existence of a constant $C_{\eta}$ such that for $\underline{x}_{A}\in
K_{\mathcal{J},\varepsilon}$,

$\displaystyle\left|\det\Sigma^{\mathcal{I}}(\underline{x}_{A})-\det\,(\Sigma^{\mathcal{I}}(\underline{x}_{A}))_{\mathcal{J}}\right|$
$\displaystyle=\left|\det\Sigma^{\mathcal{I}}(\underline{x}_{A})-\prod_{J\in\mathcal{J}}\det\Sigma^{\mathcal{I}_{J}}(\underline{x}_{J})\right|$
$\displaystyle\leq C_{\eta}\varepsilon,$

and thus

$\det\Sigma^{\mathcal{I}}(\underline{x}_{A})\geq\prod_{J\in\mathcal{J}}\det\Sigma^{\mathcal{I}_{J}}(\underline{x}_{J})-C_{\eta}\varepsilon.$

For all $J\in\mathcal{J}$, the set $J$ is a strict subset of $A$. Moreover, if
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$ then
$\underline{x}_{J}\in\Delta_{\mathcal{I}_{J},\eta}$. By induction hypothesis,
one can find a positive constant $c_{\eta}$ depending only on $\eta$ such that
$\det\Sigma^{\mathcal{I}_{J}}(\underline{x}_{J})\geq c_{\eta}$ when
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. It implies that

$\forall\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}\cap
K_{\mathcal{J},\varepsilon},\quad\det\Sigma^{\mathcal{I}}_{n}(\underline{x}_{A})\geq(c_{\eta})^{|\mathcal{J}|}-C\varepsilon.$

Taking $\varepsilon$ small enough and gathering the case $\mathcal{J}=\\{A\\}$
in and $\mathcal{J}\neq\\{A\\}$, the conclusion follows. ∎

##### Proof of Proposition 4.1.

###### Proof.

A reformulation of hypothesis $H_{1}(q)$ applied to the compact set
$[-T_{\varepsilon},T_{\varepsilon}]$ yields

$\lim_{n\rightarrow+\infty}\sup_{\begin{subarray}{c}s,t\in nU\\\ |s-t|\leq
T_{\varepsilon}\end{subarray}}\left|r_{n}^{(u,v)}(s,t)-\psi\left(\frac{s}{n}\right)r_{\infty}^{(u,v)}(s,t)\right|=0.$

The function $\psi$ is uniformly continuous by hypothesis and we can define
$\omega_{\psi}$ its uniform modulus of continuity. By hypothesis, there are
positive constants $c_{\psi}$ and $C_{\psi}$ such that for all $x\in U$,

$c_{\psi}\leq\psi(x)\leq C_{\psi}.$

Let $n\in\overline{\mathfrak{N}}$. If $s,t\in nU$ and $|t-s|>T_{\varepsilon}$
then hypothesis $H_{2}(q)$ implies that for $u,v\leq|A|-1$,

$|r_{n}^{(u,v)}(s,t)|\leq\varepsilon.$

Gathering (4.1) and (4.1), there is $n_{\varepsilon}\in\mathfrak{N}$ such that
for $n\geq n_{\varepsilon}$, and $s,t\in nU$

$\left|r_{n}^{(u,v)}(s,t)-\psi\left(\frac{s}{n}\right)r_{\infty}^{(u,v)}(s,t)\right|\leq\varepsilon(1+C_{\psi}).$

Let $n\in\overline{\mathfrak{N}}$ with $n\geq n_{\varepsilon}$ and
$\underline{x}_{A}\in\Delta_{I,\eta}$. For $I,J\in\mathcal{I}$, $a\in I$ and
$b\in J$ one has from Lemma 2.18

$\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})_{ab}=\int_{C_{a|I}\times
C_{b|J}}r_{n}^{(|\,a|I\,|-1\,,\,|\,b|J\,|-1)}\left(\sum_{i\in
a|I}s_{i}x_{i},\sum_{j\in
b|J}t_{j}x_{j}\right)\mathrm{d}m(\underline{s}_{a|I})\mathrm{d}m(\underline{t}_{b|J}).$

