For the partitions of category **II**, we bound the cumulant of size two by a constant (using Lemma **D.1**) and using Lemma **D.2**, we obtain

$$
\begin{align*}
\kappa_{\mathbf{II}} \le K \sup_{J_s} \frac{1}{|J_1||J_2|} \sum_{i_1 \in J_1, i_2 \in J_2} & \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_6 \|r(J_2)^{-\frac{d}{2}} \psi_{i_2}(\frac{\cdot}{r(J_2)})\|_6 \\
& \frac{1}{|J_3||J_4|} \sum_{i_3 \in J_3, i_4 \in J_4} \|r(J_3)^{-\frac{d}{2}} \psi_{i_3}(\frac{\cdot}{r(J_3)})\|_6^2 \|r(J_4)^{-\frac{d}{2}} \psi_{i_4}(\frac{\cdot}{r(J_4)})\|_6^2,
\end{align*}
$$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, $r(I) = r$ and $r(I_p) = r_p$. Note that $J_1$ and $J_2$ are determined by the block of size 2 of the partitions. Using Lemma **F.3** and assumption (27), we get after computations

$$
\kappa_{\mathbf{II}} \le K \sup_{J_s} \prod_{s \in \{1,2\}} \frac{\log(|W|)^{d/6}}{r(J_s)^{d/3} |J_s|^{1/9}} \prod_{s \in \{3,4\}} \frac{\log(|W|)^{d/3}}{r(J_s)^{2d/3} |J_s|^{2/9}} \le \frac{K \log(|W|)^d}{|W|^{5/3}}.
$$

For the partitions of category **III**, the block of size 3 contains one index $i_s$, $s \in [4]$ appearing twice. Accordingly, using Lemma **D.2**, we bound them as

$$
\begin{align*}
\kappa_{\mathbf{III}} \le K \sup_{J_s} \frac{1}{|J_1|} \sum_{i_1 \in J_1} \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_3 \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_5 \frac{1}{|J_2|} \sum_{i_2 \in J_2} \|r(J_2)^{-\frac{d}{2}} \psi_{i_2}(\frac{\cdot}{r(J_2)})\|_3^2 \\
\frac{1}{|J_3||J_4|} \sum_{i_3 \in J_3, i_4 \in J_4} \|r(J_3)^{-\frac{d}{2}} \psi_{i_3}(\frac{\cdot}{r(J_3)})\|_5^2 \|r(J_4)^{-\frac{d}{2}} \psi_{i_4}(\frac{\cdot}{r(J_4)})\|_5^2,
\end{align*}
$$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, $r(I) = r$ and $r(I_p) = r_p$. By Cauchy-Schwarz, we have

$$
\begin{align*}
& \frac{1}{|J_1|} \sum_{i_1 \in J_1} \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_3 \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_5 \\
& \le \sqrt{\frac{1}{|J_1|} \sum_{i_1 \in J_1} \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_3^2} \sqrt{\frac{1}{|J_1|} \sum_{i_1 \in J_1} \|r(J_1)^{-\frac{d}{2}} \psi_{i_1}(\frac{\cdot}{r(J_1)})\|_5^2}.
\end{align*}
$$

Using Lemma **F.3** and assumption (27), we get after computations

$$
\kappa_{\mathbf{III}} \le K \sup_{J_s} \frac{1}{r(J_1)^{\frac{d}{6}} |J_1|^{\frac{1}{12}}} \left( \frac{\log(|W|)^{2d/5}}{r(J_1)^{\frac{3d}{5}} |J_1|^{\frac{1}{5}}} \right)^{\frac{1}{2}} \frac{1}{r(J_2)^{\frac{d}{3}} |J_1|^{\frac{1}{6}}} \prod_{s \in \{3,4\}} \frac{\log(|W|)^{2d/5}}{r(J_s)^{\frac{3d}{5}} |J_s|^{\frac{1}{5}}} \le \frac{K \log(|W|)^d}{|W|^{\frac{7}{4}}}.
$$

For the partitions of category **IV**, the block of size 3 does not contain indexes $i_s$, $s \in [4]$ appearing twice. Accordingly, using Lemma **D.2**, we bound them as

$$
\begin{align*}
\kappa_{\mathbf{IV}} \le K \sup_{J_s} \prod_{s \in [3]} \frac{1}{|J_s|} \sum_{i_s \in J_s} \|r(J_s)^{-\frac{d}{2}} \psi_{i_s}(\frac{\cdot}{r(J_s)})\|_3 \|r(J_s)^{-\frac{d}{2}} \psi_{i_s}(\frac{\cdot}{r(J_s)})\|_5 \\
\frac{1}{|J_4|} \sum_{i_4 \in J_4} \|r(J_4)^{-\frac{d}{2}} \psi_{i_4}(\frac{\cdot}{r(J_4)})\|_5^2,
\end{align*}
$$
where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, $r(I) = r$ and $r(I_p) = r_p$. Using first the Cauchy-Schwarz inequality, then Lemma F.3 and finally assumption (27), we get after computations

$$ \kappa_{\text{III}} \leq K \sup_{J_s} \frac{\log(|W|)^{2d/5}}{r(J_4)^{\frac{3d}{5}} |J_4|^{\frac{1}{5}}} \prod_{s \in [3]} \left( \frac{1}{r(J_s)^{\frac{d}{3}} |J_s|^{\frac{1}{6}}} \frac{\log(|W|)^{2d/5}}{r(J_s)^{\frac{3d}{5}} |J_s|^{\frac{1}{5}}} \right)^{\frac{1}{2}} \leq \frac{K \log(|W|)^d}{|W|^{\frac{37}{20}}}. $$

For the partitions of category **V** or **VI**, all the indexes belong to a block of size 4. Accordingly, using Lemma D.2, we bound them as

$$ \kappa_{\mathbf{V}} + \kappa_{\mathbf{VI}} \leq K \sup_{J_s} \prod_{s \in [4]} \frac{1}{|J_s|} \sum_{i_s \in J_s} \|r(J_s)^{-\frac{d}{2}} \psi_{i_s} (\frac{\cdot}{r(J_s)})\|_4^2, $$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, $r(I) = r$ and $r(I_p) = r_p$. Using Lemma F.3 and assumption (27), we get after computations

$$ \kappa_{\mathbf{V}} + \kappa_{\mathbf{VI}} \leq K \sup_{J_s} \prod_{s \in [4]} \frac{\log(|W|)^{\frac{d}{2}}}{r(J_s)^{\frac{d}{2}} |J_s|^{\frac{1}{4}}} \leq K \frac{\log(|W|)^{2d}}{|W|^2}. $$

Gathering the previous bounds, we obtain

$$ \frac{1}{N^2} \sum_{1 \leq j_1, j_2 \leq N} \kappa_{\mathbf{I}} + \dots + \kappa_{\mathbf{VI}} \leq \frac{K \log(|W|)^d}{|W|^{\frac{5}{3}}}. $$

For the other categories of partitions, we rely on decorrelation between distinct frequencies. We start with the partitions of category **VII**. They differ from the ones of category **VIII** by the fact that the block of size 4 contains a block of 12|34|56|78, i.e. that one index $i_s$, $s \in [4]$ appears twice in this block. To fix the idea, the full list of such partitions is given by

1235|47|68, 1235|67|48, 1236|47|58, 1236|57|48, 1237|45|68, 1237|46|58, 1238|45|67
1238|46|57, 1245|37|68, 1245|67|38, 1246|37|58, 1246|57|38, 1247|35|68, 1247|36|58
1248|35|67, 1248|36|57, 1257|36|48, 1257|46|38, 1258|36|47, 1258|37|46, 1267|35|48
1267|45|38, 1268|35|47, 1268|37|45, 1345|27|68, 1345|67|28, 1346|27|58, 1346|57|28
1347|25|68, 1347|26|58, 1348|25|67, 1348|26|57, 1356|27|48, 1356|47|28, 1378|25|46
1378|26|45, 1456|27|38, 1456|37|28, 1478|25|36, 1478|26|35, 1567|23|48, 1567|24|38
1568|23|47, 1568|24|37, 1578|23|46, 1578|24|36, 1678|23|45, 1678|24|35, 2345|17|68
2345|67|18, 2346|17|58, 2346|57|18, 2347|15|68, 2347|16|58, 2348|15|67, 2348|16|57
2356|17|48, 2356|47|18, 2378|15|46, 2378|16|45, 2456|17|38, 2456|37|18, 2478|15|36
2478|16|35, 2567|13|48, 2567|14|38, 2568|13|47, 2568|14|37, 2578|13|46, 2578|14|36
2678|13|45, 2678|14|35, 3457|16|28, 3457|26|18, 3458|16|27, 3458|17|26, 3467|15|28
3467|25|18, 3468|15|27, 3468|17|25, 3567|14|28, 3567|24|18, 3568|14|27, 3568|17|24
3578|14|26, 3578|16|24, 3678|14|25, 3678|15|24, 4567|13|28, 4567|23|18, 4568|13|27
4568|17|23, 4578|13|26, 4578|16|23, 4678|13|25, 4678|15|23
Note that the cumulants related to these partitions can be written

$$ q_{\text{VII}} := \sum_{\substack{i_s \in J_s \\ s \in [4]}} \kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}, D_{i_3}) \kappa(\overline{D_{i_2}}, D_{i_4}) \kappa(\overline{D_{i_3}}, \overline{D_{i_4}}), $$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, if $i \in I$, then $D_i = C_i^1(k)$ for a proper frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and $i \in I_p$, then $D_i = C_i^2(k)$ for a proper frequency $k \in \{\pm \tilde{k}_{j_1}, \pm \tilde{k}_{j_2}\}$. Applying three times the Cauchy-Schwarz inequality, we get

$$ \begin{aligned} |q_{\text{VII}}| & \le \sum_{\substack{i_s \in J_s \\ s \in [3]}} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}, D_{i_3})| \sqrt{\sum_{i_4 \in J_4} |\kappa(\overline{D_{i_2}}, D_{i_4})|^2} \sqrt{\sum_{i_4 \in J_4} |\kappa(\overline{D_{i_3}}, \overline{D_{i_4}})|^2} \\ & \le \sum_{i_1 \in J_1} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_3 \in J_3}} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}, D_{i_3})|^2} \sqrt{\sum_{\substack{i_2 \in J_3 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_2}}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_3}}, \overline{D_{i_4}})|^2}. \end{aligned} $$

Using Lemmas D.6 and F.3, we obtain:

$$ \begin{aligned} \sum_{i_1 \in J_1} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_3 \in J_3}} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}, D_{i_3})|^2} & \le \sum_{i_1 \in J_1} \|r(J_1)\psi_{i_1}(\frac{\cdot}{r(J_1)})\|_6^2 \sqrt{\sum_{i_s \in J_s} \|r(J_s)\psi_{i_s}(\frac{\cdot}{r(J_s)})\|_6^2} + \mathcal{E} \\ & \le K \left( |J_1|(|J_2| \wedge |J_3|)^{\frac{1}{2}} |J_1|^{\frac{1}{9}} (|J_2| \wedge |J_3|)^{\frac{1}{18}} \frac{\log(|W|)^d}{|W|} + \mathcal{E} \right), \end{aligned} $$

where in the second line $s=2$ if $|J_3| \ge |J_2|$ and $s=3$ otherwise. The next step is to note that, due to the connectivity constraint, in all the partitions of category **VII**, at most one block of size 2 contains frequencies $\epsilon_1 k_j$ and $\epsilon_2 \tilde{k}_j$ for $(\epsilon_1, \epsilon_2) \in \{-1, 1\}^2$ and $j \in \{j_1, j_2\}$. The other block of size 2 contains frequencies associated to $j_1$ and $j_2$. With the $k_0$-allowed assumption and Lemma D.3, we get

$$ \begin{aligned} & \frac{1}{N^2} \sum_{1 \le j_1, j_2 \le N} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_2}}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_3}}, \overline{D_{i_4}})|^2} \\ & \le K(|J_2| \wedge |J_4|)^{\frac{1}{2}} \left( \left( \frac{(|J_2| \vee |J_4|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}}|k_0|} \right)^{\zeta} + \frac{1}{N} \right) (|J_3| \wedge |J_4|)^{\frac{1}{2}} \\ & \qquad + K(|J_3| \wedge |J_4|)^{\frac{1}{2}} \left( \left( \frac{(|J_3| \vee |J_4|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}}|k_0|} \right)^{\zeta} + \frac{1}{N} \right) (|J_2| \wedge |J_4|)^{\frac{1}{2}} + \mathcal{E}. \end{aligned} $$

This gives after computations (where we use $ab = (a \vee b)(a \wedge b)$):

$$ \begin{aligned} \frac{1}{N^2} \sum_{1 \le j_1, j_2 \le N} |\kappa_{\text{VII}}| & \le \frac{K}{|I|^{\frac{8}{9}} |I_p|^{\frac{8}{9}}} \frac{\log(|W|)^d}{|W|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}}|k_0|} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E} \\ & \le \frac{K}{|I||I_p|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}}|k_0|} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E}. \end{aligned} $$
Now, we consider the partitions of category **VIII**. They are given by

1357|24|68, 1357|26|48, 1357|46|28, 1358|24|67, 1358|26|47, 1358|27|46, 1367|24|58
1367|25|48, 1367|45|28, 1368|24|57, 1368|25|47, 1368|27|45, 1457|23|68, 1457|26|38
1457|36|28, 1458|23|67, 1458|26|37, 1458|27|36, 1467|23|58, 1467|25|38, 1467|35|28
1468|23|57, 1468|25|37, 1468|27|35, 2357|14|68, 2357|16|48, 2357|46|18, 2358|14|67
2358|16|47, 2358|17|46, 2367|14|58, 2367|15|48, 2367|45|18, 2368|14|57, 2368|15|47
2368|17|45, 2457|13|68, 2457|16|38, 2457|36|18, 2458|13|67, 2458|16|37, 2458|17|36
2467|13|58, 2467|15|38, 2467|35|18, 2468|13|57, 2468|15|37, 2468|17|35

We follow the same idea than in the study of the partitions of category **VII**, but this times the cumulants appearing are of the form

$$p := \sum_{\substack{i_s \in J_s; \\ s \in [4]}} \kappa(D_{i_1}, \overline{D_{i_2}}, D_{i_3}, D_{i_4}) \kappa(\overline{D_{i_1}}, D_{i_2}) \kappa(\overline{D_{i_3}}, \overline{D_{i_4}}),$$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, if $i \in I$, then $D_i = C_i^1(k)$ for a proper frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and $i \in I_p$, then $D_i = C_i^2(k)$ for a proper frequency $k \in \{\pm \tilde{k}_{j_1}, \pm \tilde{k}_{j_2}\}$. Using the Cauchy-Schwarz inequality, we obtain

$$|p| \le \sqrt{\sum_{\substack{i_s \in J_s; \\ s \in [4]}} |\kappa(D_{i_1}, \overline{D_{i_2}}, D_{i_3}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_2 \in J_2}} |\kappa(\overline{D_{i_1}}, D_{i_2})|^2} \sqrt{\sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_3}}, \overline{D_{i_4}})|^2}.$$

Then, using first Lemma D.6 (with one $J_s$ corresponding to $I$), then Lemma F.3 and finally assumption (27) we get

$$\sqrt{\sum_{\substack{i_s \in J_s; \\ s \in [4]}} |\kappa(D_{i_1}, \overline{D_{i_2}}, D_{i_3}, D_{i_4})|^2} \le K |I_p|^{1+\frac{1}{9}} |I|^{\frac{1}{2}+\frac{1}{18}} \frac{\log(|W|)^{d/2}}{|W|} + \mathcal{E}.$$

As for the partitions of category **VII**, at most one block of size 2 contains frequencies $\epsilon_1 k_j$ and $\epsilon_2 \tilde{k}_j$ for $(\epsilon_1, \epsilon_2) \in \{-1, 1\}^2$ and $j \in \{j_1, j_2\}$. The other block of size 2 contains frequencies associated to $j_1$ and $j_2$. Accordingly, we apply the arguments used for the category **VII**, to get

$$\frac{1}{N^2} \sum_{1 \le j_1, j_2 \le N} |\kappa_{\text{VIII}}| \le \frac{K}{|I||I_p|} \left( \left( \frac{(|I| \lor |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E}.$$

The partitions of category **IX**, **X** and **XI**, are classed depending on the number of blocks of size 3 containing a block of 12|34|56|78 (one for **IX**, two for **X** and zero for **XI**). Partitions in category **X** are the most constrained, since each of the two blocks of size 3 contains an index $i_s$, for some $s \in [4]$, that appears twice. We first bound their contribution. The proofs for the other categories
follow similar lines and can be readily adapted.

