2.1. An ASCLT for stabilizing functionals on the Poisson space

Let $\mathbb Y$ be a full-dimensional Borel set of $\mathbb R^d$ and let $\eta$ be the Poisson random measure on $\mathbb X\,:\!=\mathbb{R}^d$ with intensity measure $\mu(\mathrm{d}x)\,:\!=\ell_d(\mathrm{d}x)$, the Lebesgue measure on $\mathbb R^d$. Here and in the next subsection, we identify a counting measure $\nu\in\textbf{N}_{\mathbb R^d}$ with its support and we consider the statistic

(3)\begin{equation}H_n\,:\!=\sum_{x\in\eta_n\cap\mathbb Y}\xi_n(x,\eta_n\cap\mathbb Y),\quad n\geq 1,\end{equation}

where $\eta_n\,:\!=n^{-1/d}\eta$; i.e., $\eta_n$ is the Poisson random measure on $\mathbb R^d$ with intensity measure $n\ell_d(\mathrm d x)$. The so-called score functions $\{\xi_n\}_{n\geq 1}$ are measurable functions from $\mathbb Y\times\textbf{N}_\mathbb Y$ to $\mathbb R$ and represent the local contribution to the global statistic $H_n$. In order to introduce the notion of stabilizing functional considered in this paper, we need some definitions.

For $n\geq 1$, a measurable map $R_n\,:\,\mathbb Y\times\textbf{N}_\mathbb Y\to [0,\infty)$ is called a $\textit{radius\,of}$$\textit{stabilization}$ for the score function $\xi_n$ if, for all $(x,\nu)\in \mathbb Y\times\textbf{N}_\mathbb Y$ and any finite subset $\mathcal A\subset \mathbb Y$ with at most 7 points,

(4)\begin{equation}\xi_n(x,(\nu\cup\{x\}\cup\mathcal A)\cap B(x,R_n(x,\nu\cup\{x\})))=\xi_n(x,\nu\cup\{x\}\cup\mathcal A).\end{equation}

Here $B(x,r)\,:\!=\{y\in\mathbb Y:\,\,\|x-y\|_d\leq r\}$ denotes the closed ball of radius $r\geq 0$ centered at $x\in\mathbb Y$, and $\|\cdot\|_d$ is the Euclidean norm on $\mathbb R^d$. Loosely speaking, the notion of radius of stabilization says that the value of $\xi_n$ at x is wholly determined by the points of $\nu$ at distance at most $R_n(x,\nu\cup\{x\})$ from x. We emphasize that the assumption ‘$\mathcal{A}$ has at most 7 points’ is not required according to the classical definition of radius of stabilization. In this paper we need this extra hypothesis to place ourselves in the same framework as [Reference Lachièze-Rey, Schulte and Yukich8]. A similar comment applies to the following definitions of score functions satisfying a moment condition and score functions with exponentially fast decay.

The score functions $\{\xi_n\}_{n\geq 1}$ are called $\textit{exponentially\,\,stabilizing}$ if there exist radii of stabilization $\{R_n\}_{n\geq 1}$ and constants $C_{stab}$, $c_{stab}$, $\alpha_{stab}\in\mathbb R_+\,:\!=(0,\infty)$ such that, for $x\in \mathbb Y$, $r\geq 0$ and $n\geq 1$,

(5)\begin{equation}\mathbb{P}(R_n(x,(\eta_n\cap \mathbb Y)\cup\{x\})\geq r)\leq C_{stab}\exp\big(-c_{stab}n^{\alpha_{stab}/d}r^{\alpha_{stab}}\big).\end{equation}

Let $p'\in [0,\infty)$ be given. We say that the score functions $\{\xi_n\}_{n\geq 1}$ satisfy a $(4+p')$$\textit{th}$$\textit{moment\,\,condition}$ if there is a constant $C_{p'}\in\mathbb{R}_+$ such that for any finite subset $\mathcal{A}\subset\mathbb{Y}$ with at most 7 points,

(6)\begin{equation}\sup_{n\geq 1}\sup_{x\in\mathbb Y}\mathbb{E}\bigg[|\xi_n(x,(\eta_n\cap \mathbb Y)\cup\{x\}\cup\mathcal A)|^{4+p'}\bigg]\leq C_{p'}.\end{equation}

Let K be a Borel subset of $\mathbb Y$ and put $\mathrm{d}(x,K)\,:\!=\inf_{y\in K}\|x-y\|_d$, $x\in\mathbb Y$. We say that score functions $\{\xi_n\}_{n\geq 1}$$\textit{decay\,\,exponentially\,\,fast\,\,with\,\,the\,\,distance\,\,to\,\,K}$ if there are constants $C_K,c_K,\alpha_K\in\mathbb{R}_+$ such that for any finite subset $\mathcal{A}\subset\mathbb Y$ with at most 7 points,

(7)\begin{equation}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\}\cup\mathcal A)\neq 0)\leq C_K\exp(-c_K n^{\alpha_K/d}\mathrm{d}(x,K)^{\alpha_K})\end{equation}

for $x\in\mathbb Y$ and $n\geq 1$.