Inequality (4.1) implies

$\left|\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})_{ab}-\psi\left(\frac{x_{a}}{n}\right)\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})_{ab}\right|\leq\varepsilon(1+C_{\psi})+R_{n}(a,b),$

where

$\displaystyle R_{n}(a,b)$ $\displaystyle=\int_{C_{a|I}\times
C_{b|J}}\left|\psi\left(\frac{x_{a}}{n}\right)-\psi\left(\frac{\sum_{i\in
a|I}s_{i}x_{i}}{n}\right)\right|\mathrm{d}m(\underline{s}_{a|I})\mathrm{d}m(\underline{t}_{b|J})$
$\displaystyle\leq C\sup_{\underline{s}_{a|I}\in
C_{a|I}}\left|\psi\left(\frac{x_{a}}{n}\right)-\psi\left(\frac{\sum_{i\in
a|I}s_{i}x_{i}}{n}\right)\right|.$

For $i\in I$, one has $|x_{a}-x_{i}|\leq|A|\eta$. It implies that for any
convex combination $y$ of the variables $(x_{i})_{i\in I}$ one also have
$|x_{a}-y|\leq|A|\eta$. We deduce that

$R_{n}(a,b)\leq C\omega_{\psi}\left(\frac{|A|\eta}{n}\right).$

There is $n_{\eta,\varepsilon}$ such that for $n\geq n_{\eta,\varepsilon}$,

$R_{n}(a,b)\leq\varepsilon,$

and thus coming back to inequality (4.1),

$\left|\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})_{ab}-\psi\left(\frac{x_{a}}{n}\right)\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})_{ab}\right|\leq\varepsilon(2+C_{\psi}).$

It implies the existence of a constant $C_{\eta}$ such that for $n$ large
enough and $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$,

$\left|\det\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})-\left(\prod_{a\in
A}\psi\left(\frac{x_{a}}{n}\right)\right)\det\Sigma^{\mathcal{I}}_{\infty}(\underline{x}_{A})\right|\leq
C_{\eta}\varepsilon.$

We deduce that

$\det\Sigma_{n}^{\mathcal{I}}(\underline{x}_{A})\geq
c_{\psi}^{|A|}\det\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})-C_{\eta}\varepsilon.$

The conclusion follows from the previous Lemma 4.2 covering the stationary
case, and taking $\varepsilon$ small enough. ∎

As a consequence of the previous Proposition 4.1, we deduce the following
corollary about convergence of the Kac density associated with the process
$f_{n}$.

######  Corollary 4.3.

Assume that the sequence of processes $(f_{n})_{n\in\overline{\mathfrak{N}}}$
satisfies hypotheses $H_{1}(q)$ and $H_{2}(q)$ defined in ($\bullet$ ‣ 1.1.2)
and ($\bullet$ ‣ 1.1.2), with $q=2|A|-1$. Then there is a compact set
$K_{\eta}$ of $\mathcal{S}_{2A}^{+}(\mathbb{R})$ such that for all
$n\in\overline{\mathfrak{N}}$ and
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$,
$\Sigma^{\mathcal{I}}_{n}(\underline{x}_{A,A})\in K_{\eta}.$ In particular we
have the following convergence, uniformly for $x\in U$ and $\underline{t}_{A}$
in a compact subset of $\mathbb{R}^{A}$.
$\lim_{n\rightarrow+\infty}\rho_{n}(nx+\underline{t}_{A})=\rho_{\infty}(\underline{t}_{A})\quad\text{and}\quad\lim_{n\rightarrow+\infty}F_{A,n}(nx+\underline{t}_{A})=F_{A,\infty}(\underline{t}_{A}).$

###### Proof.

The first assertion is a direct application of Proposition 4.1 with the set
$2A$, using the fact that $\underline{x}_{A,A}\in\Delta_{\mathcal{I},\eta}$
when $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. As for the second one,
hypothesis $H_{1}(q)$ and the stationarity of the limit process $f_{\infty}$
implies that for all partition $\mathcal{I}$ of $A$, one has the following
convergence, uniformly for $x\in U$ and $\underline{t}_{A}$ in a compact
subset of $\mathbb{R}^{A}$

$\lim_{n\rightarrow+\infty}\Sigma^{\mathcal{I}}_{n}(nx+\underline{t}_{A,A})=\psi(x)\Sigma^{\mathcal{I}}_{\infty}(\underline{t}_{A,A}).$