123|567|48, 123|568|47, 123|578|46, 123|678|45, 124|567|38, 124|568|37, 124|578|36
124|678|35, 125|347|68, 125|348|67, 125|378|46, 125|478|36, 126|347|58, 126|348|57
126|378|45, 126|478|35, 127|345|68, 127|346|58, 127|356|48, 127|456|38, 128|345|67
128|346|57, 128|356|47, 128|456|37, 134|567|28, 134|568|27, 134|578|26, 134|678|25
156|347|28, 156|348|27, 156|378|24, 156|478|23, 178|345|26, 178|346|25, 178|356|24
178|456|23, 234|567|18, 234|568|17, 234|578|16, 234|678|15, 256|347|18, 256|348|17
256|378|14, 256|478|13, 278|345|16, 278|346|15, 278|356|14, 278|456|13

The cumulants related the partitions of category **X** can be written, leveraging the connectivity condition:

$$ q_{\mathbf{X}} := \sum_{\substack{i_s \in J_s \\ s \in [4]}} \kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}) \kappa(D_{i_3}, \overline{D_{i_3}}, D_{i_4}) \kappa(\overline{D_{i_2}}, \overline{D_{i_4}}), $$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$, and exactly two of the $J_s$ are equal to $I$. Moreover, if $i \in I$, then $D_i = C_i^1(k)$ for an appropriate frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and if $i \in I_p$, then $D_i = C_i^2(k)$ for a suitable frequency $k \in \{\pm \tilde{k}_{j_1}, \pm \tilde{k}_{j_2}\}$. Applying the Cauchy-Schwarz inequality twice, we obtain:

$$ \begin{aligned} |q_{\mathbf{X}}| & \le \sum_{\substack{i_s \in J_s \\ s \in \{1,3,4\}}} \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2})|^2} |\kappa(D_{i_3}, \overline{D_{i_3}}, D_{i_4})| \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_2}, D_{i_4})|^2} \\ & \le \sum_{i_1 \in J_1} \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2})|^2} \sum_{i_3 \in J_3} \sqrt{\sum_{i_4 \in J_4} |\kappa(D_{i_3}, \overline{D_{i_3}}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_4 \in J_4}} |\kappa(D_{i_2}, D_{i_4})|^2}. \end{aligned} $$

Using Lemmas D.6 and F.3, we obtain

$$ |q_{\mathbf{X}}| \le K |J_1||J_3| \frac{\log(|W|)^d}{|W|} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_4 \in J_4}} |\kappa(D_{i_2}, D_{i_4})|^2} + \mathcal{E}. $$

Two cases may arise. In the first, the cumulant $\kappa(D_{i_2}, D_{i_4})$ contains frequencies associated with the same index $j_1$ (or $j_2$). In the second, it contains frequencies associated with distinct indices $j_1$ and $j_2$. We begin with the first case. Suppose that $|J_2|^{1/(2d)}/r(J_2) \vee |J_4|^{1/(2d)}/r(J_4) \le \sqrt{2}c_2|k_0|$. By applying Lemma F.1, we obtain, for some constant $c_6 > 0$,

$$ \frac{1}{|I_p|^2|I|^2} \sum_{\substack{(i_1, i_3) \in I, \\ (i_2, i_4) \in I_p}} |q_{\mathbf{X}}| \le K e^{-c_6|k_0||W|^{1/2}} + \mathcal{E} \le \frac{K}{|k_0|^{2d}|W|^2} + \mathcal{E}. $$

Now assume that $|J_2|^{1/2d}/r(J_2) \vee |J_4|^{1/2d}/r(J_4) \ge \sqrt{2}c_2|k_0|$. Subsequently, with assumption (27), we obtain $|J_2| \wedge |J_4| \ge c_7|k_0|^d|W|$, for $c_7 > 0$. Moreover, according to Lemma D.6, we have

$$ \frac{1}{|I_p|^2|I|^2} \sum_{\substack{(i_1, i_3) \in I, \\ (i_2, i_4) \in I_p}} |q_{\mathbf{X}}| \le K \frac{\log(|W|)^d}{|W|} \frac{(|J_2| \wedge |J_4|)^{1/2}}{|J_2||J_4|} + \mathcal{E} \le K \frac{\log(|W|)^d}{|W|^2|k_0|^d} \frac{1}{(|I| \wedge |I_p|)^{1/2}} + \mathcal{E}. $$
Now we consider the second case, where the block of size two contains frequencies associated to distinct indices $j_1$ and $j_2$. With the $k_0$-allowed assumption and Lemma F.1, we obtain

$$ \frac{1}{N} \sum_{1 \le j_1, j_2 \le N} \frac{1}{|I_p|^2 |I|^2} \sum_{\substack{(i_1, i_3) \in I, \\ (i_2, i_4) \in I_p}} |q_x| \le K \frac{\log(|W|)^d}{|W||J_2||J_4|} \left( \left( \frac{(|J_2| \vee |J_4|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E}. $$

Since $|J_2| \vee |J_4| \le |I| \vee |I_p|$, we have

$$ \frac{1}{|J_2||J_4|} \left( \frac{(|J_2| \vee |J_4|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} \le \frac{1}{(|I| \wedge |I_p|)^2} \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta}. $$

This yields, using $|I| \vee |I_p| \le |W|^{2\beta/(2\beta+d)}$:

$$ \frac{1}{N} \sum_{1 \le j_1, j_2 \le N} |\kappa_x| \le K \left( \frac{\log(|W|)^d}{|W|^2 |k_0|^d} + \frac{1}{|k_0|^{2d} |W|^2} + \frac{1}{|I||I_p|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} + \frac{1}{N} \right) \right) + \mathcal{E}. $$

The partitions of category **IX** are given by

123|457|68, 123|458|67, 123|467|58, 123|468|57, 124|357|68, 124|358|67, 124|367|58
124|368|57, 125|367|48, 125|368|47, 125|467|38, 125|468|37, 126|357|48, 126|358|47
126|457|38, 126|458|37, 127|358|46, 127|368|45, 127|458|36, 127|468|35, 128|357|46
128|367|45, 128|457|36, 128|467|35, 134|257|68, 134|258|67, 134|267|58, 134|268|57
135|278|46, 135|478|26, 135|678|24, 136|278|45, 136|478|25, 136|578|24, 137|256|48
137|456|28, 137|568|24, 138|256|47, 138|456|27, 138|567|24, 145|278|36, 145|378|26
145|678|23, 146|278|35, 146|378|25, 146|578|23, 147|256|38, 147|356|28, 147|568|23
148|256|37, 148|356|27, 148|567|23, 156|237|48, 156|238|47, 156|247|38, 156|248|37
157|234|68, 157|346|28, 157|348|26, 158|234|67, 158|346|27, 158|347|26, 167|234|58
167|345|28, 167|348|25, 168|234|57, 168|345|27, 168|347|25, 178|235|46, 178|236|45
178|245|36, 178|246|35, 235|478|16, 235|678|14, 236|478|15, 236|578|14, 237|456|18
237|568|14, 238|456|17, 238|567|14, 245|378|16, 245|678|13, 246|378|15, 246|578|13
247|356|18, 247|568|13, 248|356|17, 248|567|13, 257|346|18, 257|348|16, 258|346|17
258|347|16, 267|345|18, 267|348|15, 268|345|17, 268|347|15

The cumulants related to these partitions can be written (leveraging the connectivity condition $\sigma \vee 12|34|56|78 = 1$),

$$ q_{\text{IX}} := \sum_{\substack{i_s \in J_s \\ s \in [4]}} \kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2}) \kappa(\overline{D_{i_2}}, D_{i_3}, D_{i_4}) \kappa(\overline{D_{i_3}}, \overline{D_{i_4}}), $$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, if $i \in I$, then $D_i = C_i^1(k)$ for a proper frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and $i \in I_p$, then $D_i = C_i^2(k)$ for a proper
frequency $k \in \{\pm \tilde{k}_{j_1}, \pm \tilde{k}_{j_2}\}$. Applying three times the Cauchy-Schwarz inequality, we get

$$ 
\begin{aligned}
|q_{\text{IX}}| & \le \sum_{\substack{i_s \in J_s; \\ s \in \{1,3,4\}}} \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2})|^2} \sqrt{\sum_{i_2 \in J_2} |\kappa(\overline{D_{i_2}}, D_{i_3}, D_{i_4})|^2} |\kappa(D_{i_3}, D_{i_4})| \\
& \le \sum_{\substack{i_1 \in J_1 \\ i_4 \in J_4}} \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2})|^2} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_3 \in J_3}} |\kappa(\overline{D_{i_2}}, D_{i_3}, D_{i_4})|^2} \sqrt{\sum_{i_3 \in J_3} |\kappa(D_{i_3}, D_{i_4})|^2} \\
& \le \sum_{i_1 \in J_1} \sqrt{\sum_{i_2 \in J_2} |\kappa(D_{i_1}, \overline{D_{i_1}}, D_{i_2})|^2} \sqrt{\sum_{\substack{i_s \in J_s; \\ s \in \{2,3,4\}}} |\kappa(\overline{D_{i_2}}, D_{i_3}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} |\kappa(D_{i_3}, D_{i_4})|^2}.
\end{aligned}
$$

Then, arguing as in the study of the partitions of category X, we obtain

$$ \frac{1}{N} \sum_{1 \le j_1, j_2 \le N} |\kappa_{\text{IX}}| \le K \left( \frac{\log(|W|)^d}{|W|^2 |k_0|^d} + \frac{1}{|k_0|^{2d} |W|^2} + \frac{1}{|I||I_p|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} + \frac{1}{N} \right) \right) + \mathcal{E}. $$

We consider the partitions of category XI. They are given by

135|247|68, 135|248|67, 135|267|48, 135|268|47, 135|467|28, 135|468|27, 136|247|58
136|248|57, 136|257|48, 136|258|47, 136|457|28, 136|458|27, 137|245|68, 137|246|58
137|258|46, 137|268|45, 137|458|26, 137|468|25, 138|245|67, 138|246|57, 138|257|46
138|267|45, 138|457|26, 138|467|25, 145|237|68, 145|238|67, 145|267|38, 145|268|37
145|367|28, 145|368|27, 146|237|58, 146|238|57, 146|257|38, 146|258|37, 146|357|28
146|358|27, 147|235|68, 147|236|58, 147|258|36, 147|268|35, 147|358|26, 147|368|25
148|235|67, 148|236|57, 148|257|36, 148|267|35, 148|357|26, 148|367|25, 157|236|48
157|238|46, 157|246|38, 157|248|36, 157|368|24, 157|468|23, 158|236|47, 158|237|46
158|246|37, 158|247|36, 158|367|24, 158|467|23, 167|235|48, 167|238|45, 167|245|38
167|248|35, 167|358|24, 167|458|23, 168|235|47, 168|237|45, 168|245|37, 168|247|35
168|357|24, 168|457|23, 235|467|18, 235|468|17, 236|457|18, 236|458|17, 237|458|16
237|468|15, 238|457|16, 238|467|15, 245|367|18, 245|368|17, 246|357|18, 246|358|17
247|358|16, 247|368|15, 248|357|16, 248|367|15, 257|368|14, 257|468|13, 258|367|14
258|467|13, 267|358|14, 267|458|13, 268|357|14, 268|457|13.
$$

The cumulants related to these partitions can be written (leveraging the connectivity condition $\sigma \vee 12|34|56|78 = 1$),

$$ q_{\text{XI}} := \sum_{\substack{i_s \in J_s \\ s \in [4]}} \kappa(D_{i_1}, D_{i_2}, D_{i_3}) \kappa(\overline{D_{i_1}}, \overline{D_{i_2}}, D_{i_4}) \kappa(\overline{D_{i_3}}, \overline{D_{i_4}}), $$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$ and exactly two $J_s$ are equal to $I$. Moreover, if $i \in I$, then $D_i = C_i^1(k)$ for a proper frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and $i \in I_p$, then $D_i = C_i^2(k)$ for a proper
frequency $k \in \{\pm \tilde{k}_{j_1}, \pm \tilde{k}_{j_2}\}$. Applying four times the Cauchy-Schwarz inequality, we get

$$
\begin{align*}
|q_{\mathbf{XI}}| & \le \sum_{\substack{i_s \in J_s; \\ s \in \{2,3,4\}}} \sqrt{\sum_{i_1 \in J_1} |\kappa(D_{i_1}, D_{i_2}, D_{i_3})|^2} \sqrt{\sum_{i_1 \in J_1} |\kappa(\overline{D_{i_1}}, \overline{D_{i_2}}, D_{i_4})|^2} |\kappa(D_{i_3}, D_{i_4})| \\
& \le \sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_2 \in J_2}} |\kappa(D_{i_1}, D_{i_2}, D_{i_3})|^2} \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_2 \in J_2}} |\kappa(\overline{D_{i_1}}, \overline{D_{i_2}}, D_{i_4})|^2} |\kappa(D_{i_3}, D_{i_4})| \\
& \le \sum_{i_4 \in J_4} \sqrt{\sum_{\substack{i_s \in J_s; \\ s \in \{1,2,3\}}} |\kappa(D_{i_1}, D_{i_2}, D_{i_3})|^2} \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_2 \in J_2}} |\kappa(\overline{D_{i_1}}, \overline{D_{i_2}}, D_{i_4})|^2} \sqrt{\sum_{i_3 \in J_3} |\kappa(D_{i_3}, D_{i_4})|^2} \\
& \le \sqrt{\sum_{\substack{i_s \in J_s; \\ s \in \{1,2,3\}}} |\kappa(D_{i_1}, D_{i_2}, D_{i_3})|^2} \sqrt{\sum_{\substack{i_s \in J_s; \\ s \in \{1,2,4\}}} |\kappa(\overline{D_{i_1}}, \overline{D_{i_2}}, D_{i_4})|^2} \sqrt{\sum_{i_4 \in J_4} |\kappa(D_{i_3}, D_{i_4})|^2}.
\end{align*}
$$

