbf{k,l\in R_n}\) such that \(\mathbf{k}\ne \mathbf{l}\) and \(u_{\mathbf{l},\mathbf{i}}\ge u_{\mathbf{k},\mathbf{i}}\),
$$\begin{aligned} \left| Cov\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }, \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \ll \alpha _{\mathbf {l,k},m_{l_1},m_{l_2}}+\frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
Proof
Write
$$\begin{aligned}&\left| Cov\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }, \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \\&\quad =\left| P(B_\mathbf{k}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf{R_l-R_k}))-P(B_\mathbf{k}(\mathbf{R_k}))P(B_\mathbf{l}(\mathbf R_l-R_k))\right| \\&\quad \le \left| P(B_\mathbf{k}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf{R_l-R_k}))-P(B_{\mathbf{k}}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf R_l-M_{kl}))\right| \\&\qquad +\left| P(B_\mathbf{k}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf{R_l-M_{kl}}))-P(B_{\mathbf{k}}(\mathbf{R_k}))P(B_\mathbf{l}(\mathbf R_l-M_{kl}))\right| \\&\qquad +\left| P(B_{\mathbf{k}}(\mathbf{R_k}))P(B_\mathbf{l}(\mathbf{R_l-M_{kl}}))-P(B_{\mathbf{k}}(\mathbf{R_k}))P(B_\mathbf{l}(\mathbf R_l-R_{k}))\right| \\&\quad =:I_1+I_2+I_{3}. \end{aligned}$$
Using the condition that \(\left\{ n_{1}n_{2}\max \left\{ P\left( X_{\mathbf {i}}>u_{\mathbf {n},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {n}\right\} \right\} _{\mathbf {n}\ge \mathbf {1}}\) is bounded we get
$$\begin{aligned} I_1= & {} \left| P(B_\mathbf{k}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf{R_l-R_k}))-P(B_\mathbf{k}(\mathbf{R_k})\cap B_{\mathbf{l}}(\mathbf R_l-M_{kl}))\right| \\\le & {} \left| P(B_\mathbf{l}(\mathbf{R_l-R_k}))-P(B_\mathbf{l}(\mathbf R_l-M_{kl}))\right| \\\le & {} P(\overline{B}_\mathbf{l}((\mathbf R_l-R_k)-(\mathbf R_l-M_{kl})))\\\le & {} P(\overline{B}_\mathbf{l}((\mathbf M_{kl}-R_{k})))\\\le & {} (m_{l_1}k_2+m_{l_2}k_1)\max \left\{ P\left( X_{\mathbf {i}}>u_{\mathbf {l},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {l}\right\} \\\ll & {} \frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
Similarly, we have
$$\begin{aligned} I_3 \ll \frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
Condition \(D^*(u_{\mathbf {n},\mathbf {i}})\) implies
$$\begin{aligned} I_2=\left| P(B_\mathbf{k}(\mathbf{R_k})\cap B_\mathbf{l}(\mathbf{R_l-M_{kl}}))-P(B_{\mathbf{k}}(\mathbf{R_k}))P(B_\mathbf{l}(\mathbf R_l-M_{kl}))\right| \le \alpha _{\mathbf {l},m_{l_1},m_{l_2}}. \end{aligned}$$
Noticing \(\alpha _{\mathbf {l,k},m_{l_1},m_{l_2}}\ll (\log l_1 \log l_2)^{-(\epsilon +1)}\), we obtain
$$\begin{aligned} \left| Cov\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }, \mathbbm {1}_{\left\{ \bigcap _\mathbf{i\in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \ll \alpha _{\mathbf {l},m_{l_1},m_{l_2}}+\frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
Lemma 4.2
Let \(\mathbf{X}\) be a nonstationary random field such that \(\big \{ n_{1}n_{2}\max \big \{ P\left( X_{\mathbf {i}}>u_{\mathbf {n},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {n}\big \} \big \} _{\mathbf {n}\ge \mathbf {1}}\) is bounded. Then, for \(\mathbf k,l\in R_n\) such that \(\mathbf k\ne \mathbf l\) and \(u_\mathbf{l, i}\ge u_\mathbf{k, i}\),
$$\begin{aligned} E\left| \mathbbm {1}_{\left\{ \cap _\mathbf{i\in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }-\mathbbm {1}_{\left\{ \cap _\mathbf{i\in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right| \le \frac{l_1l_2-\sharp (\mathbf R_l-R_k)}{l_1l_2}. \end{aligned}$$
Proof
Using the condition that \(\left\{ n_{1}n_{2}\max \left\{ P\left( X_{\mathbf {i}}>u_{\mathbf {n},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {n}\right\} \right\} _{\mathbf {n}\ge \mathbf {1}}\) is bounded we get
$$\begin{aligned}&E\left| \mathbbm {1}_{\left\{ \cap _\mathbf{i\in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }-\mathbbm {1}_{\left\{ \cap _\mathbf{i\in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right| \\&\quad =P\left( \bigcap _{\mathbf{i}\in \mathbf{R_l}-\mathbf{R_k}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right) -P\left( \bigcap _{\mathbf{i}\in \mathbf{R_l}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right) \\&\quad \le \sum _\mathbf{i\in R_l-(R_l-R_k)}P(X_\mathbf{i}>u_\mathbf{l,i})\\&\quad \le \left[ l_1l_2-\sharp (\mathbf R_l-R_k)\right] \max \left\{ P\left( X_{\mathbf {i}}>u_{\mathbf {l},\mathbf {i} }\right) :\mathbf {i}\le \mathbf {l}\right\} \\&\quad \ll \frac{l_1l_2-\sharp (\mathbf R_l-R_k)}{l_1l_2}. \end{aligned}$$
The following lemma is from Tan and Wang [16].