The conclusion follows from the alternative expression for $\rho_{n}$ given by
Lemma 3.6. Note that the function $\rho_{\infty}$ does not depends on the
function $\psi(x)$, by a change of variable. ∎

###  4.2 Asymptotics of the cumulants

Let $A$ be a finite set of cardinal $p$. We assume that the sequence of
processes $(f_{n})_{n\in\overline{\mathfrak{N}}}$ satisfies hypotheses
$H_{1}(q)$ and $H_{2}(q)$ defined in ($\bullet$ ‣ 1.1.2) and ($\bullet$ ‣
1.1.2), with $q=2p-1$. We then choose $\eta=\frac{\varepsilon}{2p}$ where
$\varepsilon$ is the parameter of hypothesis $H_{2}(q)$, so that

$g_{\varepsilon}(x)=\sup_{|u|\leq 2\eta p}g(x+u).$

####  4.2.1 Decay of the cumulant Kac density

Let us now translate Lemma 3.22 to the cumulant Kac density $F_{A,n}$. The
previous Corollary 4.3 ensures that the matrix
$\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$ lives in a compact subset of
$\mathcal{S}_{2A}^{+}(\mathbb{R})$ when
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$.

######  Lemma 4.4.

There is a constant $C$ such that for all $\underline{x}_{A}\in(nU)^{A}$,
$\left|F_{A,n}(\underline{x}_{A})\right|\leq
C\sum_{G\in\mathcal{G}_{A}}\prod_{\\{a,b\\}\in
E(G)}g_{\varepsilon}(x_{a}-x_{b}).$

###### Proof.

Let $\mathcal{I}$ be a partition of $A$ and
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. According to Corollary 4.3,
the matrix $\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$ lives in a compact
subset of $\mathcal{S}_{2A}^{+}(\mathbb{R})$ depending only on the parameter
$\eta$. By Proposition 2.17, the element
$(\widetilde{M}^{\mathcal{I}}(\underline{x}_{A}),\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A}))$
also lives in a compact subset of
$\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R})$ that depends only on $\eta$. We
can then apply Lemma 3.22 with
$\Sigma=\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$ and
$(\widetilde{M},\widetilde{Q})=(\widetilde{M}^{\mathcal{I}}(\underline{x}_{A}),\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A}))$.
Given the representation formula for $F_{A}$ given by Lemma 3.12, we deduce
the existence of a constant $C_{\eta}$ such that for all
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$,

$\left|F_{A}(\underline{x}_{A})\right|\leq
C_{\eta}\sum_{G\in\mathcal{G}_{\mathcal{I}}}\prod_{\\{I,J\\}\in
E(G)}\|(\Sigma^{\mathcal{I}}(\underline{x}_{A,A}))_{I,J}\|.$

Let $H\in\mathcal{G}_{\mathcal{I}}$. According to Lemma 3.3.1, there is a
graph $G\in\mathcal{G}_{A}$ such that $G_{\mathcal{I}}=H$. If we remove the
edges $\\{a,b\\}$ of $G$ such that $[a]_{\mathcal{I}}=[b]_{\mathcal{I}}$, then
there is a bijection between the edges of $G$ and the edges of $H$ given by
the mapping

$\\{a,b\\}\longrightarrow\\{[a]_{\mathcal{I}},[b]_{\mathcal{I}}\\}.$

Let $\\{a,b\\}$ an be edge of the graph $G$.

  * •

If $[a]_{\mathcal{I}}=[b]_{\mathcal{I}}$ then $|x_{a}-x_{b}|\leq A\eta$. We
deduce that

$0<r_{\infty}(0)\leq g(0)\leq g_{\varepsilon}(x_{a}-x_{b}).$

  * •

If $[a]_{\mathcal{I}}\neq[b]_{\mathcal{I}}$ then from the Hermite-Genocchi
formula and Lemma 2.18,

$\|(\Sigma^{\mathcal{I}}(\underline{x}_{A,A}))_{I,J}\|\leq\sup_{i\in[a]_{\mathcal{I}},\,j\in[b]_{\mathcal{I}}}g(x_{i}-x_{j})\leq
g_{\varepsilon}(x_{a}-x_{b}).$