Arguing as in the case of the partitions of category **X**, we obtain

$$
\frac{1}{N} \sum_{1 \le j_1, j_2 \le N} |\kappa_{\mathbf{XI}}| \le K \left( \frac{\log(|W|)^d}{|W|^2 |k_0|^d} + \frac{1}{|k_0|^{2d} |W|^2} + \frac{1}{|I||I_p|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k_0|} \right)^{\zeta} + \frac{1}{N} \right) \right) + \mathcal{E}.
$$

It remains to consider the partitions of category **XII**. They consist in

16|28|35|47, 68|14|27|35, 16|27|48|35, 16|27|38|45, 38|27|15|46, 28|14|36|57
58|27|13|46, 45|67|18|23, 16|23|58|47, 25|46|38|17, 24|68|15|37, 14|68|25|37
38|14|57|26, 58|13|47|26, 58|46|17|23, 16|28|45|37, 18|46|23|57, 45|68|17|23
27|15|48|36, 38|67|14|25, 67|15|23|48, 38|24|15|67, 16|23|48|57, 18|35|47|26
16|24|37|58, 58|14|27|36, 67|24|18|35, 13|48|57|26, 68|15|23|47, 38|17|45|26
45|28|13|67, 24|18|36|57, 28|13|46|57, 58|14|37|26, 16|24|38|57, 68|13|25|47
28|15|46|37, 68|27|13|45, 67|13|25|48, 67|28|14|35, 24|68|17|35, 28|36|15|47
18|25|36|47, 45|18|37|26, 58|24|36|17, 25|36|17|48, 17|48|35|26, 18|46|25|37

The cumulants related to these partitions can be written, leveraging the connectivity condition
$\sigma \lor 12|34|56|78 = 1$,

$$
q_{\mathbf{XII}} := \sum_{\substack{i_s \in J_s \\ s \in [4]}} \kappa(D_{i_1}, D_{i_2}) \kappa(\overline{D_{i_2}}, D_{i_3}) \kappa(\overline{D_{i_3}}, D_{i_4}) \kappa(\overline{D_{i_4}}, \overline{D_{i_1}}).
$$

where $(J_s)_{s \in [4]}$ takes values in $\{I, I_p\}$, and exactly two of the $J_s$ are equal to $I$. Moreover, if $i \in I$,
then $D_i = C_i^1(k)$ for an appropriate frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$, and if $i \in I_p$, then $D_i = C_i^2(k)$
for a suitable frequency $k \in \{\pm k_{j_1}, \pm k_{j_2}\}$. Applying the Cauchy-Schwarz inequality four times, we
obtain

$$
|q_{\mathbf{XII}}| \le \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_2 \in J_2}} |\kappa(D_{i_1}, D_{i_2})|^2} \sqrt{\sum_{\substack{i_2 \in J_2 \\ i_3 \in J_3}} |\kappa(\overline{D_{i_2}}, D_{i_3})|^2} \sqrt{\sum_{\substack{i_3 \in J_3 \\ i_4 \in J_4}} |\kappa(\overline{D_{i_3}}, D_{i_4})|^2} \sqrt{\sum_{\substack{i_1 \in J_1 \\ i_4 \in J_4}} |\kappa(D_{i_1}, D_{i_4})|^2}.
$$
Note that, due to the connectivity condition, at least one block contains two frequencies, one indexed by $j_1$ and the other by $j_2$. Since the partition is connected to $12|34|56|78$, the difference between these two frequencies is at least $|k|c_7$ with $c_7 > 0$. Accordingly, we use the $k_0$-assumption and Lemma D.3 to get

$$ \frac{1}{N^2} \sum_{1 \le j_1, j_2 \le N} |q_{\mathbf{X}\mathbf{I}}| \le K \prod_{s=1}^{4} (|J_s| \wedge |J_{s+1}|)^{\frac{1}{2}} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k|^d} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E}, $$

with $J_5 := J_1$. Using $\prod_{s=1}^4 (|J_s| \wedge |J_{s+1}|)^{\frac{1}{2}} \le |I||I_p|$, we obtain

$$ \frac{1}{N^2} \sum_{1 \le j_1, j_2 \le N} |\kappa_{\mathbf{X}\mathbf{I}}| \le K \frac{1}{|I||I_p|} \left( \left( \frac{(|I| \vee |I_p|)^{\frac{1}{d}}}{|W|^{\frac{1}{d}} |k|^d} \right)^{\zeta} + \frac{1}{N} \right) + \mathcal{E}. $$

Gathering all the previous bounds, we conclude the proof of this lemma. $\square$

With the previous lemmas, are now in position to prove Lemma C.1.

*Proof of Lemma C.1.* By Jensen's inequality, we have

$$ \mathbb{E} \left[ \max_{I \in \mathcal{I}} |\Delta_I| \right] \le \mathbb{E} \left[ \max_{I \in \mathcal{I}} |\Delta_I|^2 \right]^{\frac{1}{2}} \le \sqrt{|\mathcal{I}| \max_{I \in \mathcal{I}} (\mathbb{E}[\Delta_I]^2 + \mathrm{Var}[\Delta_I])}. $$

We prove that $\max_{I \in \mathcal{I}} (\mathbb{E}[\Delta_I]^2 + \mathrm{Var}[\Delta_I]) = o(|W|^{-4\beta/(2\beta+d)})$. Throughout the proof, $K$ denotes a generic constant that may vary from line to line. We start with $\max_{I \in \mathcal{I}} \mathbb{E}[\Delta_I]^2$. According to Lemma C.2, we have

$$ |\mathbb{E}[\Delta_I]| \le K \left( \frac{(|I||I_p|)^{\frac{\beta}{d}}}{|W|^{2\frac{\beta}{d}}} + \frac{\log(|W|)^d}{|W|} + \frac{1}{|W||k_0|^d} + \mathcal{E}(|I|, |I_p|) \right). $$

Since $|I| \wedge |I_p| \ge \log(|W|)^{2d/(3\theta)+\eta}$, we have

$$ \mathcal{E} \le K e^{-\frac{1}{K} \log(|W|)^{1+3\theta\eta/(2d)}} = o\left(|W|^{-\frac{4\beta}{2\beta+d}}\right). \quad (68) $$

We always have $\log(|W|)^d/|W| = o(|W|^{-2\beta/(2\beta+d)})$. Moreover,

$$ \frac{1}{|W||k_0|^d} = A_W^{-d} \frac{1}{|W||W|^{-d/(2\beta+d)}} = A_W^{-d} \frac{1}{|W|^{2\beta/(2\beta+d)}} = o(|W|^{-2\beta/(2\beta+d)}), $$

since $A_W \to \infty$ as $|W| \to \infty$. Since $|I| \le c_5|W|^{2\beta/(2\beta+d)}$ and $|I_p| = o(|W|^{2\beta/(2\beta+d)})$, we have

$$ \frac{(|I||I_p|)^{\frac{\beta}{d}}}{|W|^{2\frac{\beta}{d}}} = o\left(\left(\frac{|W|^{\frac{4\beta}{2\beta+d}}}{|W|^2}\right)^{\frac{\beta}{d}}\right) = o(|W|^{-2\beta/(2\beta+d)}). $$

All previous $o$ are uniform over $I \in \mathcal{I}$. Accordingly $\max_{I \in \mathcal{I}} \mathbb{E}[\Delta_I]^2 = o(|W|^{-4\beta/(2\beta+d)})$. Concerning the variance, using Corollary G.1, we obtain

$$ \mathrm{Var}[\Delta_I] = \frac{4}{N^2} \sum_{1 \le j_1, j_2 \le N} (A + B + C), $$
with

$$A := \kappa(\hat{S}_I(k_{j_1}), \hat{S}_0(\tilde{k}_{j_1}), \hat{S}_I(k_{j_2}), \hat{S}_0(\tilde{k}_{j_2})),$$

$$B := \mathbb{E}[\hat{S}_I(k_{j_1}) - S_{\frac{1}{2}}(k_{j_1})] \kappa(\hat{S}_0(\tilde{k}_{j_1}), \hat{S}_I(k_{j_2}), \hat{S}_0(\tilde{k}_{j_2})) + \mathbb{E}[\hat{S}_0(\tilde{k}_{j_1}) - S_{\frac{1}{2}}(\tilde{k}_{j_1})] \kappa(\hat{S}_I(k_{j_1}), \hat{S}_I(k_{j_2}), \hat{S}_0(\tilde{k}_{j_2})) \\ + \mathbb{E}[\hat{S}_I(k_{j_2}) - S_{\frac{1}{2}}(\tilde{k}_{j_2})] \kappa(\hat{S}_I(k_{j_1}), \hat{S}_0(\tilde{k}_{j_1}), \hat{S}_0(\tilde{k}_{j_2})) - \mathbb{E}[\hat{S}_0(\tilde{k}_{j_2}) - S_{\frac{1}{2}}(\tilde{k}_{j_2})] \kappa(\hat{S}_I(k_{j_1}), \hat{S}_0(\tilde{k}_{j_1}), \hat{S}_I(k_{j_2})),$$

$$C := \kappa(\hat{S}_I(k_{j_1}), \hat{S}_I(k_{j_2}))\kappa(\hat{S}_0(\tilde{k}_{j_1}), \hat{S}_0(\tilde{k}_{j_2})) + \kappa(\hat{S}_I(k_{j_1}), \hat{S}_0(\tilde{k}_{j_2}))\kappa(\hat{S}_0(\tilde{k}_{j_1}), \hat{S}_I(k_{j_2})).$$

According to Lemma C.3,

$$\frac{4}{N^2} \sum_{1 \le j_1, j_2 \le N} |C| \le K \left( a_1 + \frac{\log(|W|)^{2d}}{|W|^2} + \mathcal{E} \right),$$

with

$$a_1 := \frac{1}{|I||I_p|} \left( \left( \frac{(|I_p| \lor |I|)^{\frac{1}{d}}}{|W|^{\frac{d}{d}} |k_0|} \right)^{d-1} + \frac{1}{N} \right).$$

By (68), we have $\mathcal{E} = o(|W|^{-4\beta/(2\beta+d)})$. Moreover, we have $\log(|W|)^{2d}/|W|^2 = o(|W|^{-4\beta/(2\beta+d)})$. Using the assumptions concerning the cardinal of the tapers index set of $\mathcal{I}$, we get

$$\begin{align*}
a_1 &\le K \left( \frac{1}{(|I| \wedge |I_p|)^{1+1/d}} \left( \frac{1}{|k||W|^d} \right)^{d-1} + \frac{1}{N(|I| \wedge |I_p|)^2} \right) \\
&\le K \left( \frac{A_W^{d-1-\eta(1+1/d)}}{|W|^{\frac{2\beta}{2\beta+d} + \frac{d+1}{d}}} \left( \frac{1}{A_W |W|^{\frac{1}{d} - \frac{1}{2\beta+d}}} \right)^{d-1} + \frac{A_W^{\frac{2d(d-1)}{d+1}-\eta(1+1/d)}}{N |W|^{4\beta/(2\beta+d)}} \right) \\
&\le K A_W^{-\eta(1+1/d)} |W|^{-\frac{4\beta}{2\beta+d}} = o(|W|^{-\frac{4\beta}{2\beta+d}}),
\end{align*}$$

since $N \ge A_W^{2d(d-1)/(d+1)}$ and $A_W \to \infty$ as $|W| \to \infty$. According to Lemma C.4,

$$\frac{4}{N^2} \sum_{1 \le j_1, j_2 \le N} |B| \le K (b(I, I_p) + b(I_p, I) + b + \mathcal{E}(|I|, |I_p|)),$$

where $b(I, I_p)$, $b(I_p, I)$ and $b$ are defined in the statement of Lemma C.4. For $b$, we have, using the fact that $4\beta/(2\beta+d) \le 4/3 < 5/3$ when $\beta \le 2$ and $d \ge 2$:

$$\begin{align*}
b &= \frac{\log(|W|)^d}{|W|^2 |k_0|^d} + \frac{\log(|W|)^d}{|W|^{\frac{5}{3}}} + \frac{1}{|k_0|^{2d} |W|^2} \\
&= \frac{\log(|W|)^d |W|^{\frac{d}{2\beta+d}}}{|W|^2 A_W^d} + \frac{|W|^{\frac{2d}{2\beta+d}}}{A_W^{2d} |W|^2} + o(|W|^{-\frac{4\beta}{2\beta+d}}) \\
&= \frac{\log(|W|)^d}{|W|^{\frac{d}{2\beta+d}}} |W|^{-\frac{4\beta}{2\beta+d}} + A_W^{-2d} |W|^{-\frac{4\beta}{2\beta+d}} + o(|W|^{-\frac{4\beta}{2\beta+d}}) = o(|W|^{-\frac{4\beta}{2\beta+d}}).
\end{align*}$$
We bound $b(I, I_p)$. We have $b(I, I_p) \le K(b_1 + b_2)$, with

$$b_1 := \left( \frac{|I|}{|W|} \right)^{\frac{\beta}{d}} |I_p|^{\frac{1}{2}} \frac{1}{|I||I_p|} \left( \frac{|I_p|^{\frac{1}{d}}}{|W|^{\frac{1}{d}}|k_0|} \right)^{d-1},$$ $$b_2 := \left( \frac{|I|}{|W|} \right)^{\frac{\beta}{d}} \left( \frac{1}{|W| \log(|W|)^{\eta}} + \frac{1}{N|I||I_p|^{\frac{1}{2}}} \right).$$