Lemma 4.3
Let \(\eta _\mathbf{i}\), \(\mathbf{i}\in \mathbb {Z}_+^2\), be uniformly bounded variables. Assume that
$$\begin{aligned} \mathrm{Var}\left( \frac{1}{\log n_1 \log n_2}\sum _\mathbf{k \in R_n}\frac{1}{k_1k_2}\eta _\mathbf{k}\right) \ll \frac{1}{(\log n_1 \log n_2)^{\epsilon +1}}. \end{aligned}$$
Then
$$\begin{aligned} \frac{1}{\log n_1 \log n_2}\sum _{\mathbf{k}\in \mathbf{R_n}}\frac{1}{k_1k_2}(\eta _\mathbf{k}-E(\eta _\mathbf{k}))\rightarrow 0 \ \ \hbox {a.s.} \end{aligned}$$
Proof of Theorem 2.1
Let \(\eta _\mathbf{k}=\mathbbm {1}_{\left\{ \bigcap _{\mathbf{i}\le \mathbf{k}}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }-E\left( \mathbbm {1}_{\left\{ \bigcap _{\mathbf{i}\le \mathbf{k}}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }\right) \). Then
$$\begin{aligned}&\hbox {Var}\left( \frac{1}{\log n_1 \log n_2}\sum _\mathbf{k \in R_n}\frac{1}{k_1k_2}\mathbbm {1}_{\left\{ \bigcap _{\mathbf{i}\le \mathbf{k}}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} }\right) \\&\quad =E\left( \frac{1}{\log n_1 \log n_2}\sum _\mathbf{k \in R_n}\frac{\eta _\mathbf{k}}{k_1k_2}\right) ^2\\&\quad =\frac{1}{\log ^2n_1 \log ^2n_2}\left( \sum _\mathbf{k \in R_n}\frac{E(\eta _\mathbf{k}^2)}{k_1^2k_2^2}+ \sum _{\mathbf{k,l \in R_n},\mathbf{k}\ne \mathbf{l}}\frac{E(\eta _\mathbf{k}\eta _\mathbf{l})}{k_1k_2l_1l_2}\right) \\&\quad =T_1+T_2. \end{aligned}$$
Since \(|\eta _\mathbf{k}|\le 1\), it follows that
$$\begin{aligned} T_1\le \frac{1}{\log ^2n_1 \log ^2n_2}\sum _{\mathbf{k}\in \mathbf{R_n}}\frac{1}{k_1^2 k_2^2}\le \frac{K}{\log ^2n_1 \log ^2n_2}. \end{aligned}$$
Note that for \(\mathbf{k}\ne \mathbf{l}\) such that \(u_\mathbf{k,i}<u_\mathbf{l,i}\),
$$\begin{aligned} |E(\eta _\mathbf{k}\eta _\mathbf{l})|= & {} \left| \hbox {Cov}\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} },\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \\\le & {} \left| \hbox {Cov}\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} },\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }-\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_{k}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \\&+\left| \hbox {Cov}\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} },\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_{k}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| \\\le & {} E\left| \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }-\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_{k}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right| \\&+\left| \hbox {Cov}\left( \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} },\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right) \right| . \end{aligned}$$
By Lemma 4.2 we get
$$\begin{aligned} E\left| \mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }-\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_{k}}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} }\right| \le \frac{l_1l_2-\sharp (\mathbf{R_l-R_k})}{l_1l_2} \end{aligned}$$
and from Lemma 4.1. we obtain
$$\begin{aligned} \left| Cov(\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} },\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_l-R_k}\left\{ X_\mathbf{i}\le u_\mathbf{l,i}\right\} \right\} })\right| \ll \alpha _{\mathbf {l},m_{l_1},m_{l_2}}+\frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