From here on, we consider stabilizing functionals $H_n$ and, for a fixed ${p''}\in (0,p')$, the quantity

(8)\begin{equation}I_{K,n}\,:\!=n\int_{\mathbb Y}\exp\left(-\frac{{p''}\,c_{p'}n^{\alpha/d}\mathrm{d}(x,K)^\alpha}{2^{2\alpha+3}(4+{p''})}\right)\,\mathrm d x,\quad n\geq 1,\end{equation}

where

(9)\begin{equation}c_{p'}\,:\!=p'(c_{stab}\wedge c_K),\quad\alpha\,:\!=\alpha_{stab}\wedge\alpha_K,\end{equation}

and we denote by $a\wedge b$ the minimum of $a,b\in\mathbb R$. Throughout this paper, we use the standard Landau notation; i.e., given two sequences $\{a_n\}_{n\geq 1},\{b_n\}_{n\geq 1}\subset\mathbb R_+$, we write $a_n=O(b_n)$ if $\limsup_{n\to\infty}\frac{a_n}{b_n}<\infty$.

The following theorem holds.

Theorem 2.2. If the functionals $H_n$, $n\geq 1$, defined by (3) are stabilizing, and $n^\tau=O(\mathbb{V}ar(H_n))$ and $I_{K,n}=O(n^\tau)$ for some $\tau\in (0,2)$, then $\{F_n\}_{n\geq 1}$, where

\begin{equation*}F_n\,:\!=(H_n-\mathbb{E}[H_n])/\sqrt{\mathbb{V}ar(H_n)},\end{equation*}

satisfies the ASCLT.

We remark that in the statement of Theorem 2.2 the quantities $\mathbb{V}ar(H_n)$ and $I_{K,n}$ are finite and strictly positive for any $n\geq 1$ (this is an implicit consequence of the Landau notation, as defined above, which is employed to compare sequences in $\mathbb R_+$). We also emphasize that, although Theorem 2.2 holds for any scaling regime $\tau\in(0,2)$, all the examples considered in this paper fall either in the class of the volume order scaling (i.e. $\tau=1$) or in the class of the surface area order scaling (i.e. $\tau=1-1/d$).

2.2. ASCLTs for stochastic geometry models

Theorem 2.2 finds a natural application to models that arise in stochastic geometry, such as the total edge length of the k-nearest neighbors random graph, the clique count in random geometric graphs, and the volume of the set approximation via the Poisson–Voronoi tessellation.

2.2.1 An ASCLT for the total edge length of the k-nearest neighbors random graph

Let $\mathbb{Y}$ be a full-dimensional Borel set of $\mathbb R^d$ and $n,k\geq 1$ two integers. For $x\in\eta_n\cap\mathbb Y$, we denote by $V_k(x,\eta_n\cap\mathbb Y)$ the set of the k nearest neighbors of x, i.e., the k closest points to x in $(\eta_n\cap\mathbb Y)\setminus\{x\}$. The (undirected) k-nearest neighbors random graph $NG_k(\eta_n\cap\mathbb Y)$ is the random graph with vertex set $\eta_n\cap\mathbb Y$ obtained by including an edge $\{x,y\}$ if $y\in V_k(x,\eta_n\cap \mathbb Y)$ and/or $x\in V_k(y,\eta_n\cap \mathbb Y)$. For all $m\geq 0$, we set

\begin{equation*}\xi^{(m)}(x,\eta_n\cap\mathbb Y)\,:\!=\sum_{y\in V_k(x,\eta_n\cap\mathbb Y)}\rho^{(m)}(x,y),\end{equation*}

where

\begin{align}\rho^{(m)}(x,y)&\,:\!=\textbf{1}\{x\in V_k(y,\eta_n\cap\mathbb Y)\}\frac{\|x-y\|_d^m}{2}+\textbf{1}\{x\notin V_k(y,\eta_n\cap\mathbb Y)\}\|x-y\|_d^m.\nonumber\end{align}

The total edge length of the (undirected) k-nearest neighbors random graph on $\eta_n\cap\mathbb Y$ with mth-power weighted edges is

\begin{equation*}L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)\,:\!=\sum_{x\in\eta_n\cap\mathbb Y}\xi^{(m)}(x,\eta_n\cap\mathbb Y).\end{equation*}

Note that $L_{NG_k}^{(0)}(\eta_n\cap\mathbb Y)$ is the number of edges and $L_{NG_k}^{(1)}(\eta_n\cap\mathbb Y)$ is the total edge length. When $\mathbb Y$ is a full-dimensional compact and convex subset of $\mathbb R^d$, the central limit theorem and a quantitative central limit theorem (in the Kolmogorov distance) for the sequence $\{L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)\}_{n\geq 1}$ are proved in [Reference Penrose and Yukich16] and [Reference Last, Peccati and Schulte11], respectively. Here, we provide an ASCLT for such a sequence.