Using the computations for $a_1$ and

$$\left(\frac{|I|}{|W|}\right)^{\frac{\beta}{d}} |I_p|^{\frac{1}{2}} \le K |W|^{-\frac{\beta}{d} \frac{d}{2\beta+d}} |W|^{\frac{\beta}{2\beta+d}} = K,$$

we obtain $b_1 = o(|W|^{-4\beta/(2\beta+d)})$. Concerning $b_2$, we have

$$b_2 \le \frac{K}{|W|^{\frac{3\beta+d}{2\beta+d}} \log(|W|)^{\eta}} + \frac{K}{|W|^{\frac{\beta}{2\beta+d}}} \frac{A_W^{\frac{3}{2} \frac{d(d-1)}{d+1}}}{A_W^{\frac{2d(d-1)}{d+1}} |W|^{\frac{3}{2} \frac{2\beta}{2\beta+d}}} = o(|W|^{-\frac{4\beta}{2\beta+d}}),$$

since $\beta \le d$ and $A_W \to \infty$ as $|W| \to \infty$. The bound for $b(I_p, I)$ is obtained by exchanging $I$ and $I_p$. It remains to consider $A$. According to Lemma C.5:

$$\frac{4}{N^2} \sum_{1 \le j_1, j_2 \le N} |A| \le K (a_1 + b) + \mathcal{E}.$$

We have already proved that $a_1 + b + \mathcal{E} = o(|W|^{-4\beta/(2\beta+d)})$. Since all the previous $o$ are uniform over $I \in \mathcal{I}$, we have obtained $\max_{I \in \mathcal{I}} \mathrm{Var}[\Delta_I]^2 = o(|W|^{-4\beta/(2\beta+d)})$. This concludes the proof. $\square$

# D Technical results concerning joint cumulants of linear statistics of point processes

This section gathers bounds on the joint cumulants of linear statistics of point processes. Since Section 6 requires handling joint statistics involving thinnings of a given point process, the results presented here are established in this general framework. The structure of the section is as follows. Lemma D.1 is related to second order cumulants. Lemma D.2 provides a general but coarse bound on high-order cumulants, without using the orthogonality of the tapers or any decorrelation between frequencies. Lemmas D.3 and D.6 exploit the orthogonality of the tapers, while Lemmas D.4 and D.5 address decorrelation between frequencies.

**Lemma D.1.** Let $\Phi$ be a stationary point process with intensity $\lambda$, pair correlation function $g$ and structure factor $S$. Let $p \in [0, 1]$. Let $\Phi_p$ and $\Phi'_p$ be two independent p-thinning of $\Phi$. We denote $\Phi_{1-p} := \Phi \setminus \Phi_p$ and $\Phi'_{1-p} := \Phi \setminus \Phi'_p$. Let $f_1, f_2 \in L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$. Then,
$$a := \operatorname{Cov} \left[ \sum_{x \in \Phi_p} f_1(x), \sum_{x \in \Phi_p} f_2(x) \right] = \lambda p \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} (1 - p + pS(k)) dk,$$

$$b := \operatorname{Cov} \left[ \sum_{x \in \Phi_p} f_1(x), \sum_{x \in \Phi_{1-p}} f_2(x) \right] = (2\pi)^{\frac{d}{2}} \lambda^2 p(1-p) \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} \mathcal{F}[g-1](k) dk,$$

$$c := \operatorname{Cov} \left[ \sum_{x \in \Phi_p} f_1(x), \sum_{x \in \Phi'_p} f_2(x) \right] = \lambda^2 p^2 \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} S(k) dk,$$

$$d := \operatorname{Cov} \left[ \sum_{x \in \Phi_p} f_1(x), \sum_{x \in \Phi'_{1-p}} f_2(x) \right] = \lambda^2 p(1-p) \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} S(k) dk.$$

*Proof.* The proof of this lemma proceeds by conditioning on $\Phi$ and applying the law of total covariance. The argument for $a$, which is a standard result, follows the same steps as those used for $b$, $c$, and $d$, and is therefore omitted. We focus on the latter three, which are less classical. Note that

$$b := \operatorname{Cov} \left[ \sum_{x \in \Phi} B_x f_1(x), \sum_{x \in \Phi} (1 - B_x) f_2(x) \right],$$

for i.i.d. Bernoulli random variables $(B_x)_{x \in \Phi}$ of parameter $p$. Using the law of the total covariance, we obtain

$$b = \mathbb{E} \left[ \operatorname{Cov} \left[ \sum_{x \in \Phi} B_x f_1(x), \sum_{x \in \Phi} (1 - B_x) f_2(x) | \Phi \right] \right] + \operatorname{Cov} \left[ \mathbb{E} \left[ \sum_{x \in \Phi} B_x f_1(x) | \Phi \right], \mathbb{E} \left[ \sum_{x \in \Phi} (1 - B_x) f_2(x) | \Phi \right] \right] \\ = \mathbb{E} \left[ \sum_{x,y \in \Phi} f_1(x) \overline{f_2(x)} \operatorname{Cov}(B_x, 1 - B_y) \right] + p(1-p) \operatorname{Cov} \left[ \sum_{x \in \Phi} f_1(x), \sum_{x \in \Phi} f_2(x) \right].$$

According to the Plancherel theorem and using (4), we have

$$b = (2\pi)^{\frac{d}{2}} \lambda p(1-p) \left( - \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} dk + \int_{\mathbb{R}^d} \mathcal{F}[f_1] \overline{\mathcal{F}[f_2]}(1 + \lambda \mathcal{F}[g-1]) dk \right) \\ = (2\pi)^{\frac{d}{2}} \lambda^2 p(1-p) \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2](k)} \mathcal{F}[g-1](k) dk,$$

which concludes the proof for this term. Concerning $c$, we have

$$c := \operatorname{Cov} \left[ \sum_{x \in \Phi} B_x f_1(x), \sum_{x \in \Phi} B'_x f_2(x) \right],$$

for i.i.d. Bernoulli random variables $(B_x, B'_x)_{x \in \Phi}$ of parameter $p$. In the law of the total covariance, the first term vanishes due to the independence between $(B_x)_{x \in \Phi}$ and $(B'_x)_{x \in \Phi}$. Hence

$$c = p^2 \operatorname{Cov} \left[ \sum_{x \in \Phi} f_1(x), \sum_{x \in \Phi} f_2(x) \right],$$

which gives the result for $c$ using (4). The proof of $d$ consists in considering i.i.d. Bernoulli $(B'_x)_{x \in \Phi}$ of parameter $1-p$ instead of $p$. $\square$
**Lemma D.2.** Let $\Phi$ be a stationary point process with intensity $\lambda$ and $p \in [0, 1]$. Let $m \ge 2$. Denote by $(\Phi_p^j)_{1 \le j \le m}$ p-independent-thinnings of $\Phi$, that are independent between them. Moreover, let $\Phi_{1-p}^j = \Phi \setminus \Phi_p^j$. Consider $(f_i)_{i=1,\dots,m} \in L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$. Let $\tau : [m] \to \{p, 1-p\}$ and $\varphi : [m] \to [m]$. Then,

$$ \left| \kappa \left( \sum_{x \in \Phi_{\tau(s)}^{\varphi(s)}} f_s(x); s \in [m] \right) \right| \le K \prod_{s=1}^{m} \|f_s\|_m, $$

where $K < \infty$ depends only on $p$, $\tau$, $m$, $\lambda$ and $|\gamma_{red}^{(n)}|(\mathbb{R}^{d(n-1)})$ for $2 \le n \le m$. In particular if $p=1$, $\tau \equiv 1$ and $\varphi \equiv 1$, we have $K = \sum_{\pi \in \Pi[m]} |\gamma_{red}^{(|\pi|)}|(\mathbb{R}^{d(|\pi|-1)})$, with the convention $|\gamma_{red}^{(1)}|(\mathbb{R}^0) = \lambda$.

*Proof.* Let $(B_x^j)_{x \in \Phi, j \in [m]}$ be i.i.d. Bernoulli random variables of parameter $p$. We use the notation, $\tau_s(y) = y1_{\tau_s=p} + (1-y)1_{\tau_s=1-p}$. According to the law of the total cumulance, we have

$$ \begin{aligned} \kappa &:= \kappa \left( \sum_{x \in \Phi_{\tau(s)}^{\varphi(s)}} f_s(x); s \in [m] \right) = \sum_{\pi \in \Pi[m]} \kappa \left( \left( \sum_{x \in \Phi} \tau_s(B_x^{\varphi(s)}) f_s(x) \mid \Phi; s \in b \right); b \in \pi \right) \\ &= \sum_{\pi \in \Pi[m]} \kappa \left( \sum_{\substack{x_s \in \Phi; \\ s \in b}} \kappa(\tau_s(B_{x_s}^{\varphi(s)}); s \in b) f_s(x_s); b \in \pi \right). \end{aligned} $$

Using the independence of $(B_x^j)_{x \in \Phi}$, for $j \in [m]$, we obtain

$$ \sum_{\substack{x_s \in \Phi; \\ s \in b}} \kappa(\tau_s(B_{x_s}^{\varphi(s)}); s \in b) f_s(x_s) = 1_{\#\{\varphi(s)|s \in b\}=1} \sum_{\substack{x_s \in \Phi; \\ s \in b}} \kappa(\tau_s(B_{x_s}^{\varphi(s_0)}); s \in b) f_s(x_s), $$

where $s_0 = \min(b)$. Moreover, we have

$$ \kappa(\tau_s(B_{x_s}^{\varphi(s_0)}); s \in b) = (-1)^{\#\{\tau_s=1|s \in b\}} \kappa(B_{x_s}^{\varphi(s_0)}; s \in b) = (-1)^{\#\{\tau_s=1|s \in b\}} 1_{\#\{x_s|s \in b\}=1} \kappa|_{b}(B), $$

where $B$ is a Bernoulli random variable of parameter $p$. This yields

$$ \kappa = \sum_{\pi \in \Pi[m]} A \times B, \qquad (69) $$

with

$$ A := (-1)^{\#\{\tau_s=1|s \in [m]\}} \prod_{b \in \pi} 1_{\#\{\varphi(s)|s \in b\}=1} \kappa|_{b}(B), \quad B := \kappa \left( \sum_{x \in \Phi} \prod_{s \in b} f_s(x); b \in \pi \right). $$

The quantity $A$ does not depend on the functions $(f_i)_{1 \le i \le m}$. Accordingly, it remains to bound $B$. Denote $g_b(x) := \prod_{s \in b} f_s(x)$. Then,

$$ B = \sum_{\pi' \in \Pi\{b; b \in \pi\}} \int_{\mathbb{R}^{d|\pi'|}} \prod_{b' \in \pi'} \prod_{b \in b'} g_b(x_{b'}) \gamma^{(|\pi'|)}(x_{b'}; b' \in \pi') =: \sum_{\pi' \in \Pi\{b; b \in \pi\}} B_{\pi'}, $$
where $\Pi\{b; b \in \pi\}$ denotes the set of partitions of $\{b; b \in \pi\}$. If $|\pi'| = 1$, we have using the Hölder's inequality,

$$|B_{\pi'}| \le \lambda \int_{\mathbb{R}^d} \prod_{i \in [m]} |f_i(x)| dx \le \lambda \prod_{s=1}^{m} \|f_s\|_m.$$

We use the same argument when $|\pi| > 1$. Let $b'_0 \in \pi'$ be a fixed block. By stationarity, and using again the Hölder's inequality, we obtain:

$$ \begin{aligned} |B_{\pi'}| &\le \int_{\mathbb{R}^d(|\pi'|-1)} \int_{\mathbb{R}^d} \prod_{b \in b_0'} |g_b(x_{b_0'})| \prod_{b' \in \pi' \setminus \{b_0'\}} \prod_{b \in b'} |g_b(x_{b'} + x_{b_0'})| dx_{b_0'} |\gamma_{red}^{(|\pi'|)}|(dx_b'; b \in \pi \setminus \{b_0'\}) \\ &\le \int_{\mathbb{R}^d(|\pi'|-1)} \prod_{b \in b_0'} \prod_{s \in b} \|f_s\|_m \prod_{b' \in \pi' \setminus \{b_0'\}} \prod_{b \in b'} \prod_{s \in b} \|f_s(\cdot + x_b')\|_m |\gamma_{red}^{(|\pi'|)}|(dx_b'; b' \in \pi' \setminus \{b_0'\}) \\ &= |\gamma_{red}^{(|\pi'|)}|(\mathbb{R}^d(|\pi'|-1)) \prod_{s=1}^{m} \|f_s\|_m. \end{aligned} \quad (70) $$

This concludes the proof. To obtain the value of the constant when $p = 1$, note that in that case $A$ is non zero only if all the blocks of $\pi$ are of size 1. $\square$

**Lemma D.3.** Let $\Phi$ be a stationary point process with intensity $\lambda$ and pair correlation function $g$ and $p \in [0, 1]$. Denote by $(\Phi_p^j)_{j \in [2]}$ two p-independent-thinnings of $\Phi$, that are independent between them. Denote $\Phi_{1-p}^j = \Phi \setminus \Phi_p^j$. Let $I_1$ and $I_2$ be two discrete subsets of $\mathbb{N}^d$. Let $(f_i)_{i \in \mathbb{N}^d}$ and $(h_i)_{i \in \mathbb{N}^d}$ be two orthogonal families of $L^2(\mathbb{R}^d)$. Let $\tau[2] \mapsto \{p, 1-p\}^2$ and $\varphi: [2] \mapsto [2]$. Then, we have for all set $W \subset \mathbb{R}^d$,

$$ \frac{1}{|I_1||I_2|} \sum_{\substack{i_1 \in I_1 \\ i_2 \in I_2}} \left| \operatorname{Cov} \left( \sum_{x \in \Phi_{\tau(1)}^{\varphi(1)} \cap W} f_{i_1}(x), \sum_{x \in \Phi_{\tau(2)}^{\varphi(2)} \cap W} h_{i_2}(x) \right) \right|^2 \le \frac{K}{|I_1| \vee |I_2|} \left( 1 + \mathbb{1}_{|I_1| \ge |I_2|} \left( \sum_{i_1 \in I_1} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2^2 \right)^{\frac{1}{2}} + \mathbb{1}_{|I_1| < |I_2|} \left( \sum_{i_2 \in I_2} \|h_{i_2} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2^2 \right)^{\frac{1}{2}} \right)^2. $$

where $K < \infty$ depends only on $p$, $\tau$, $\varphi$, $\lambda$, $\|S\|_{\infty}$ and $\|g-1\|_1$. In particular if $p = 1$, $\tau \equiv 1$ and $\varphi \equiv 1$, we have $K = (\lambda \|S\|_{\infty})^2$.