We deduce that

$\displaystyle\prod_{\\{I,J\\}\in
E(H)}\|(\Sigma^{\mathcal{I}}(\underline{x}_{A,A}))_{I,J}\|$
$\displaystyle\leq\prod_{\begin{subarray}{c}\\{a,b\\}\in E(G)\\\
[a]_{\mathcal{I}}\neq[b]_{\mathcal{I}}\end{subarray}}g_{\varepsilon}(x_{a}-x_{b})$
$\displaystyle\leq C\prod_{\\{a,b\\}\in E(G)}g_{\varepsilon}(x_{a}-x_{b}).$

We deduce the existence of a constant $C_{\eta}$ such that for
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$,

$\left|F_{A}(\underline{x}_{A})\right|\leq
C_{\eta}\sum_{G\in\mathcal{G}_{A}}\prod_{\\{a,b\\}\in
E(G)}g_{\varepsilon}(x_{a}-x_{b}).$

The inequality is true for every partition $\mathcal{I}$ of $A$ and the
conclusion follows. ∎

####  4.2.2 Convergence of the error term towards zero

Recall from Definition (1.1.2) that $\nu_{n}$ is the random counting measure
of the zero set of the Gaussian process $f_{n}(n\,.\,)$ defined on $U$. The
previous Lemma 4.4 and the formula for the $p$-th cumulant given by
Proposition 3.5 shows that the convergence of the cumulant reduces to the
behavior of the quantity

$I_{n}(G)=\int_{(nU)^{A}}\prod_{a\in
A}\left|\phi_{a}\left(\frac{\underline{x}_{A}}{n}\right)\right|\prod_{e=\\{i,j\\}\in
G}g_{e}(x_{i}-x_{j})\mathrm{d}\underline{x}_{A},$

where $G$ is a $2$-edge connected graph with set of vertices $A$ and set of
edges $E(G)$, $(\phi_{a})_{a\in A}$ are functions in $L^{(\infty)}(U)$ and
$(g_{e})_{e\in E(G)}$ are functions in $L^{2}\cap L^{\infty}$.

The quantity $I_{n}(G)$, in the context of cumulants asymptotics, is somehow
reminiscent of a theorem of Szegő (see [6] and the references therein), where
this kind of integral received a thorough treatment and Hölder bounds that
depends on the structure of the graph were given. Nevertheless our setting is
not exactly the same, and we were able to give a simple and self contained
argument, that relies only on a basic interpolation inequality for Hölder
norms, which proves an Hölder type bound for the quantity $I_{n}(G)$. The
result is contained in the following lemma.

######  Lemma 4.5.

Assume that for all $e\in E(G)$, $g_{e}\in L^{\frac{p}{p-1}}$. Then for every
$e\in E(G)$, there is a number $\alpha_{e}\geq p/(p-1)$ such that
$\frac{1}{n}|I_{n}(G)|\leq\left(\prod_{a\in
A}\|\phi_{a}\|_{p}\right)\left(\prod_{e\in
E(G)}\|g_{e}\|_{\alpha_{e}}\right).$ Assume that $p\geq 3$ and $g_{e}\in
L^{2}\cap L^{\infty}$. Then
$\lim_{n\rightarrow+\infty}\frac{1}{n^{p/2}}I_{n}(G)=0.$

###### Proof.

Let $(T_{a})_{a\in A}$ be the family of spanning trees of $G$ given by Lemma
3.15. For $a\in A$, the vector

$(x_{a},(x_{b}-x_{c})_{\\{b,c\\}\in E(T_{a})})$

is a basis of $\mathbb{R}^{A}$. For $e\notin T_{a}$ we bound the term
$g_{e}(x_{i}-x_{j})$ in $I_{G}(n)$ by $\|g_{e}\|_{\infty}$, and for $b\neq a$,
the function $\phi_{b}$ by $\|\phi\|_{\infty}$. By a change of variable, we
get

$I_{n}(G)\leq n\|\phi_{a}\|_{1}\left(\prod_{b\neq
a}\|\phi_{b}\|_{\infty}\right)\left(\prod_{e\in
T_{a}}\|g_{e}\|_{1}\right)\left(\prod_{e\notin
T_{a}}\|g_{e}\|_{\infty}\right).$

Since this inequality is true for all $a\in A$, one can interpolate these
inequalities to obtain