Hence
$$\begin{aligned} |E(\eta _\mathbf{k}\eta _\mathbf{l})|\ll \frac{l_1l_2-\sharp (\mathbf{R_l-R_k})}{l_1l_2}+\alpha _{\mathbf {l},m_{l_1},m_{l_2}}+\frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}. \end{aligned}$$
In order to consider \(T_2\), we define \(A_\mathbf{m}=\big \{\left( \mathbf{k},\mathbf{l}\right) \in \mathbf{R_n\times R_n}:(2m_j-1)(k_j-l_j)\ge 0, \mathbf{k}\ne \mathbf{l} \big \}\) for \(\mathbf{m}\in \Lambda =\left\{ (m_1,m_2):m_1,m_2\in \left\{ 0,1\right\} , \mathbf{m\ne 1}\right\} \). Then, we have
$$\begin{aligned} T_2\le & {} \frac{1}{(\log n_1 \log n_2)^2}\sum _{{\mathbf{m} \in \Lambda }}\sum _{(\mathbf{k},\mathbf{l})\in A_\mathbf{m}}\frac{l_1l_2-\sharp (\mathbf R_l-R_k)}{l_1^2l_2^2k_1k_2}\\&+\frac{1}{(\log n_1 \log n_2)^2}\sum _{{\mathbf{m} \in \Lambda }}\sum _{(\mathbf{k},\mathbf{l})\in A_\mathbf{m}}\frac{\alpha _{\mathbf {l},m_{l_1},m_{l_2}}+\frac{m_{l_1}k_{2}}{l_1l_{2}}+\frac{m_{l_2}k_{1}}{l_{1}l_2}}{k_1k_2l_1l_2}=:T_{21}+T_{22}. \end{aligned}$$
Since
$$\begin{aligned} T_{21}= & {} \frac{1}{\log ^2n_1 \log ^2n_2}\underset{\underset{1\le k_2\le l_2\le n_2, \mathbf{k\ne l}}{1\le k_1\le l_1\le n_1}}{\sum }\bigg [\frac{k_1k_2}{l_1l_2}\times \frac{1}{k_1k_2l_1l_2}+\frac{1}{k_1k_2l_1l_2} \times \frac{k_1}{l_1}+\frac{1}{k_1k_2l_1l_2}\times \frac{k_2}{l_2}\bigg ]\\\le & {} \frac{K}{\log ^2n_1 \log ^2n_2}\bigg [\prod _{i=1}^2\underset{1\le k_i\le l_i\le n_i}{\sum }\frac{1}{l_i^2}+\underset{1\le k_1< l_1\le n_1}{\sum }\frac{1}{l_1^2}\underset{1\le l_2< k_2\le n_2}{\sum }\frac{1}{k_2l_2}\\&+\underset{1\le k_2< l_2\le n_2}{\sum }\frac{1}{l_2^2}\underset{1\le l_1< k_1\le n_1}{\sum }\frac{1}{k_1l_1}\bigg ]\\\le & {} K\left( \frac{1}{\log n_1 \log n_2}+\frac{\log n_2}{\log n_1 \log n_2}+\frac{\log n_1}{\log n_1 \log n_2}\right) \end{aligned}$$
and
$$\begin{aligned} T_{22}= & {} \frac{K}{(\log n_1 \log n_2)^2}\bigg [\underset{\underset{1\le k_2\le l_2\le n_2, \mathbf{k\ne l}}{1\le k_1\le l_1\le n_1}}{\sum }\frac{1}{k_1k_2l_1l_2(\log l_1 \log l_2)^{\epsilon _1+1}}\\&+\underset{1\le k_2\le l_2\le n_2}{\sum }\frac{1}{k_2l_2(\log l_2)^{\epsilon _1}}\underset{1\le l_1\le k_1\le n_1}{\sum }\frac{1}{k_1l_1(\log l_1)^{\epsilon _1+1}}\\&+\underset{1\le k_1\le l_1\le n_1}{\sum }\frac{1}{k_1l_1(\log l_1)^{\epsilon _1}}\underset{1\le l_2\le k_2\le n_2}{\sum }\frac{1}{k_2l_2(\log l_2)^{\epsilon _1+1}}\bigg ]\\\le & {} K(\log n_1 \log n_2)^{-(\epsilon _1+1)} \end{aligned}$$
we have
$$\begin{aligned} T_2\le K\left( \frac{1}{\log n_1 \log n_2}+\frac{\log n_2}{\log n_1 \log n_2}+\frac{\log n_1}{\log n_1 \log n_2}+\frac{1}{(\log n_1 \log n_2)^{\epsilon _1+1}}\right) \end{aligned}$$
and hence
$$\begin{aligned} T_2\le K \frac{1}{(\log n_1\log n_2)^{\epsilon +1}}, \ \ {\text{ for } \text{ some }} \ \epsilon >0. \end{aligned}$$
So
$$\begin{aligned} \hbox {Var}\left( \frac{1}{\log n_1 \log n_2}\sum _\mathbf{k \in R_n}\frac{1}{k_1k_2}\mathbbm {1}_{\left\{ \bigcap _\mathbf{i \in R_k}\left\{ X_\mathbf{i}\le u_\mathbf{k,i}\right\} \right\} } \right) \le \frac{K}{(\log n_1\log n_2)^{\epsilon +1}}. \end{aligned}$$
The result follows by Lemma 4.3 and Proposition 1.2.