*Proof.* According to Lemma D.1, we have

$$ \operatorname{Cov} \left( \sum_{x \in \Phi_{\tau(1)}^{\varphi(1)} \cap W} f_{i_1}(x), \sum_{x \in \Phi_{\tau(2)}^{\varphi(2)} \cap W} h_{i_2}(x) \right) = \int_{\mathbb{R}^d} \mathcal{F}[f_{i_1} \mathbb{1}_W](k) \overline{\mathcal{F}[h_{i_2} \mathbb{1}_W]}(k) s(k) dk, $$

where $s$ is bounded. Assume that $|I_1| \ge |I_2|$. We use the decomposition

$$ I_{i_1, i_2} := \int_{\mathbb{R}^d} \mathcal{F}[f_{i_1} \mathbb{1}_W](k) \overline{\mathcal{F}[h_{i_2} \mathbb{1}_W]}(k) s(k) dk = \int_{\mathbb{R}^d} \mathcal{F}[f_{i_1}](k) \overline{\mathcal{F}[\mathbb{1}_W h_{i_2}]}(k) s(k) dk + \delta, $$
where

$$ \delta = \int_{\mathbb{R}^d} \mathcal{F}[f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}](k) \overline{\mathcal{F}[h_{i_2} \mathbb{1}_W]}(k) s(k) dk. $$

Using the Cauchy-Schwarz inequality and the Plancherel theorem, we get

$$ |\delta| \le \|s\|_{\infty} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2 \|h_{i_2} \mathbb{1}_W\|_2 \le \|s\|_{\infty} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2. $$

Accordingly, using the fact that $(\mathcal{F}[f_i])_{i \in \mathbb{N}^d}$ is orthogonal, we obtain

$$ \begin{aligned} \left( \sum_{\substack{i_1 \in I_1 \\ i_2 \in I_2}} |I_{i_1, i_2}|^2 \right)^{\frac{1}{2}} & \le \left( \sum_{\substack{i_1 \in I \\ i_2 \in I}} |\langle \mathcal{F}[f_{i_1}], \mathcal{F}[\mathbb{1}_W h_{i_2}] s \rangle|^2 \right)^{\frac{1}{2}} + \left( \sum_{\substack{i_1 \in I_1 \\ i_2 \in I_2}} \delta^2 \right)^{\frac{1}{2}} \\ & \le \left( \sum_{i_2 \in I_2} \|\mathcal{F}[\mathbb{1}_W h_{i_2}] s\|_2^2 \right)^{\frac{1}{2}} + |I_2|^{\frac{1}{2}} \|s\|_{\infty} \left( \sum_{i_1 \in I_1} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2^2 \right)^{\frac{1}{2}} \\ & \le |I_2|^{\frac{1}{2}} \|s\|_{\infty} \left( 1 + \left( \sum_{i_1 \in I_1} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2^2 \right)^{\frac{1}{2}} \right). \end{aligned} $$

This yields

$$ \frac{1}{|I_1||I_2|} \sum_{\substack{i_1 \in I_1 \\ i_2 \in I_2}} |I_{i_1, i_2}|^2 \le \frac{\|s\|_{\infty}^2}{|I_1| \vee |I_2|} \left( 1 + \left( \sum_{i_1 \in I_1} \|f_{i_1} \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2^2 \right)^{\frac{1}{2}} \right)^2. $$

The proof for the case $|I_1| < |I_2|$ follows by symmetry. $\square$

**Lemma D.4.** Let $\Phi$ be a stationary point process with intensity $\lambda$ and pair correlation function $g$ and $p \in [0, 1]$. Denote by $(\Phi_p^j)_{j \in [2]}$ two p-independent-thinnings of $\Phi$, that are independent between them. Denote $\Phi_{1-p} = \Phi \setminus \Phi_p^j$. Let $f_1, f_2$ be in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$ with $\|f_1\|_2 = \|f_2\|_2 = 1$ and $k_1, k_2 \in \mathbb{R}^d$. Let $\tau[2] \mapsto \{p, 1-p\}^2$ and $\varphi: [2] \mapsto [2]$. Then, we have for all set $W \subset \mathbb{R}^d$,

$$ \left| \mathrm{Cov} \left( \sum_{x \in \Phi_{\tau(1)}^{(1)} \cap W} f_1(x) e^{-ik_1 \cdot x}, \sum_{x \in \Phi_{\tau(2)}^{(2)} \cap W} f_2(x) e^{-ik_2 \cdot x} \right) \right| \\ \le K \sum_{s=1}^{2} \|f_s \mathbb{1}_{\mathbb{R}^d \setminus W}\|_2 + \|\mathcal{F}[f_s] \mathbb{1}_{|\cdot| \ge |k_1 - k_2|/2}\|_2, $$

where $K < \infty$ depends only on $p, \tau, \varphi, \lambda, \|S\|_{\infty}$ and $\|g-1\|_1$. In particular if $p=1, \tau \equiv 1$ and $\varphi \equiv 1$, we have $K = \lambda \|S\|_{\infty}$.

**Proof.** According to Lemma D.1, we have

$$ \mathrm{Cov} \left( \sum_{x \in \Phi_{\tau(1)}^{(1)} \cap W} f_1(x) e^{-ik_1 \cdot x}, \sum_{x \in \Phi_{\tau(2)}^{(2)} \cap W} f_2(x) e^{-ik_2 \cdot x} \right) \\ = \int_{\mathbb{R}^d} \mathcal{F}[f_1 \mathbb{1}_W e^{-ik_1 \cdot x}] (k) \overline{\mathcal{F}[f_2 \mathbb{1}_W e^{-ik_2 \cdot x}]} (k) s(k) dk, $$
where $s$ is bounded. The first step is to remove the two indicator functions $1_W$. We have

$$ \int_{\mathbb{R}^d} \mathcal{F}[f_1 1_W e^{-ik_1 \cdot}](k) \overline{\mathcal{F}[f_2 1_W e^{-ik_2 \cdot}]}(k) s(k) dk = \int_{\mathbb{R}^d} \mathcal{F}[f_1 e^{-ik_1 \cdot}](k) \overline{\mathcal{F}[f_2 e^{-ik_2 \cdot}]}(k) s(k) dk + \delta, $$

with

$$ \delta = - \int_{\mathbb{R}^d} \mathcal{F}[f_1 1_{\mathbb{R}^d \setminus W} e^{-ik_1 \cdot}](k) \overline{\mathcal{F}[f_2 e^{-ik_2 \cdot}]}(k) s(k) dk - \int_{\mathbb{R}^d} \mathcal{F}[f_1 1_W e^{-ik_1 \cdot}](k) \overline{\mathcal{F}[f_2 1_{\mathbb{R}^d \setminus W} e^{-ik_2 \cdot}]}(k) s(k) dk. $$

By Cauchy-Schwarz inequality, we have

$$ \begin{aligned} |\delta| &\le \|s\|_{\infty} (\|\mathcal{F}[f_1 1_{\mathbb{R}^d \setminus W}]\|_2 \|\mathcal{F}[f_2]\|_2 + \|\mathcal{F}[f_1 1_W]\|_2 \|\mathcal{F}[f_2] 1_{\mathbb{R}^d \setminus W}\|_2) \\ &\le \|s\|_{\infty} (\|\mathcal{F}[f_1 1_{\mathbb{R}^d \setminus W}]\|_2 + \|\mathcal{F}[f_2] 1_{\mathbb{R}^d \setminus W}\|_2). \end{aligned} $$

It remains to upper bound

$$ I := \int_{\mathbb{R}^d} \mathcal{F}[f_1 e^{-ik_1 \cdot}](k) \overline{\mathcal{F}[f_2 e^{-ik_2 \cdot}]}(k) s(k) dk = \int_{\mathbb{R}^d} \mathcal{F}[f_1](k) \overline{\mathcal{F}[f_2]}(k + k_2 - k_1) s(k - k_1) dk. $$

We split $\mathbb{R}^d = A_1 \cup A_2$ with $A_1 := \{k \in \mathbb{R}^d | |k - k_1 + k_2| \le |k_1 - k_2|/2\}$ and $A_2 := \mathbb{R}^d \setminus A_1$. Since for $k \in A_1$, we have $|k - k_1 + k_2| \ge |k_1 - k_2| - |k| \ge |k_1 - k_2|/2$, we obtain using the Cauchy-Schwarz's inequality and the Plancherel theorem

$$ |I| \le \|s\|_{\infty} (\|\mathcal{F}[f_1] 1_{|\cdot| \ge |k_1 - k_2|/2}\|_2 + \|\mathcal{F}[f_2] 1_{|\cdot| \ge |k_1 - k_2|/2}\|_2), $$

which concludes the proof. □

**Lemma D.5.** Let $\Phi$ be a stationary point process with intensity $\lambda$ and pair correlation function $g$ and $p \in [0, 1]$. Denote by $(\Phi_p^j)_{j \in [3]}$ three p-independent-thinnings of $\Phi$, that are independent between them. Denote $\Phi_{1-p}^j = \Phi \setminus \Phi_p^j$. Let $f_1, f_2, f_3$ be $L^1(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d)$ functions and $k_1, k_2, k_3 \in \mathbb{R}^d$. Let $\tau[3] \mapsto \{p, 1-p\}^3$ and $\varphi: [3] \mapsto [3]$. Then, we have for all set $W \subset \mathbb{R}^d$,

$$ \left| \kappa \left( \sum_{x \in \Phi_{\tau(s)}^{\varphi(s)} \cap W} f_s(x) e^{-ik_s \cdot x}; s \in [3] \right) \right| \le K \sum_{s=1}^{3} \|f_s 1_{\mathbb{R}^d \setminus W}\|_3 \prod_{s' \in [3] \setminus \{s\}} \|f_{s'}\|_3 \\ + K \sum_{s=1}^{3} \|\mathcal{F}[f_s] 1_{|\cdot| \ge |k_1+k_2+k_3|/3}\|_1 \prod_{s' \in [3] \setminus \{s\}} \|f_{s'}\|_2, \quad (71) $$

where $K < \infty$ depends only on $p, \tau, \varphi, \lambda$ and $|\gamma_{red}^{(m)}|(\mathbb{R}^{d(m-1)})$ for $m \in \{2, 3\}$. In particular if $p = 1$, $\tau \equiv 1$ and $\varphi \equiv 1$, we have $K = \sum_{\pi \in \Pi[3]} |\gamma_{red}^{(\pi)}|(\mathbb{R}^{d(|\pi|-1)})$, with the convention $|\gamma_{red}^{(1)}|(\mathbb{R}^0) = \lambda$.

**Proof.** Denote by $|\kappa_W|$ the left hand-side of (71). According to (69) for $m = 3$, we have

$$ \kappa_W = \sum_{\pi \in \Pi[3]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} \int_{\mathbb{R}^d|\pi'|} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b} 1_W(x_{b'}) f_s(x_{b'}) \gamma^{(|\pi'|)}(x_{b'}; b' \in \pi'), $$
with, denoting by B a Bernoulli random variable of parameter p,

$$a_{\pi} := (-1)^{\#\{\tau_s=1|s \in [m]\}} \prod_{b \in \pi} 1_{\#\{\varphi(s)|s \in b\}=1} \kappa_{|b|}(B).$$

The first step is to remove the truncation to W. Note that

$$1_{\mathbb{R}^d|\pi'|\setminus W|\pi'|}(x'_b; x'_b \in \pi') \leq \sum_{b'_0 \in \pi'} 1_{\mathbb{R}^d\setminus W}(x_{b'_0}).$$

Denote $\Delta := |\kappa_W - \kappa_{\mathbb{R}^d}|$. Subsequently,

$$\Delta \leq \sum_{\pi \in \Pi[3]} |a_{\pi}| \sum_{\pi' \in \Pi\{b; b \in \pi\}} \sum_{b'_0 \in \pi'} \int_{\mathbb{R}^d|\pi'|} 1_{\mathbb{R}^d \setminus W}(x_{b'_0}) \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b} |f_s(x_{b'})| \gamma^{(|\pi'|)}(x_{b'}; b' \in \pi').$$

By stationarity and using the Hölder's inequality, as in (70), we obtain

$$\Delta \leq \sum_{\pi \in \Pi[3]} |a_{\pi}| \sum_{\pi' \in \Pi\{b; b \in \pi\}} |\gamma_{red}^{(|\pi'|)}| (\mathbb{R}^{d(|\pi'|-1)}) \sum_{b'_0 \in \pi'} \prod_{i \in b'_0} \|f_i 1_{\mathbb{R}^d \setminus W}\|_3 \prod_{i \notin b'_0} \|f_i\|_3.$$

Since

$$\prod_{i \in b'_0} \|f_i 1_{\mathbb{R}^d \setminus W}\|_3 \prod_{i \notin b'_0} \|f_i\|_3 \leq \sum_{s=1}^3 \|f_s 1_{\mathbb{R}^d \setminus W}\|_3 \prod_{s' \in [3] \setminus \{s\}} \|f_{s'}\|_3,$$

we have

$$\Delta \leq \left( \sum_{\pi \in \Pi[3]} |a_{\pi}| \sum_{\pi' \in \Pi\{b; b \in \pi\}} |\gamma_{red}^{(|\pi'|)}| (\mathbb{R}^{d(|\pi'|-1)}) |\pi'| \right) \sum_{s=1}^{3} \|f_s 1_{\mathbb{R}^d \setminus W}\|_3 \prod_{s' \in [3] \setminus \{s\}} \|f_{s'}\|_3.$$

Now, we focus on

$$\begin{aligned} \kappa_{\mathbb{R}^d} &= \sum_{\pi \in \Pi[3]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} \int_{\mathbb{R}^d|\pi'|} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b} f_s(x_{b'}) \gamma^{(|\pi'|)}(x_{b'}; b' \in \pi') \\ &= \sum_{\pi \in \Pi[3]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} I_{\pi'}. \end{aligned}$$

Each of the terms $I_{\pi'}$ is bounded using the same idea, but the details differ. We start with with the case $|\pi'| = 1$. Using the Fourier inversion theorem [29] we get

$$|I_{\pi'}| = \left| \int_{\mathbb{R}^d} \prod_{i=1}^3 f_i(x) e^{-ik_i \cdot x} dx \right| \leq \int_{\mathbb{R}^{2d}} |\mathcal{F}[f_1](k) \mathcal{F}[f_2](k') \mathcal{F}[f_3](k_1 + k_2 + k_3 - k - k')| dk dk' := I_{\infty}.$$