$I_{n}(G)\leq n\left(\prod_{a\in A}\|\phi_{a}\|_{p}\right)\left(\prod_{e\in
E(G)}\|g_{e}\|_{p_{e}}\right),\quad\text{with}\quad\frac{1}{p_{e}}=\frac{1}{p}\sum_{a\in
A}\mathds{1}_{e\in E(T_{a})}.$

Since for all $e\in E(G)$, there is a tree $T_{a_{e}}$ that does not contain
the edge $e$, one must have $p_{e}\geq\frac{p}{p-1}$, and the first part of
the lemma follows. For the second part, note that we have the crude bound

$|I_{n}(G)|\leq n^{p}\left(\prod_{a\in
A}\|\phi_{a}\|_{1}\right)\left(\prod_{e\in E(G)}\|g_{e}\|_{\infty}\right).$

We interpolate this inequality with inequality (4.2.2) to get

$\frac{1}{n^{p/2}}|I_{n}(G)|\leq\left(\prod_{a\in
A}\|\phi_{a}\|_{2}\right)\left(\prod_{e\in
E(G)}\|g_{e}\|_{q_{e}}\right),\quad\text{with}\quad q_{e}\geq 2.$

In remains show that the left hand side of (4.2.2) converges towards zero for
$p\geq 3$. By a density argument, we can assume that the functions
$(g_{e})_{e\in E(G)}$ are bounded and compactly supported. In that case,
inequality (4.2.2) implies the convergence towards zero of the left hand side
of (4.2.2) when $p\geq 3$. ∎

In the following, $(\phi_{a})_{a\in A}$ is a family of functions in
$L^{(\infty)}(U)$. The previous Lemma 4.5 and the convergence of the Kac
density given by Corollary 4.3 imply the following lemma.

######  Lemma 4.6.

For $p\geq 3$,
$\lim_{n\rightarrow+\infty}\;\frac{1}{n^{p/2}}\int_{(nU)^{A}}\underline{\phi}_{A}^{\otimes}\left(\frac{\underline{x}_{A}}{n}\right)F_{A}(\underline{x}_{A})\mathrm{d}\underline{x}_{A}=0.$
If $g_{\varepsilon}\in L^{\frac{p}{p-1}}$ then
$\lim_{n\rightarrow+\infty}\frac{1}{n}\int_{(nU)^{A}}\underline{\phi}_{A}^{\otimes}\left(\frac{\underline{x}_{A}}{n}\right)F_{A}(\underline{x}_{A})\mathrm{d}\underline{x}_{A}=\left(\int_{U}\prod_{a\in
A}\phi_{a}(y)\mathrm{d}y\right)\left(\int_{\mathbb{R}^{p-1}}F_{A,\infty}(\underline{x})\mathrm{d}\underline{x}\right).$

###### Proof.

According to Lemma 4.4, there is a constant $C$ such that

$\left|\int_{(nU)^{A}}\underline{\phi}_{A}^{\otimes}\left(\frac{\underline{x}_{A}}{n}\right)F_{A}(\underline{x}_{A})\mathrm{d}\underline{x}_{A}\right|\leq\sum_{G\in\mathcal{G}_{A}}I_{n}(G),$

where $I_{n}(G)$ is defined in $\eqref{eq:61}$ with functions
$g_{e}=g_{\varepsilon}$. The first part of the corollary is an immediate
consequence of the first part of Lemma 4.5. As for the second part, the bound
given in Lemma 4.5 implies that one can assume by a density argument that the
functions $(\phi_{a})_{a\in A}$ are continuous and compactly supported. In
that case, pick $a_{0}\in A$. We define $y_{a_{0}}=0$ and we make the change
of variables

$x_{a_{0}}=ny\quad\text{and}\quad\forall a\in
A\setminus\\{a_{0}\\},\;\;x_{a}=ny+y_{a}.$

Then we have the following uniform convergence

$\lim_{n\rightarrow+\infty}\phi_{a}\left(y+\frac{y_{a}}{n}\right)=\phi_{a}(y),$

and according to Corollary 4.3,

$\lim_{n\rightarrow+\infty}F_{A,n}\left(y+\frac{\underline{y}_{A}}{n}\right)=F_{A,\infty}(\underline{y}_{A}).$