Appendix 2: Proofs for Sect. 3
The proof of Theorem 3.2 will be given through a technical lemma showing that (2) implies that
$$\begin{aligned} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n}}(\mathbf {R_{k}},\mathbf {R}_\mathbf{n}):= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\sum \limits _{\mathop {\mathbf {i}\in {\mathbf {R_{k}}},\mathbf {j}\in {\mathbf {R_{n}}}}\limits _{\mathbf {i}\le \mathbf {j}, \mathbf {i}\ne \mathbf {j}}}\left| r_{\mathbf {i},\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k},\mathbf {i}}^{2}+u_{\mathbf {n},\mathbf {j}}^{2}\right) }{ 1+\left| r_{{\mathbf {i},\mathbf {j}}}\right| }\right) \nonumber \\\ll & {} (\log n_1 n_2)^{-(1+\epsilon )} \end{aligned}$$
(3)
Lemma 4.4
Suppose that the covariance functions \(r_{\mathbf{i}, \mathbf{j}}\) satisfy \(\left| r_{\mathbf{i}, \mathbf{j}}\right| <\rho _{\left| \mathbf{i}-\mathbf{j}\right| }\) for some sequence \(\left\{ \rho _\mathbf{n}\right\} _{\mathbf{n}\in \mathbb {N}^2-\left\{ \mathbf 0\right\} }\) that verifies (2) for some \(\epsilon >0\). Let the constants \(\{u_{\mathbf {n,i}},\mathbf {i}\le \mathbf {n}\}_{\mathbf {n}\ge \mathbf {1}}\) be such that \(\left\{ n_{1}n_{2}(1-\Phi (\lambda _{\mathbf {n}}))\right\} _{n\,\ge \,1}\) is bounded, where \(\lambda _{\mathbf {n}}=\min _{\mathbf {i}\in \mathbf {R_{n}}}u_{\mathbf {n,i}}\). Then (3) holds.
We omit the proof, since it follows similar arguments to those of Lemmas 3.3–3.5 of Tan and Wang [16].
Proof of Theorem 3.2
We will denote the event \(\left\{ X_{\mathbf {i}}\le u_{\mathbf {n,i} }\right\} \) by \(A_{\mathbf {i,n}}\). Using the normal comparison lemma we obtain
$$\begin{aligned}&\alpha _{\mathbf {n,k},m_{n_1},m_{n_2}} \\&\quad =\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\underset{\left( \mathbf {I},\mathbf {J}\right) \in S(m_{n_{1}},m_{n_{2}})}{\sup }\left| P\left( \underset{\mathbf {i}\in \mathbf {I\wedge j} \in \mathbf {J}}{\bigcap }A_{\mathbf {i,k}}A_{\mathbf {j,n}}\right) -P\left( \underset{\mathbf {i}\in \mathbf {I}}{\bigcap }A_{\mathbf {i,k}}\right) P\left( \underset{\mathbf {j}\in \mathbf {J}}{\bigcap }A_{\mathbf {j,n}}\right) \right| \\&\quad \le \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\underset{\left( \mathbf {I},\mathbf {J}\right) \in S(m_{n_{1}},m_{n_{2}})}{\sup }\underset{\mathbf {i}\in {\mathbf {I}},\mathbf {j}\in {\mathbf {J}}}{ \sum }\left| r_{\mathbf {i},\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k},\mathbf {i}}^{2}+u_{\mathbf {n},\mathbf {j}}^{2}\right) }{ 1+\left| r_{{\mathbf {i},\mathbf {j}}}\right| }\right) \\&\quad \le \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\underset{\left( \mathbf {I},\mathbf {J}\right) \subseteq \mathbf {R_{k}}\times \mathbf {R_{n}}}{\sup }\underset{\mathbf {i}\in {\mathbf {I}},\mathbf {j}\in {\mathbf {J}}}{ \sum }\left| r_{\mathbf {i},\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k},\mathbf {i}}^{2}+u_{\mathbf {n},\mathbf {j}}^{2}\right) }{ 1+\left| r_{{\mathbf {i},\mathbf {j}}}\right| }\right) \\&\quad \le C\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\sum \limits _{\mathop {\mathbf {i}\in {\mathbf {R_{k}}},\mathbf {j}\in {\mathbf {R_{n}}}}\limits _{\mathbf {i}\le \mathbf {j}, \mathbf {i}\ne \mathbf {j}}}\left| r_{\mathbf {i},\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k},\mathbf {i}}^{2}+u_{\mathbf {n},\mathbf {j}}^{2}\right) }{ 1+\left| r_{{\mathbf {i},\mathbf {j}}}\right| }\right) \\&\quad =C\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n}}(\mathbf {R_{k}},\mathbf {R}_\mathbf{n}), \end{aligned}$$