Denote $\Sigma := k_1 + k_2 + k_3$. Next, we split the integrals over the domains:

$$D_1 := \{|k| \geq |\Sigma|/3\}, D_2 = \{|k| \leq |\Sigma|/3, |k'| \geq |\Sigma|/3\}, D_3 := \{|k| \leq |\Sigma|/3, |k'| \leq |\Sigma|/3\}.$$
Note that on $D_3$, we have $|\Sigma - k - k'| \ge |\Sigma| - |k| - |k'| \ge |\Sigma|/3$. Hence, the triangular inequality and then the Young's inequality for the convolution yield

$$|I_{\pi'}| \le I_{\infty} \le \sum_{s=1}^{3} \|\mathcal{F}[f_s]\mathbf{1}_{|\cdot| \ge |\Sigma|/3}\|_{1} \prod_{s' \in [3] \setminus \{s\}} \|\mathcal{F}[f_{s'}]\|_{2}. \quad (72)$$

For the other partitions, we adapt the previous technique, leveraging on the Brillinger-mixing condition. Define

$$\forall k \in \mathbb{R}^{d(m-1)}, \quad \mathcal{F}[\gamma_{\text{red}}^{(m)}](k) := \int_{\mathbb{R}^{d(m-1)}} e^{-ik \cdot x} \gamma_{\text{red}}^{(m)}(dx).$$

Note that $\|\mathcal{F}[\gamma_{\text{red}}^{(m)}]\|_{\infty} \le |\gamma_{\text{red}}^{(m)}|(\mathbb{R}^{d(m-1)}) < \infty$ by Brillinger-mixing. We consider the case when $|\pi'| = 3$. Using the Fourier inversion theorem and the Fubini's theorem, we have

$$\begin{aligned}
I_{\pi'} &= \int_{\mathbb{R}^{3d}} \prod_{i \in [3]} f_i(x_i) e^{-ik_i \cdot x_i} \gamma^{(3)}(dx_1, dx_2, dx_3) \\
&= \int_{\mathbb{R}^{3d}} f_1(x_1) e^{-ix_1 \cdot k_1} \prod_{s=2,3} f_s(x_1 + x_s) e^{-i(x_1+x_s) \cdot k_s} dx_1 \gamma_{\text{red}}^{(3)}(dx_2, dx_3) \\
&= \int_{\mathbb{R}^{2d}} \mathcal{F}[f_1 e^{-ik_1 \cdot x}] (k'_2 + k'_3) \prod_{s=2,3} \mathcal{F}[f_s e^{-ik_s \cdot x}] (-k'_s) \mathcal{F}[\gamma_{\text{red}}^{(3)}(k'_2, k'_3) dk'_2 dk'_3.
\end{aligned}$$

By change of variables, we obtain

$$\begin{aligned}
I_{\pi'} &= \int_{\mathbb{R}^{2d}} \mathcal{F}[f_1] (k_1 + k'_1 + k'_2) \prod_{s=2,3} \mathcal{F}[f_s] (k_s - k'_s) \mathcal{F}[\gamma_{\text{red}}^{(3)}(k'_2, k'_3) \\
&= \int_{\mathbb{R}^{2d}} \mathcal{F}[f_1] \left( \sum_{s=1}^{3} k_s - k'_1 - k'_2 \right) \prod_{s=2,3} \mathcal{F}[f_s] (k'_s) \mathcal{F}[\gamma_{\text{red}}^{(3)}](-k'_1 + k_1, -k'_2 + k_2) dk'_2 dk'_3.
\end{aligned}$$

Accordingly, $|I_{\pi'}| \le |\gamma_{\text{red}}^{(3)}|(\mathbb{R}^{2d})I_{\infty}$, and we apply (72). Finally, assume that $|\pi'| = 2$. By renumbering, it suffices to upper bound:

$$\begin{aligned}
I_{\pi'} &= \int_{\mathbb{R}^{2d}} f_1(x) e^{-ik_1 \cdot x} f_2(x) e^{-ik_2 \cdot x} f_3(x') e^{-ik_3 \cdot x'} \gamma^{(2)}(dx, dx') \\
&= \int_{\mathbb{R}^d} \mathcal{F}[f_1 e^{-ik_1 \cdot x} f_2 e^{-ik_2 \cdot x}] (k) \mathcal{F}[f_3 e^{-ik_3 \cdot x}] (-k) \mathcal{F}[\gamma_{\text{red}}^{(2)}(k)] dk \\
&= \int_{\mathbb{R}^{2d}} \mathcal{F}[f_1](k' + k_1) \mathcal{F}[f_2](k + k_2 - k') \mathcal{F}[f_3](-k + k_3) \mathcal{F}[\gamma_{\text{red}}^{(2)}(k)] dk dk',
\end{aligned}$$

where we used the Fourier inversion theorem. By change of variables, we get $|I_{\pi'}| \le |\gamma_{\text{red}}^{(2)}|(\mathbb{R}^d)I_{\infty}$. Recalling (72), this concludes the proof. To obtain the value of the constant when $p = 1$, note that in that case A is non zero only if all the blocks of $\pi$ are of size 1. $\square$

**Lemma D.6.** Let $\Phi$ be a stationary point process with intensity $\lambda$ and $p \in [0, 1]$. Let $m \ge 2$ and denote by $(\Phi_p^j)_{j \in [m]}$ three p-independent-thinnings of $\Phi$, that are independent between them. Denote $\Phi_{1-p}^j = \Phi \setminus \Phi_p^j$. Let $(f_i)_{i \in \mathbb{N}^d}$, $(h_s)_{s=2,...,m}$ be $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$ functions. Assume that $(f_i)_{i \in \mathbb{N}^d}$ is
orthogonal. Let $\tau[m] \mapsto \{p, 1-p\}^m$ and $\varphi : [m] \mapsto [m]$. Then, we have for all set $W \subset \mathbb{R}^d$,

$$ \sum_{i \in \mathbb{N}^d} \left| \kappa \left( \sum_{x \in \Phi_{\tau(1)}^{\varphi(1)} \cap W} f_i(x), \sum_{x \in \Phi_{\tau(s)}^{\varphi(s)} \cap W} h_s(x); 2 \le s \le m \right) \right|^2 \\ \le K \prod_{s=2}^{m} \|h_2\|_{2(m-1)}^2 + K \sum_{i \in I} \sum_{s=1}^{m} \|h_s\|_{1\mathbb{R}^d \setminus W}\|_m \prod_{s' \in [m]\setminus\{s'\}} \|h_s\|_m. $$

where, we use the notation $h_1 := f_i$ in the second line and where $K < \infty$ depends only on $p, \tau, \varphi, \lambda$ and $|\gamma_{red}^{(n)}|(\mathbb{R}^{d(m-1)})$ for $2 \le n \le m$.

*Proof.* The first part of the proof is identical to the beginning of the proof of Lemma D.5. Denote $h_1 = f_i$, $\Delta := |\kappa_W - \kappa_{\mathbb{R}^d}|$, where

$$ \kappa_W := \kappa \left( \sum_{x \in \Phi_{\tau(s)}^{\varphi(s)} \cap W} h_s(x); 1 \le s \le m \right). $$

To remove the truncation to $W$, we use the method of Lemma D.5 to get

$$ \Delta \le \left( \sum_{\pi \in \Pi[m]} |a_{\pi}| \sum_{\pi' \in \Pi\{b; b \in \pi\}} |\gamma_{red}^{(|\pi'|)}(\mathbb{R}^{d(|\pi'|-1)})|^{\pi'} \right) \sum_{s=1}^{m} \|h_s\|_{1\mathbb{R}^d \setminus W}\|_m \prod_{s' \in [m]\setminus\{s\}} \|h_{s'}\|_m. $$

Now, we focus on

$$ \kappa_{\mathbb{R}^d} = \sum_{\pi \in \Pi[m]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} \int_{\mathbb{R}^d|\pi'|} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b} h_s(x_{b'}) \gamma^{(|\pi'|)}(x_{b'}; b' \in \pi') =: \sum_{\pi \in \Pi[m]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} I_{\pi'}. $$

where, denoting by $B$ a Bernoulli random variable of parameter $p$,

$$ a_{\pi} := (-1)^{\#\{\tau_s=1|s \in [m]\}} \prod_{b \in \pi} 1_{\#\{\varphi(s)|s \in b\}=1} \kappa_{|b|}(B). $$

By triangular inequality, we have

$$ \sum_{i \in I} |\kappa_{\mathbb{R}^d}|^2 \le \left( \sum_{\pi \in \Pi[m]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} \left( \sum_{i \in I} |I_{\pi'}|^2 \right)^{1/2} \right)^2. $$

As in Lemma D.5, we start by considering the partition $\pi'$ of size 1. We have

$$ I_{\pi'} = \lambda \int_{\mathbb{R}^d} f_i(x) \prod_{s=2}^{m} h_s(x) dx = \lambda \langle f_i, \overline{\prod_{s=2}^{m} h_s} \rangle. $$

Since $(f_i)_{i \in \mathbb{N}^d}$ is orthogonal and using the Hölder's inequality, we obtain

$$ \sum_{i \in \mathbb{N}^d} |I_{\pi'}|^2 \le \lambda^2 \int_{\mathbb{R}^d} \prod_{s=2}^{m} |h_s(x)|^2 dx \le \prod_{s=2}^{m} \|h_2\|_{2(m-1)}^2. $$
For the other partitions, we leverage on the Brillinger-mixing condition to adapt this argument. Let $\pi'$ with $|\pi'| \ge 2$. Denote by $b'_0 \in \pi'$ the block containing 1. By stationarity, we have

$$ I_{\pi} = \int_{\mathbb{R}^d} f_i(x_{b'_0}) \left( \int_{\mathbb{R}^{d(|\pi'|-1)}} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b \setminus \{1\}} h_s(x_{b'} + x_{b'_0}) \gamma_{red}^{(|\pi'|)}(dx_{b'}; b' \in \pi' \setminus \{b'_0\}) \right) dx_{b'_0} \\ =: \langle f_i, \bar{F} \rangle_{L^2(\mathbb{R}^d)}. $$

Since $(f_i)_{i \in \mathbb{N}^d}$ is orthogonal and using the Hölder's inequality, we have

$$ \begin{aligned} \sum_{i \in \mathbb{N}^d} |I_{\pi'}|^2 &\le \int_{\mathbb{R}^d} |F(x)|^2 dx = \int_{\mathbb{R}^{2d(|\pi'|-1)}} \int_{\mathbb{R}^d} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b \setminus \{1\}} h_s(x'_{b'} + x_{b'_0}) \overline{h_s(y'_{b'} + x_{b'_0})} \\ &\qquad \gamma_{red}^{(|\pi'|)}(dx_{b'}; b' \in \pi \setminus \{b'_0\}) \overline{\gamma_{red}^{(|\pi'|)}(dy_{b'}; by \in \pi \setminus \{b'_0\})} \\ &\le \int_{\mathbb{R}^{2d(|\pi'|-1)}} \prod_{b' \in \pi'} \prod_{b \in b'} \prod_{s \in b \setminus \{1\}} \|h_s(x'_{b'} + \cdot)\|_{2(m-1)} \|h_s(y'_{b'} + \cdot)\|_{2(m-1)} \\ &\qquad |\gamma_{red}^{(|\pi'|)}|(dx_{b'}; b' \in \pi \setminus \{b'_0\})|\gamma_{red}^{(|\pi'|)}|(dy_{b'}; b' \in \pi \setminus \{b'_0\}) \\ &= |\gamma_{red}^{(|\pi'|)}|(\mathbb{R}^{d(|\pi'|-1)})^2 \prod_{s=2}^{m} \|h_s\|_{2(m-2)}^2. \end{aligned} $$

Accordingly,

$$ \sum_{i \in I} |\kappa_{\mathbb{R}^d}|^2 \le \left( \sum_{\pi \in \Pi[m]} a_{\pi} \sum_{\pi' \in \Pi\{b; b \in \pi\}} |\gamma_{red}^{(|\pi|)}|(\mathbb{R}^{d(|\pi'|-1)}) \right)^2 \prod_{s=2}^{m} \|h_s\|_{2(m-2)}^2, $$

which concludes the proof.

<figure><img src="image_12.png" /></figure>

## E Generalization to multitaper estimators with plug-in intensity

In this section, we consider the practical setting where the intensity is unknown. In that case, we adapt the multitaper estimators of the structure factor as follows.

**Definition 8.** Let $\Phi$ be a stationary point process of $\mathbb{R}^d$, $W$ be a compact set of $\mathbb{R}^d$ and $I$ be a subset of $\mathbb{N}^d$. Let $k_0 \in \mathbb{R}^d$ and $(f_i)_{i \in I} \in L^2(\mathbb{R}^d)^{|I|}$. The multitaper estimator with plug-in intensity is defined as

$$ \hat{S}^{\hat{\lambda}}(k_0) := \frac{1}{\hat{\lambda}|I|} \sum_{i \in I} \hat{S}_i^{\hat{\lambda}}(k_0) 1_{\hat{\lambda}>0} + 01_{\hat{\lambda}=0} \quad (73) $$

where

$$ \hat{S}_i^{\hat{\lambda}}(k_0) := \left| \sum_{x \in \Phi \cap W} e^{-ik_0 \cdot x} f_i(x) - \hat{\lambda} \int_W f_i(x) e^{-ik \cdot x} dx \right|^2, \quad \hat{\lambda} := \frac{\#\{\Phi \cap W\}}{|W|}. $$
**Remark 3.** Note that for estimating the spectra $\lambda S$ instead of the structure factor $S$ (see the discussion following equation (3)), we can use the following estimator

$$ \hat{\lambda} S^{\hat{\lambda}}(k_0) := \frac{1}{|I|} \sum_{i \in I} \hat{S}_i^{\hat{\lambda}}(k_0). $$

The results concerning $\hat{S}^{\hat{\lambda}}$ can be readily adapted to the estimator $\hat{\lambda} \hat{S}^{\hat{\lambda}}$.

The following proposition shows that replacing the true intensity $\lambda$ with its estimator $\hat{\lambda}$ in $\hat{S}^{\hat{\lambda}}$ introduces an additional main error term of order $|I|^{-1/2}$. This stems from the fact that the Fourier transform of the tapers $(\mathcal{F}[f_i])_{i \in I}$ converge to $\delta_0$, causing the integrals $\int_W f_i(x) e^{-ik \cdot x} dx$ to diverge for small frequencies $k$. Note that under the assumptions of Corollary 5.1, this result implies that $\hat{S}^{\hat{\lambda}}$ still remains minimax optimal for the $L^1(\mathbb{P})$ risk.