We can bound the functions $(\phi_{a})_{a\in A}$ by the indicator of a compact
set. In that case, dominated convergence and the second part of Lemma 4.5
imply the claimed convergence, which is valid for $|A|\geq 1$. ∎

Given the expression of cumulants given by Proposition 3.5 and the previous
Lemma 4.6, we then deduce the following theorem concerning the convergence of
cumulants associated with the linear statistics of the zeros counting measure
of the sequence of processes $(f_{n})_{n\in\overline{\mathfrak{N}}}$. We
define the Stirling number of the second kind

$\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}:=\mathrm{Card}\left\\{\mathcal{I}\in\mathcal{P}_{A}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ |\mathcal{I}|=k\right\\}.$

######  Theorem 4.7.

Let $p\geq 2$ and assume that the sequence of processes
$(f_{n})_{n\in\overline{\mathfrak{N}}}$ satisfies the hypotheses $H_{1}(q)$
and $H_{2}(q)$ with $q=2p-1$. If $p\geq 3$ then
$\lim_{n\rightarrow+\infty}\frac{1}{n^{p/2}}\kappa_{p}(\langle\nu,\phi\rangle)=0.$
Moreover when $g_{\varepsilon}\in L^{\frac{p}{p-1}}$,
$\lim_{n\rightarrow+\infty}\frac{1}{n}\kappa_{p}(\langle\nu,\phi\rangle)=\left(\int_{U}\phi^{p}(y)\mathrm{d}y\right)\sum_{k=1}^{p}\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}\left(\int_{\mathbb{R}^{k-1}}\\!\\!\\!F_{k,\infty}(\underline{x})\mathrm{d}\underline{x}\right).$

###### Proof.

Recall from Proposition 3.5 that

$\kappa_{p}(\langle\nu,\phi\rangle)=\sum_{\mathcal{I}\in\mathcal{P}_{A}}\int_{\mathbb{R}^{\mathcal{I}}}\left(\prod_{I\in\mathcal{I}}\phi(\underline{x}_{I})^{|I|}\right)F_{\mathcal{I}}(\underline{x}_{\mathcal{I}})\mathrm{d}\underline{x}_{\mathcal{I}}.$

According to the previous Lemma 4.6, one has

$\frac{1}{n^{p/2}}\left|\kappa_{p}(\langle\nu,\phi\rangle)\right|\leq\sum_{\mathcal{I}\in\mathcal{P}_{A}}\frac{1}{n^{|\mathcal{I}|/2}}\left|\int_{\mathbb{R}^{\mathcal{I}}}\left(\prod_{I\in\mathcal{I}}\phi(\underline{x}_{I})^{|I|}\right)F_{\mathcal{I}}(\underline{x}_{\mathcal{I}})\mathrm{d}\underline{x}_{\mathcal{I}}\right|\underset{n\rightarrow+\infty}{\longrightarrow}0,$

which proves the first assertion. As for the second assertion, it is again a
consequence of Lemma 4.6, which implies that

$\lim_{n\rightarrow+\infty}\frac{1}{n}\kappa_{p}(\langle\nu,\phi\rangle)=\left(\int_{U}\phi^{p}(y)\mathrm{d}y\right)\sum_{\mathcal{I}\in\mathcal{P}_{A}}\left(\int_{\mathbb{R}^{|\mathcal{I}|-1}}\\!\\!\\!F_{|\mathcal{I}|,\infty}(\underline{x})\mathrm{d}\underline{x}\right).$

∎

The proof of the main Theorem 1.6 is a reformulation of the previous Theorem
4.7, with

$\forall p\geq 1,\quad\gamma_{p}=\sum_{k=1}^{p}\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}\left(\int_{\mathbb{R}^{k-1}}\\!\\!\\!F_{k,\infty}(\underline{x})\mathrm{d}\underline{x}\right).$

In particular, one has

$\gamma_{1}=\frac{1}{\pi}\sqrt{\frac{r_{\infty}^{\prime\prime}(0)}{r(0)}}\quad\text{and}\quad\gamma_{2}=\gamma_{1}+\int_{\mathbb{R}}F_{2,\infty}(x)\mathrm{d}x.$

It has been shown for instance in [15] that under our assumptions on the
process $f_{\infty}$, the constant $\gamma_{2}$ is positive, from which
follows the central limit theorem for the linear statistic associated with the
zeros counting measure.

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