where C is a constant. So, \(D^*(u_{\mathbf {n},\mathbf {i}})\) follows from Lemma 4.4. Next, we show condition \(D'(u_{\mathbf {n},\mathbf {i}})\) holds. To that end, let \(\mathbf I \in \mathcal {E}(u_{\mathbf {n},\mathbf {i}})\). Then, we have
$$\begin{aligned}&k_{n_{1}}k_{n_{2}}\underset{\mathbf {i,j}\in \mathbf {I}}{\sum }P(\overline{A }_{\mathbf {i,n}}\overline{A}_{\mathbf {j,n}}) \\&\quad \le k_{n_{1}}k_{n_{2}}\underset{\mathbf {i,j}\in \mathbf {I}}{\sum }\left| P(\overline{A}_{\mathbf {i,n}}\overline{A}_{\mathbf {j,n}})-P(\overline{A}_{ \mathbf {i,n}})P(\overline{A}_{\mathbf {j,n}})\right| +k_{n_{1}}k_{n_{2}} \underset{\mathbf {i,j}\in \mathbf {I}}{\sum }P(\overline{A}_{\mathbf {i,n}})P( \overline{A}_{\mathbf {j,n}}) \\&\quad \le k_{n_{1}}k_{n_{2}}S_{\mathbf {n}}(\mathbf {I},\mathbf {I})+k_{n_{1}}k_{n_{2}} \underset{\mathbf {i,j}\in \mathbf {I}}{\sum }\left( 1-\Phi (u_{\mathbf {n}, \mathbf {i}}) \right) \left( 1-\Phi (u_{\mathbf {n},\mathbf {j}}) \right) \\&\quad \le k_{n_{1}}k_{n_{2}}S_{\mathbf {n}}(\mathbf {R_{n}},\mathbf {R_{n}})+k_{n_{1}}k_{n_{2}}\left( \underset{\mathbf {i}\in \mathbf {R_{n}}}{\sum }\left( 1-\Phi (u_{\mathbf {n}, \mathbf {i}}\right) )\right) ^{2} \\&\quad \le k_{n_{1}}k_{n_{2}}S_{\mathbf {n}}(\mathbf {R_{n}},\mathbf {R_{n}})+\frac{1}{k_{n_{1}}k_{n_{2}}} \left( \underset{\mathbf {i}\le \mathbf {n}}{\sum }\left( 1-\Phi (u_{\mathbf {n}, \mathbf {i}}\right) )\right) ^{2}\xrightarrow [\mathbf {n}\rightarrow {\varvec{\infty }}]{}0, \end{aligned}$$
which completes the proof of Theorem 3.2.
We need the following facts to prove Example 3.1, which are from Choi [5]. The covariance function \(\gamma _{n}\) satisfies the following facts
$$\begin{aligned} \sum _{m=0}^{n}|\gamma _{m}|^{2}\le C n^{1-1/\log _{2}^{3}}\ \ \ \text{ and }\ \ \ \sum _{m=0}^{n}|\gamma _{m}|^{2}\ge C \frac{n^{1-1/\log _{2}^{3}}}{\log n} \end{aligned}$$
(4)
for some constants C whose value may change from place to place. From (4) and the definition of \(\gamma _{\mathbf {n}}\), it is easy to see that
$$\begin{aligned} \sum _{\mathbf {m}\in \mathbf {R_{n}}}|\gamma _{\mathbf {m}}|^{2}\le C (n_{1}n_{2})^{(1-1/\log _{2}^{3})}\ \ \ \text{ and }\ \ \ \sum _{\mathbf {m}\in \mathbf {R_{n}}}|\gamma _{\mathbf {m}}|^{2}\ge C \frac{n_{1}^{1-1/\log _{2}^{3}}}{\log n_{1}}\frac{n_{2}^{1-1/\log _{2}^{3}}}{\log n_{2}}. \end{aligned}$$
(5)
Proof of Example 3.1
We only need to show that conditions \(D'\mathbb {(}u_{\mathbf {n}}\mathbb {)}\) and \(D^*\mathbb {(}u_{\mathbf {n}}\mathbb {)}\) hold. The checking of condition \(D'\mathbb {(}u_{\mathbf {n}}\mathbb {)}\) is the same as it was given in the proof of Theorem 3.2, so we omit it. We will denote the event \(\left\{ X_{\mathbf {i}}\le u_{\mathbf {n} }\right\} \) by \(B_{\mathbf {i,n}}\). Using the normal comparison lemma, as for the proof Theorem 3.2, we obtain
$$\begin{aligned}&\alpha _{\mathbf {n},m_{n_1},m_{n_2}}\\&\quad =\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\alpha _{\mathbf {n,k},m_{n_1},m_{n_2}}\\&\quad =\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\underset{\left( \mathbf {I},\mathbf {J}\right) \in S(m_{n_{1}},m_{n_{2}})}{\sup }\left| P\left( \underset{\mathbf {i}\in \mathbf {I\wedge j}\in \mathbf {J}}{\bigcap }B_{\mathbf {i,k}}B_{\mathbf {j,n}}\right) -P\left( \underset{\mathbf {i}\in \mathbf {I}}{\bigcap }B_{\mathbf {i,k}}\right) P\left( \underset{\mathbf {j}\in \mathbf {J}}{\bigcap }B_{\mathbf {j,n}}\right) \right| \\&\quad \le C\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}\sum \limits _{\mathop {\mathbf {i}\in {\mathbf {R_{k}}},\mathbf {j}\in {\mathbf {R_{n}}}}\limits _{\mathbf {i}\le \mathbf {j}, \mathbf {i}\ne \mathbf {j}}}\left| \gamma _{\mathbf {i},\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2}\left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{{\mathbf {i},\mathbf {j}}}\right| }\right) \\&\quad \le C\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\underset{\mathbf {0}\le \mathbf {j}\le \mathbf {n}, \mathbf {j}\ne \mathbf {0}}{ \sum }\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2}\left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{{\mathbf {j}}}\right| }\right) \\&\quad =:C\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n}}^{*}(\mathbf {R_{k}},\mathbf {R_{n}}). \end{aligned}$$
Let \(\delta =\sup _{\mathbf {m}\ge \mathbf {0}, \mathbf {m}\ne \mathbf {0}}|\gamma _{\mathbf {m}}|<1\) and \(\theta _{\mathbf {n}}=\exp (\alpha u_{\mathbf {n}}^{2})\), where \(\alpha \) is a constant satisfying \(0<\alpha <(1-\delta )/4(1+\delta )\). Split the term \(S_{\mathbf {n}}^{*}(\mathbf {R_{k}},\mathbf {R_{n}})\) into two parts as:
$$\begin{aligned} S_{\mathbf {n}}^{*}(\mathbf {R_{k}},\mathbf {R_{n}})=\sum \limits _{\mathop {\mathbf {0}\le \mathbf {j}\le \mathbf {n}, \mathbf {j}\ne \mathbf {0},}\limits _{ \chi (\mathbf {|j-i|})\le \theta _{\mathbf {n}}}} +\sum \limits _{\mathop {\mathbf {0}\le \mathbf {j}\le \mathbf {n}, \mathbf {j}\ne \mathbf {0},}\limits _{\chi (\mathbf {|j-i|})> \theta _{\mathbf {n}}}}=:S_{\mathbf {n},1}^{*}+S_{\mathbf {n},2}^{*}, \end{aligned}$$
where \(\chi (\mathbf {j})=\max (j_{1},1)\times \max (j_{2},1)\). For sufficiently large \(\mathbf {n}\), we have
$$\begin{aligned} \exp \left( -\frac{u_{\mathbf {n}}^{2}}{2}\right) \thicksim C\frac{u_{\mathbf {n}}}{n_{1}n_{2}}\ \ \ \text{ and }\ \ \ u_{\mathbf {n}}\thicksim \sqrt{2\log (n_{1}n_{2})}, \end{aligned}$$
(6)
and (6) will be extensively used in the following proof. For the term \(S_{\mathbf {n},1}^{*}\), using (6), we have
$$\begin{aligned} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n},1}^{*}= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum \limits _{\mathop {\mathbf {0}\le \mathbf {j}\le \mathbf {n}, \mathbf {j}\ne \mathbf {0},}\limits _{\chi (\mathbf {j})\le \theta _{\mathbf {n}}}} \left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\\le & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum \limits _{\mathop {\mathbf {0}\le \mathbf {j}\le \mathbf {n}, \mathbf {j}\ne \mathbf {0},}\limits _{\chi (\mathbf {j})\le \theta _{\mathbf {n}}}} \delta \exp \left( -\frac{\frac{1}{2}\left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{1+\delta }\right) \\\ll & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\theta _{\mathbf {n}}^{2}\exp \left( -\frac{u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}}{2}\right) ^{1/(1+\delta )} \\\ll & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\theta _{\mathbf {n}}^{2}\left( \frac{u_{\mathbf {k}}}{k_{1}k_{2}}\frac{u_{\mathbf {n}}}{n_{1}n_{2}}\right) ^{1/(1+\delta )}\\\le & {} (n_{1}n_{2})^{1+4\alpha -2/(1+\delta )}(\log n_{1}n_{2})^{1/(1+\delta )}. \end{aligned}$$
Since \(1+4\alpha -2/(1+\delta )<0\), we get \(S_{1}\le (n_{1}n_{2})^{-\kappa }\) for some \(\kappa >0\).
We split the term \(S_{\mathbf {n},2}^{*}\) into three parts, the first for \(\mathbf {j}>\mathbf {0}\), the second for \(j_{1}=0\wedge j_{2}>0\), the third for \(j_{2}=0\wedge j_{1}>0\). We will denote them by \(\mathbf {S}^{*}_{\mathbf {n},2i}\), \(i=1,2,3,\) respectively.