**Proposition E.1.** Let $\Phi$ be a stationary point process satisfying Assumption 3.1. Let $W$ be a subset of $\mathbb{R}^d$, $I$ be a discrete subset of $\mathbb{N}^d$. We consider a family of orthonormal functions $(f_i)_{i \in I}$ of $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$. Then,

$$ \sup_{k \in \mathbb{R}^d} \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k) - S(k)| \right] \le a_1 \sup_{k \in \mathbb{R}^d} \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k) - S(k)|^2 \right]^{\frac{1}{2}} + \frac{a_2}{|I|^{\frac{1}{2}}} + \frac{a_4}{|W|^{\frac{1}{2}}}, \quad (74) $$

where $(a_n)_{n \in [4]}$ are finite constants (see (78)) provided that $|\gamma_{red}^4| := \lambda + 7|\gamma_{red}^{(2)}|(\mathbb{R}^d) + 6|\gamma_{red}^{(3)}|(\mathbb{R}^{2d}) + |\gamma_{red}^{(4)}|(\mathbb{R}^{3d})$ is finite.

*Proof.* Let $k_0 \in \mathbb{R}^d$. By triangular inequality

$$ \begin{aligned} \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k_0) - S(k_0)| \right] &\le \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k_0) - S(k_0)| \right] + \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k_0) - \hat{S}^{\hat{\lambda}}(k_0)| \right] \\ &\le \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k_0) - S(k_0)| \right] + \mathbb{E} \left[ |\hat{S}^{\hat{\lambda}}(k_0) - \hat{S}^{\hat{\lambda}}(k_0)| 1_{\hat{\lambda}>0} \right] + \mathbb{P}[\hat{\lambda}=0] \mathbb{E}[\hat{S}^{\hat{\lambda}}(k_0)]. \end{aligned} $$

We bound $\mathbb{P}[\hat{\lambda}=0] \mathbb{E}[\hat{S}^{\hat{\lambda}}(k_0)]$. Using (4), we obtain

$$ \mathbb{P}(\hat{\lambda}=0) \le \mathbb{P}[|\hat{\lambda}-\lambda| \ge \lambda] \le \frac{\mathbb{E}[|\hat{\lambda}-\lambda|^2]}{\lambda^2} = \frac{1}{\lambda|W|^2} \int_{\mathbb{R}^d} |\mathcal{F}[1_W](k)|^2 S(k) dk \le \frac{\|S\|_{\infty}}{\lambda|W|}. $$

Moreover, using the Plancherel theorem we get

$$ \mathbb{E}[\hat{S}^{\hat{\lambda}}(k_0)] = \frac{1}{|I|} \sum_{i \in I} \int_{\mathbb{R}^d} |\mathcal{F}[f_i e^{-ik_0 \cdot}](k)|^2 S(k) dk \le \|S\|_{\infty}. $$

Accordingly $\mathbb{P}[\hat{\lambda}=0] \mathbb{E}[\hat{S}^{\hat{\lambda}}(k_0)] \le \|S\|_{\infty}^2 / (\lambda|W|)$. We consider $\mathbb{E}[|\hat{S}^{\hat{\lambda}}(k_0) - \hat{S}^{\hat{\lambda}}(k_0)| 1_{\hat{\lambda}>0}]$. By writing $\hat{\lambda} = \hat{\lambda} - \lambda + \lambda$ in $\hat{S}_i^{\hat{\lambda}}(k_0)$, we obtain when $\hat{\lambda} > 0$, $\hat{S}_i^{\hat{\lambda}}(k_0) - \hat{S}^{\hat{\lambda}}(k_0) = A + B + C$, where

$$ A := \left( \frac{\lambda}{\hat{\lambda}} - 1 \right) \hat{S}^{\hat{\lambda}}(k_0), \quad B = 2 \left( \frac{\lambda}{\hat{\lambda}} - 1 \right) \frac{1}{|I|} \sum_{i \in I} \Re \left( C_i(k_0) \cdot \overline{E_i(k_0)} \right), $$

$$ C := \frac{(\hat{\lambda} - \lambda)^2}{\hat{\lambda}} \frac{1}{|I|} \sum_{i \in I} |E_i(k_0)|^2, \quad E_i := \int_W f_i(x) e^{-ik \cdot x} dx. $$
and where $C_i(k_0)$ is defined in (9). We start with $C$. Since $(f_i)_{i \in I}$ is orthogonal, we have

$$ \frac{1}{|I|} \sum_{i \in I} |E_i(k_0)|^2 \le \frac{1}{|I|} \|1_W\|_2^2 = \frac{|W|}{|I|}. \quad (75) $$

Moreover, by Cauchy-Schwarz inequality, we have

$$ \mathbb{E} \left[ \frac{(\hat{\lambda} - \lambda)^2}{\hat{\lambda}} 1_{\hat{\lambda} > 0} \right] \le \mathbb{E} \left[ (\hat{\lambda} - \lambda)^4 \right]^{\frac{1}{2}} \mathbb{E} \left[ \frac{1}{\hat{\lambda}^2} 1_{\hat{\lambda} > 0} \right]^{\frac{1}{2}}. $$

If $\hat{\lambda} > 0$ then $\#\{\Phi \cap W\} \ge 1$, whence $\hat{\lambda} \ge |W|^{-1}$. Accordingly,

$$ \begin{aligned} \mathbb{E} \left[ \frac{1}{\hat{\lambda}^2} 1_{\hat{\lambda} > 0} \right] &\le \mathbb{E} \left[ (2/\lambda)^2 1_{\hat{\lambda} \ge \lambda/2} \right] + \mathbb{E} \left[ |W|^2 1_{\hat{\lambda} \le \lambda/2} \right] \\ &\le \frac{4}{\lambda^2} + |W|^2 \mathbb{P}[|\hat{\lambda} - \lambda| \ge \lambda/2] \\ &\le \frac{4}{\lambda^2} + \frac{16}{\lambda^4} |W|^2 \mathbb{E}[|\hat{\lambda} - \lambda|^4]. \end{aligned} $$

We express the centered moment of order four as $\mathbb{E}[|\hat{\lambda} - \lambda|^4] = 3\mathbb{E}[|\hat{\lambda} - \lambda|^2]^2 + \kappa_4(\hat{\lambda})$. Using Lemma D.2 we get

$$ \mathbb{E}[|\hat{\lambda} - \lambda|^4] \le \frac{3\lambda^2 \|S\|_{\infty}^2}{|W|^2} + \frac{|\gamma_{\text{red}}^4| \|1_W\|_4^4}{|W|^4} = \frac{3\lambda^2 \|S\|_{\infty}^2}{|W|^2} + \frac{|\gamma_{\text{red}}^4|}{|W|^3}. \quad (76) $$

Accordingly,

$$ \mathbb{E} \left[ \frac{1}{\hat{\lambda}^2} 1_{\hat{\lambda} > 0} \right] \le \frac{48 \|S\|_{\infty}^2}{\lambda^2} + \frac{16 |\gamma_{\text{red}}^4|}{\lambda^4 |W|} + \frac{4 \|S\|_{\infty}^2}{\lambda^2 |W|^2} + \frac{4 |\gamma_{\text{red}}^4|}{\lambda^2 |W|^3} := b_1 \quad (77) $$

This yields,

$$ C \le \frac{1}{|I|} \left( 3\lambda^2 \|S\|_{\infty}^2 + \frac{|\gamma_{\text{red}}^4|}{|W|} \right)^{\frac{1}{2}} b_1^{\frac{1}{2}}. $$

Now, we consider A. Using the Hölder-inequality we have

$$ \begin{aligned} A &\le \|S\|_{\infty} \mathbb{E} \left[ \frac{|\hat{\lambda} - \lambda|}{\hat{\lambda}} 1_{\hat{\lambda} > 0} \right] + \mathbb{E} \left[ |\hat{S}^{\lambda}(k_0) - S(k_0)| \frac{|\hat{\lambda} - \lambda|}{\hat{\lambda}} 1_{\hat{\lambda} > 0} \right] \\ &\le \|S\|_{\infty} \mathbb{E} \left[ |\hat{\lambda} - \lambda|^2 \right]^{\frac{1}{2}} \mathbb{E} \left[ \frac{1_{\hat{\lambda} > 0}}{\hat{\lambda}^2} \right]^{\frac{1}{2}} + \left[ |\hat{\lambda} - \lambda|^4 \right]^{\frac{1}{4}} \mathbb{E} \left[ \frac{1_{\hat{\lambda} > 0}}{\hat{\lambda}^4} \right]^{\frac{1}{4}} \mathbb{E} \left[ |\hat{S}^{\lambda}(k_0) - S(k_0)|^2 \right]^{\frac{1}{2}}. \end{aligned} $$

For the first term, we have using (77)

$$ \mathbb{E} \left[ |\hat{\lambda} - \lambda|^2 \right]^{\frac{1}{2}} \mathbb{E} \left[ \frac{1_{\hat{\lambda} > 0}}{\hat{\lambda}^2} \right]^{\frac{1}{2}} \le \frac{\|S\|_{\infty}}{\lambda^{\frac{1}{2}} |W|^{\frac{1}{2}}} b_1^{\frac{1}{2}}. $$

For the second one, using again the fact that if $\hat{\lambda} > 0$, then $\hat{\lambda} \ge |W|^{-1}$, we obtain

$$ \mathbb{E} \left[ \frac{1}{\hat{\lambda}^4} 1_{\hat{\lambda} > 0} \right] \le \frac{16}{\lambda^4} \left( 1 + |W|^4 \mathbb{E}[|\hat{\lambda} - \lambda|^4] \right). $$
Using equation (76), we obtain after computations

$$ \mathbb{E}\left[\frac{1}{\hat{\lambda}^4} \mathbf{1}_{\hat{\lambda}>0}\right] \mathbb{E}[|\hat{\lambda} - \lambda|^4] \le 48 \|S\|_{\infty}^4 + \frac{96 \|S\|_{\infty}^2 |\gamma_{\text{red}}^4|}{\lambda^2 |W|} + \frac{48 \|S\|_{\infty}^2}{\lambda^2 |W|^2} + \frac{16 |\gamma_{\text{red}}^4|^2}{|W|^2 \lambda^4} + \frac{16 |\gamma_{\text{red}}^4|}{\lambda^4 |W|^3}. $$

This yields, after numerical evaluations

$$ A \le (3\|S\|_{\infty} + \varepsilon_W) \mathbb{E} \left[ |\hat{S}^{\lambda}(k_0) - S(k_0)|^2 \right]^{\frac{1}{2}} + \frac{\|S\|_{\infty}}{\lambda^{\frac{1}{2}}|W|^{\frac{1}{2}}} b_1^{\frac{1}{2}} $$

where

$$ \varepsilon_W := \frac{4 \|S\|_{\infty}^{\frac{1}{4}} |\gamma_{\text{red}}^4|^{\frac{1}{4}}}{\lambda^{\frac{1}{2}} |W|^{\frac{1}{4}}} + \frac{3 \|S\|_{\infty}^{\frac{1}{2}}}{\lambda^{\frac{1}{2}} |W|^{\frac{1}{2}}} + \frac{2 |\gamma_{\text{red}}^4|^{\frac{1}{2}}}{|W|^{\frac{1}{2}} \lambda} + \frac{2 |\gamma_{\text{red}}^4|^{\frac{1}{4}}}{\lambda |W|^{\frac{3}{4}}}. $$

It remains to bound $B$. Using twice the Hölder-inequality, we obtain

$$ \begin{aligned} B &\le 2 \mathbb{E} \left[ \frac{1}{\hat{\lambda}^2} \mathbf{1}_{\hat{\lambda}>0} \right]^{\frac{1}{2}} \mathbb{E} [ |\hat{\lambda} - \lambda|^4 ]^{\frac{1}{4}} \mathbb{E} \left[ \left( \frac{1}{|I|} \sum_{i \in I} |C_i(k_0)| |E_i(k_0)| \right)^4 \right]^{\frac{1}{4}} \\ &\le 2 \mathbb{E} \left[ \frac{1}{\hat{\lambda}^2} \mathbf{1}_{\hat{\lambda}>0} \right]^{\frac{1}{2}} \mathbb{E} [ |\hat{\lambda} - \lambda|^4 ]^{\frac{1}{4}} \mathbb{E} \left[ \left( \frac{1}{|I|} \sum_{i \in I} |C_i(k_0)|^2 \right)^2 \right]^{\frac{1}{4}} \left( \frac{1}{|I|} \sum_{i \in I} |E_i(k_0)|^2 \right)^{\frac{1}{2}}. \end{aligned} $$

Recalling (75), (76) and (77), we obtain

$$ \begin{aligned} B &\le \frac{2 \mathbb{E}[\hat{S}^{\lambda}(k_0)^2]^{\frac{1}{4}}}{|I|^{\frac{1}{2}}} \left( 3\lambda^2 \|S\|_{\infty}^2 + \frac{|\gamma_{\text{red}}^4|}{|W|} \right)^{\frac{1}{4}} b_1^{\frac{1}{2}} \\ &\le \left( 3\lambda^2 \|S\|_{\infty}^2 + \frac{|\gamma_{\text{red}}^4|}{|W|} \right)^{\frac{1}{4}} b_1^{\frac{1}{2}} \left( \frac{2^{\frac{3}{2}} \|S\|_{\infty}^{\frac{1}{2}}}{|I|^{\frac{1}{2}}} + 2^{\frac{1}{2}} \mathbb{E}[|\hat{S}^{\lambda}(k_0) - S(k_0)|^2]^{\frac{1}{2}} + \frac{2^{\frac{1}{2}}}{|I|} \right). \end{aligned} $$

Gathering the bounds on $A$, $B$ and $C$, we conclude the proof of (74) with

$$ \begin{aligned} b_1 &:= \frac{48 \|S\|_{\infty}^2}{\lambda^2} + \frac{16 |\gamma_{\text{red}}^4|}{\lambda^4 |W|} + \frac{4 \|S\|_{\infty}^2}{\lambda^2 |W|^2} + \frac{4 |\gamma_{\text{red}}^4|}{\lambda^2 |W|^3}, & b_2 &:= 3\lambda^2 \|S\|_{\infty}^2 + \frac{|\gamma_{\text{red}}^4|}{|W|}, \\ a_1 &:= 1 + 3 \|S\|_{\infty} + \varepsilon_W + \sqrt{2} b_2^{\frac{1}{4}} b_1^{\frac{1}{2}}, & a_2 &:= 2^{\frac{3}{2}} \|S\|_{\infty}^{\frac{1}{2}} b_2^{\frac{1}{4}} b_1^{\frac{1}{2}} + (b_2^{\frac{1}{2}} + \sqrt{2} b_2^{\frac{1}{4}}) b_1^{\frac{1}{2}}, \\ a_4 &:= \frac{\|S\|_{\infty} b_1^{\frac{1}{2}}}{\lambda^{\frac{1}{2}}} + \frac{\|S\|_{\infty}}{\lambda |W|^{\frac{1}{2}}}. \end{aligned} \quad (78) $$

□

# F Results concerning the Hermite functions

The next lemma provides bounds on the tail of the Hermite functions.
**Lemma F.1.** Consider a finite set of indexes I of $\mathbb{N}^d$ and denote $M_I = \max\{i_1, \dots, i_d | i = (i_1, \dots, i_d) \in I\}$. Let $\rho \in (0, \infty)$ be such that $\rho > \sqrt{2(M_I + 1) + (M_I + 1)^{1/3+\theta}}$, for $\theta > 0$. Denote $W = [-\rho, \rho]^d$. Then, there exist two constants $0 < c, C < \infty$ such that

$$ \sup_{i \in I} \|\psi_i \mathbf{1}_{W^c}\|_1 + \sup_{i \in I} \|\psi_i \mathbf{1}_{W^c}\|_2^2 \le Ce^{-c\rho^{3\theta}}, \quad (79) $$

*Proof.* Recall that the $d$-dimensional Hermite functions are defined as the product of one-dimensional ones. Using the inequality:

$$ \forall x = (x_1, \dots, x_d) \in \mathbb{R}^d, \quad 1 - \prod_{i=1}^d \mathbf{1}_{|x_i| \le \rho} \le \sum_{l=1}^d \mathbf{1}_{|x_l| > \rho}, $$

and the fact that one-dimensional Hermite functions have a unit $L^2(\mathbb{R}^d)$-norm, we reduce the proof to the case $d=1$. Let $n \le M_I$. We start with the case $\rho \ge \sqrt{2\sqrt{2n+1}}$. According to Theorem B of [3], we have for all $x \in \mathbb{R}$ with $|x| \ge \rho$, $|\psi_n(x)| \le Ce^{-cx^2}$, for some finite positive constants $c$ and $C$, that will change from line to line in the following. Using the standard upper bound of the tail of a Gaussian random variable, we obtain:

$$ \|\psi_n \mathbf{1}_{|\cdot| \ge \rho}\|_1 + \|\psi_n \mathbf{1}_{|\cdot| \ge \rho}\|_2^2 \le Ce^{-c\rho^2}. $$

Consider now the case $\rho \le \sqrt{2\sqrt{2n+1}}$. Since $\rho \ge \sqrt{2(n+1) + (n+1)^{1/3+\theta}}$, according to the bound on equation (2.13) of the proof of Theorem 2.3 of [2], we have

$$ \|\psi_n \mathbf{1}_{|\cdot| \ge \rho}\|_1 + \|\psi_n \mathbf{1}_{|\cdot| \ge \rho}\|_2^2 \le Ce^{-cn^{3\theta/2}} \le Ce^{-c\rho^{3\theta}}. $$

This concludes the proof.

<figure><img src="image_13.png" /></figure>

The next lemma studies the Sobolev norms of the Hermite functions.

**Lemma F.2.** Assume that $I = \{i \in \mathbb{N}^d | |i|_{\infty} < i_{\max}\}$ with $i_{\max} \ge 1$. Let $\beta \in (0, 2]$. Then

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_{H^{\beta/2}}^2 \le |I|^{\beta/(2d)}. \quad (80) $$

*Proof.* According to Hölder's inequality, we have, since $\beta \le 2$,

$$ \begin{aligned} \int_{\mathbb{R}^d} |\psi_i(k)|^2 |k|^{\beta} dk &\le \|\psi_i\|_2^{1-\beta/2} \left( \int_{\mathbb{R}^d} |\psi_i(k)|^2 |k|^2 dk \right)^{\beta/2} \\ &= \left( \frac{1}{2} \sum_{s=1}^d (2i_s + 1) \right)^{\beta/2} \le \frac{1}{2^{\beta/2}} \sum_{s=1}^d (2i_s + 1)^{\beta/2}, \end{aligned} $$

where we used the inclusion of $\ell^1([d])$ in $\ell^{2/\beta}([d])$ (since $2/\beta \ge 1$). This yields (80):

$$ \begin{aligned} \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_{H^{\frac{1}{2}}}^2 &\le \frac{1}{2^{\beta/2} |I|} \sum_{0 \le i_1, \dots, i_d < i_{\max}} \sum_{s=1}^d (2i_s + 1)^{\beta/2} = \frac{i_{\max}^{d-1}}{2^{\beta/2} |I|} \sum_{s=1}^{i_{\max}-1} (2s + 1)^{\beta/2} \\ &\le \frac{i_{\max}^{d-1}}{2^{\beta/2} |I|} \sum_{s=1}^{i_{\max}-1} (2i_{\max})^{\beta/2} = |I|^{\beta/(2d)}. \end{aligned} $$

<figure><img src="image_14.png" /></figure>
The following lemma provides upper bounds on averages of $L^p(\mathbb{R}^d)$ norms of the Hermite functions. The key point is to control the scaling with respect to both the number of tapers $|I|$ and the order $p$ of the norms.

**Lemma F.3.** Assume that $I = \{i \in \mathbb{N}^d | |i|_{\infty} < i_{\max}\}$ with $i_{\max} \ge 1$. Then, there exists a numerical constant $c_{\psi} \in (0, \infty)$ such that,

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_3^2 \le c_{\psi}^d \frac{1}{|I|^{\frac{1}{6}}}. \quad (81) $$

Moreover, for $p \ge 4$, we have

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_p^2 \le c_{\psi}^d \frac{\log(|I|^{\frac{1}{d}})^{\frac{2d}{p}}}{|I|^{\frac{1}{3p}+\frac{1}{6}}}, \quad (82) $$

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_p \le c_{\psi}^d \frac{\log(|I|^{\frac{1}{d}})^{\frac{d}{p}}}{|I|^{\frac{1}{6p}+\frac{1}{12}}}. \quad (83) $$

*Proof.* The proof of this lemma relies on the following results (see equation (3.2) of [2]). For all $p < 4$, there exists $c_p < \infty$ such that

$$ \forall n \ge 0, \quad \int_{\mathbb{R}} |\psi_n(x)|^p dx \le \frac{c_p}{(n+1)^{\frac{p}{4}-\frac{1}{2}}}, \quad (84) $$

for $p = 4$, there exists $c_4 < \infty$ such that

$$ \forall n \ge 0, \quad \int_{\mathbb{R}} |\psi_n(x)|^4 dx \le \frac{c_4 \log(n+1)}{(n+1)^{\frac{1}{2}}}, \quad (85) $$

and for $p > 4$, there exists $c_p < \infty$ such that

$$ \forall n \ge 0, \quad \int_{\mathbb{R}} |\psi_n(x)|^p dx \le \frac{c_p}{(n+1)^{\frac{p}{12}+\frac{1}{6}}}, \quad (86) $$

We start by proving (81). According to (84) we have

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_3^2 \le c_3^{\frac{2d}{3}} \frac{1}{|I|} \sum_{i \in I} \prod_{s=1}^{d} \frac{1}{(i_s + 1)^{1/6}} = c_3^{\frac{2d}{3}} \frac{1}{|I|} \left( \sum_{s=0}^{i_{\max}-1} \frac{1}{(s+1)^{1/6}} \right)^d $$

A comparison sum-integral yields the result:

$$ \frac{1}{|I|} \sum_{i \in I} \|\psi_i\|_4^2 \le c_3^{\frac{2d}{3}} \frac{1}{|I|} \left( \int_0^{i_{\max}} \frac{ds}{(s+1)^{1/6}} \right)^d \le c_4^{\frac{2d}{3}} \frac{1}{|I|} \left( \frac{6}{5} (2i_{\max})^{5/6} \right)^d \le c_{\psi}^d \frac{1}{|I|^{1/4}}, $$

where $c_{\psi} \ge 62^{5/6} c_2^{2/3}/5$. The subtlety of (82) is that the constant $c_{\psi}$ does not depend on $p$. Theorem 1 of [13] ensures that there exists $D \in (1, \infty)$ such that

$$ \forall n \ge 0, \quad \|\psi_n\|_{\infty} \le \frac{D}{(n+1)^{1/12}}. \quad (87) $$
Hence, for $n \ge 0$ and $p \ge 4$, using (85) and (87), we have

$$ \int_{\mathbb{R}} |\psi_n(x)|^p dx = \int_{\mathbb{R}} |\psi_n(x)|^{p-4} |\psi_n(x)|^4 dx \\ \le D^{p-4} c_2 \frac{\log(n+1)}{(n+1)^{(p-4)/12}(n+1)^{1/2}} \le (Dc_2)^p \frac{\log(n+1)}{(n+1)^{p/12+1/6}}. $$

Finally, concluding as in the proof of (81) we obtain (82) and (83) with $c_{\psi} = (1 + Dc_2)^2$. $\square$

# G Results related to cumulants

## G.1 Isserlis-Wick type formulae

We denote by $\mathbf{1}_n$ the single block partition of $[n]$, i.e. $\mathbf{1}_n := 12...n$. For two partitions $\sigma$ and $\tau$, $\sigma \vee \tau$ denotes their least upper bound (refer to Chapter 3 of [54] or Chapter 2 of [61]). The next result is a direct application of equation (3.3) of [54].

**Corollary G.1.** Let $Y_1, \dots, Y_m$ be a family of complex valued random variables. Then,

$$ \kappa_m(|Y_1|^2, \dots, |Y_m|^2) = \sum_{\substack{\sigma \in \Pi[2m], \\ \sigma \vee \tau = \mathbf{1}_{2m}, |\sigma| \ge 2}} \prod_{b \in \sigma} \kappa(Z_i; i \in b), \quad (88) $$

where the sum runs over the partition of $[2m]$ and for $i \in \{1, \dots, m\}$, $Z_{2i-1} = Y_i$ and $Z_{2i} = \bar{Y}_i$ and $\tau = 12|34|\dots|(2m-1)2m$.

## G.2 Extension of the result of Bentkus and Rudzkis under multiple Statulevičius conditions

The proof of the concentration inequality stated in Theorem 5.2 relies on the following lemma, which provides a tool to derive non-asymptotic deviation inequalities from cumulant bounds. This result is a straightforward extension of a classical inequality [10], corresponding to the case $J = 1$. Although this generalization is natural and may be known to experts, we have not found it explicitly stated in the literature. Such an extension is required to handle the three distinct contributions to the fluctuations of the multitaper estimator of the structure factor, as discussed in Section 5.3.

**Theorem G.1.** Let $X$ be a real random variable. Suppose that $\mathbb{E}[X] = 0$. Let $J \ge 1$, $(\gamma_i)_{1 \le i \le J} \in [0, \infty)^J$, $(H_i, \Delta_i)_{1 \le i \le J} \in (0, \infty)^{2J}$. Assume that, for all $m \ge 2$,

$$ |\kappa_m(X)| \le \sum_{j=1}^{J} \left(\frac{m!}{2}\right)^{1+\gamma_j} \frac{H_j}{\Delta_j^{m-2}}. \quad (89) $$

Then, for all $x > 0$, we have

$$ \mathbb{P}(X \ge x) \le C \max_{j=1,\dots,J} \exp \left( -\frac{1}{2J} \frac{x^2}{H_j + x^{2-\alpha_j/\Delta_j^{\alpha_j}}} \right), \quad (90) $$

where $\alpha_j := 1/(1 + \gamma_j)$ and $C < \infty$.
Proof. For simplicity, we follow the proof of Theorem 2.5 of Section 5 of [25] to prove (90). At a price of more technicality, the following proof could be adapted to prove (90) with $C = 1$ (see [71], proof of Lemma 2.4, pages 37-41). For $n \ge 1$, we denote the truncated exponential sum as

$$ \exp_n(x) := \sum_{m=0}^{n} \frac{1}{m!} x^m. $$

Let $h > 0$ and $n \ge 1$ that will be specified during the proof. Since $\exp_{2n}$ is increasing over $\mathbb{R}^+$, we have by Markov's inequality

$$ \mathbb{P}(X \ge x) \le \frac{\mathbb{E}[\exp_{2n}(hX)]}{\exp_{2n}(hx)}. $$

Then, according to equation (5.2) of [25],

$$ \mathbb{E}[\exp_{2n}(hX)] \le \exp_n \left( \sum_{m=2}^{2n} \frac{h^m}{m!} |\kappa_m(X)| \right) := \exp_n(A). $$

Let $h_j = x/(H_j + x^{2-\alpha_j}/\Delta_j^{\alpha_j})$ and $h = (\sum_{j=1}^J 1/h_j)^{-1}$. Using (89), we have

$$ A \le \frac{h^2}{2} \sum_{j=1}^{J} H_j \sum_{m=2}^{2n} \left( \frac{m!}{2} \right)^{\gamma_j} \left( \frac{h}{\Delta_j} \right)^{m-2}. $$

Then, assume that $2n \le hx$. We use $m!/2 \le (2n)^{m-2}$ for $m \le 2n$ (see [25]), to get

$$ A \le \frac{h^2}{2} \sum_{j=1}^{J} H_j \sum_{m=2}^{2n} \left( (hx)^{\gamma_j} \frac{h}{\Delta_j} \right)^{m-2} \le \frac{h^2}{2} \sum_{j=1}^{J} H_j \frac{1}{1 - (hx)^{\gamma_j} \frac{h}{\Delta_j}}. $$

Note that

$$ (hx)^{\gamma_j} \frac{h}{\Delta_j} = \frac{(xh)^{1+\gamma_j}}{x\Delta_j} \le \frac{(xh_j)^{1+\gamma_j}}{x\Delta_j} = \left( \frac{x^2}{x^2 + H_j(\Delta_j x)^{1/(1+\gamma_j)}} \right)^{1+\gamma_j} < 1. $$

Hence,

$$ 1 - (hx)^{\gamma_j} \frac{h}{\Delta_j} \ge 1 - \frac{x^2}{x^2 + H_j(\Delta_j x)^{1/(1+\gamma_j)}} = \frac{H_j(\Delta_j x)^{1/(1+\gamma_j)}}{x^2 + H_j(\Delta_j x)^{1/(1+\gamma_j)}} = \frac{H_j h_j}{x}. $$

Subsequently,

$$ A \le \frac{h^2}{2} \sum_{j=1}^{J} \frac{x}{h_j} = \frac{hx}{2}, $$

and

$$ \mathbb{P}(X \ge x) \le \frac{\exp_n(hx/2)}{\exp_{2n}(hx)}. $$

Arguing exactly as in the end of proof of Theorem 2.5 of [25] (from equation (5.5)), we get

$$ \mathbb{P}(X \ge x) \le C e^{-hx/2} \le C \max_{j=1, \dots, J} \exp \left( -\frac{1}{2J} \frac{x^2}{H_j + x^{2-\alpha_j}/\Delta_j^{\alpha_j}} \right), $$

for some numerical constant $C > 0$. $\square$