To deal with the first case \(\mathbf {j}>\mathbf {0}\), let
$$\begin{aligned} \mathbf {A}_{\mathbf {n}}=\left\{ \mathbf {m}|\mathbf {1}\le \mathbf {m}\le \mathbf {n}, \chi (\mathbf {m})>\theta _{\mathbf {n}}, |\gamma _{\mathbf {m}}|>\frac{1}{(\log m_{1}m_{2})^{3}}\right\} . \end{aligned}$$
Now, we have
$$\begin{aligned} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n},21}^{*}= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {A}_{\mathbf {n}}^{c}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\&+\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {A}_{\mathbf {n}}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\=: & {} S_{1}+S_{2}. \end{aligned}$$
Since
$$\begin{aligned} \max _{\mathbf {j}\in \mathbf {A}_{\mathbf {n}}^{c}}|\gamma _{\mathbf {j}}|\le \frac{1}{(\log \theta _{\mathbf {n}})^{3}}, \end{aligned}$$
by the same arguments as for \(\mathbf {S}^{*}_{\mathbf {n},1}\), we have
$$\begin{aligned} S_{1}\le & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}n_{1}n_{2}\frac{1}{(\log \theta _{\mathbf {n}})^{3}}\exp \left( -\frac{u_{k}^{2}+u_{n}^{2}}{2(1+\frac{1}{(\log \theta _{\mathbf {n}})^{3}})}\right) \\\ll & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}n_{1}n_{2}\frac{1}{u_{\mathbf {n}}^{6}}\left( \frac{u_{\mathbf {k}}}{k_{1}k_{2}}\frac{u_{\mathbf {n}}}{n_{1}n_{2}}\right) ^{1+\frac{1}{\alpha ^{3}u_{\mathbf {n}}^{6}}}\\\ll & {} (n_{1}n_{2})^{-\frac{2}{\alpha ^{3}u_{\mathbf {n}}^{6}}}(u_{\mathbf {n}})^{-4+\frac{2}{\alpha ^{3}u_{\mathbf {n}}^{6}}}\\\ll & {} (\log n_{1}n_{2})^{-2}. \end{aligned}$$
Now we consider the term \(S_{2}\). Let \(\beta =1-1/(2\log _{2}^{3})\). Form the definition of \(\gamma _{\mathbf {m}}\), we have
$$\begin{aligned} \delta ':= & {} \sup _{\mathbf {m}\in \mathbf {A}_{\mathbf {n}}}|\gamma _{\mathbf {m}}|\le \sup _{\mathbf {m}\in \mathbf {A}_{\mathbf {n}}}\left( \frac{1}{\log m_{1}\log m_{2}}\right) ^{1/2}\nonumber \\\le & {} \sup _{\mathbf {m}\in \mathbf {A}_{\mathbf {n}}}\left( \frac{1}{\log m_{1}m_{2}}\right) ^{1/2}\le \left( \frac{1}{\log \theta _{\mathbf {n}}}\right) ^{1/2} \end{aligned}$$
As in Choi [5], we claim that \(\hbox {card}(\mathbf {A}_{\mathbf {n}})=O((n_{1}n_{2})^{\beta })\). If not, \(|\gamma _{\mathbf {m}}|>\frac{1}{(\log m_{1}m_{2})^{3}}\) on a set of size \(O((n_{1}n_{2})^{\beta })\) and thus
$$\begin{aligned} \sum _{\mathbf {m}\in \mathbf {R_{n}}}|\gamma _{\mathbf {m}}|^{2}\ge \sum _{\mathbf {m}\in \mathbf {A}_{\mathbf {n}}}|\gamma _{\mathbf {m}}|^{2}\ge C\frac{(n_{1}n_{2})^{\beta }}{(\log n_{1}n_{2})^{6}} \end{aligned}$$
contradicting (5). Hence
$$\begin{aligned} S_{2}= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {A}_{\mathbf {n}}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\\le & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}(n_{1}n_{2})^{\beta }\frac{1}{(\log \theta _{\mathbf {n}})^{1/2}}\exp \left( -\frac{u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}}{2(1+\delta ')}\right) \\\ll & {} (n_{1}n_{2})^{1+\beta -\frac{2}{1+\delta '}}(u_{\mathbf {n}})^{\frac{2}{1+\delta '}-1}\\\ll & {} (n_{1}n_{2})^{2-\frac{1}{2\log _{2}^{3}}-\frac{2}{1+\delta '}}(u_{\mathbf {n}})^{\frac{2}{1+\delta '}-1}\\\ll & {} (n_{1}n_{2})^{-\varepsilon }, \end{aligned}$$
for some \(\varepsilon >0\).
Next, we deal with the second case \(j_{1}=0\wedge j_{2}>0\). If \(n_{2}\le \theta _{\mathbf {n}}\), by the same argument as for \(\mathbf {S}^{*}_{\mathbf {n},1}\), we can show
$$\begin{aligned} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n},22}^{*}\le (n_{1}n_{2})^{-\varepsilon } \end{aligned}$$
for some \(\varepsilon >0\). If \(n_{2}>\theta _{\mathbf {n}}\), let
$$\begin{aligned} \mathbf {B}_{\mathbf {n}}=\left\{ (0,m_{2})|1\le m_{2}\le n_{2}, m_{2}>\theta _{\mathbf {n}}, |\gamma _{(0,m_{2})}|>\frac{1}{(\log m_{2})^{3}}\right\} . \end{aligned}$$
Now, we have
$$\begin{aligned} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}S_{\mathbf {n},22}^{*}= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {B}_{\mathbf {n}}^{c}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \nonumber \\&+\sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {B}_{\mathbf {n}}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2} \left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\=: & {} S_{3}+S_{4}. \end{aligned}$$
Since
$$\begin{aligned} \max _{\mathbf {j}\in \mathbf {B}_{\mathbf {n}}^{c}}|\gamma _{\mathbf {j}}|\le \frac{1}{(\log \theta _{\mathbf {n}})^{3}}, \end{aligned}$$
by the same arguments as for \(S_{1}\), we have
$$\begin{aligned} S_{3}\le & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}n_{2}\frac{1}{(\log \theta _{\mathbf {n}})^{3}}\exp \left( -\frac{u_{k}^{2}+u_{n}^{2}}{2(1+\frac{1}{(\log \theta _{\mathbf {n}})^{3}})}\right) \\< & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}n_{1}n_{2}\frac{1}{u_{\mathbf {n}}^{6}}\left( \frac{u_{\mathbf {k}}}{k_{1}k_{2}}\frac{u_{\mathbf {n}}}{n_{1}n_{2}}\right) ^{1+\frac{1}{\alpha ^{3}u_{\mathbf {n}}^{6}}}\\\ll & {} (n_{1}n_{2})^{-\frac{2}{\alpha ^{3}u_{\mathbf {n}}^{6}}}(u_{\mathbf {n}})^{-4+\frac{2}{\alpha ^{3}u_{\mathbf {n}}^{6}}}\\\ll & {} (\log n_{1}n_{2})^{-2}. \end{aligned}$$
Now we consider the term \(S_{4}\). Noting that \(\gamma _{\mathbf {m}}=\gamma _{m_{1}}\gamma _{m_{2}}\) and \(\gamma _{0}=1\), we have
$$\begin{aligned} \delta '':=\sup _{\mathbf {m}\in \mathbf {B}_{\mathbf {n}}}|\gamma _{\mathbf {m}}|\le \sup _{\mathbf {m}\in \mathbf {B}_{\mathbf {n}}}\left( \frac{1}{\log m_{2}}\right) ^{1/2}\le \left( \frac{1}{\log \theta _{\mathbf {n}}}\right) ^{1/2} \end{aligned}$$
As in Choi [5], we claim that \(\hbox {card}(\mathbf {B}_{\mathbf {n}})=O((n_{2})^{\beta })\). If not, \(|\gamma _{\mathbf {m}}|>\frac{1}{(\log m_{2})^{3}}\) on a set of size \(O((n_{2})^{\beta })\) and thus
$$\begin{aligned} \sum _{m_{2}=1}^{n_{2}}|\gamma _{m_{2}}|^{2}=\sum _{m_{2}=1}^{n_{2}}|\gamma _{(0,m_{2})}|^{2}\ge \sum _{\mathbf {m}\in \mathbf {B}_{\mathbf {n}}}|\gamma _{\mathbf {m}}|^{2}\ge C\frac{(n_{2})^{\beta }}{(\log n_{2})^{6}} \end{aligned}$$
contradicting (4). Hence
$$\begin{aligned} S_{4}= & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}\sum _{\mathbf {j}\in \mathbf {B}_{\mathbf {n}}}\left| \gamma _{\mathbf {j}}\right| \exp \left( -\frac{\frac{1}{2}\left( u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}\right) }{ 1+\left| \gamma _{\mathbf {j}}\right| }\right) \\\le & {} \sup _{\mathbf {1}\le \mathbf {k}\le \mathbf {n}}k_{1}k_{2}(n_{2})^{\beta }\frac{1}{(\log \theta _{\mathbf {n}})^{1/2}}\exp \left( -\frac{u_{\mathbf {k}}^{2}+u_{\mathbf {n}}^{2}}{2(1+\delta ')}\right) \\< & {} (n_{1}n_{2})^{1+\beta -\frac{2}{1+\delta '}}(u_{\mathbf {n}})^{\frac{2}{1+\delta '}-1}\\\ll & {} (n_{1}n_{2})^{2-\frac{1}{2\log _{2}^{3}}-\frac{2}{1+\delta '}}(u_{\mathbf {n}})^{\frac{2}{1+\delta '}-1}\\\ll & {} (n_{1}n_{2})^{-\varepsilon }, \end{aligned}$$
for some \(\varepsilon >0\). Likewise we can bound the third case \(j_{2}=0\wedge j_{1}>0\). Thus, condition \(D^*\mathbb {(}u_{\mathbf {n}}\mathbb {)}\) holds.