Set

(10)\begin{equation}F_n\,:\!=\frac{L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)-\mathbb{E}\Big[L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)\Big]}{\sqrt{\mathbb{V}ar\Big(L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)\Big)}},\quad n\geq 1.\end{equation}

The following corollary holds.

Corollary 2.3. If $\mathbb Y$ is a full-dimensional compact and convex subset of $\mathbb R^d$, then $\{F_n\}_{n\geq 1}$ defined by (10) satisfies the ASCLT.

2.2.2 An ASCLT for the clique count in random geometric graphs

Let $\mathbb{Y}$ be a full-dimensional subset of $\mathbb{R}^d$ and consider the random geometric graph $G(\eta_n\cap\mathbb Y,r)$, $n\geq 1$, $r\in\mathbb{R}_+$, where two nodes $x,y\in\eta_n\cap\mathbb Y$ are joined with an edge if $\|x-y\|_d\leq r$. We recall that $k+1$ nodes of $G(\eta_n\cap\mathbb Y,r)$ form a clique of order $k+1$ if each pair of them is connected by an edge. The number of cliques of order $k+1$ in $G(\eta_n\cap\mathbb Y,r)$, denoted by $\mathcal{C}_k(\eta_n\cap\mathbb Y,r)$, is a central statistic in topological data analysis. A quantitative central limit theorem (in the Kolmogorov distance) for the sequence $\{\mathcal{C}_k(\eta_n\cap\mathbb Y,rn^{-1/d})\}_{n\geq 1}$ has been proved in [14, Chapter 3]. Here, we give an ASCLT for such a sequence.

Set

(11)\begin{equation}F_n\,:\!=\frac{\mathcal{C}_k\big(\eta_n\cap\mathbb Y,rn^{-1/d}\big)-\mathbb{E}\big[\mathcal{C}_k(\eta_n\cap\mathbb Y,rn^{-1/d})\big]}{\sqrt{\mathbb{V}ar(\mathcal{C}_k(\eta_n\cap\mathbb Y,rn^{-1/d}))}},\quad n\geq 1.\end{equation}

The following corollary holds.

Corollary 2.4. If $\mathbb Y$ is a full-dimensional subset of $\mathbb R^d$ such that $\ell_d(\mathbb Y)<\infty$, then $\{F_n\}_{n\geq 1}$ defined by (11) satisfies the ASCLT.

2.2.3. An ASCLT for the volume of the set approximation via the Poisson–Voronoi tessellation

Let $\mathbb Y\,:\!=[-1/2,1/2]^d$, $d\geq 2$, and let $A\subset (-1/2,1/2)^d$ be a full-dimensional subset of $\mathbb R^d$. Assume that we observe $\eta_n\cap\mathbb Y$ and that the only information about A at our disposal is which points of $\eta_n\cap\mathbb Y$ lie in A; i.e., we have the partition of the process $\eta_n\cap\mathbb Y$ into $\eta_n\cap A$ and $(\eta_n\cap\mathbb Y)\setminus A$. To reconstruct the set A just from the information at our disposal, we approximate A by its Poisson–Voronoi approximation $A_n$, i.e. the set of all points in $\mathbb Y$ which are closer to $\eta_n\cap A$ than to $(\eta_n\cap\mathbb Y)\setminus A$. Formally,

\begin{equation*}A_n\,:\!=\bigcup_{x\in\eta_n\cap A}\mathrm{C}(x,\eta_n\cap\mathbb Y),\end{equation*}

where $\mathrm{C}(x,\eta_n\cap\mathbb Y)$ is the Poisson–Voronoi cell generated by $\eta_n\cap\mathbb Y$ with nucleus $x\in\eta_n\cap\mathbb Y$; i.e.,

\begin{equation*}\mathrm{C}(x,\eta_n\cap\mathbb Y)\,:\!=\{y\in\mathbb Y:\,\,\|y-x\|_d\leq\|y-z\|_d\,\,\forall\,\,z\in\eta_n\cap\mathbb Y\}.\end{equation*}

Note that $\mathrm{C}(x,\eta_n\cap\mathbb Y)$ is a random convex polytope and that the Poisson–Voronoi tessellation (or mosaic) of $\mathbb Y$, i.e. the family $\{\mathrm{C}(x,\eta_n\cap\mathbb Y)\}_{x\in\eta_n\cap\mathbb Y}$, is a partition of $\mathbb Y$.

If A is compact and convex, it follows from [Reference Penrose15] and [Reference Reitzner, Spodarev and Zhaporazhets18] respectively that as $n\to\infty$, $\ell_d(A_n)\to\ell_d(A)$ almost surely, and $\mathbb{E}[\ell_d(A_n)]\to\ell_d(A)$. A quantitative central limit theorem (in the Kolmogorov distance) for the sequence $\{\ell_d(A_n)\}_{n\geq 1}$ is proved in [Reference Lachièze-Rey, Schulte and Yukich8]. Here, we give an ASCLT for such a sequence.

Put

(12)\begin{equation}F_n\,:\!=\frac{\ell_d(A_n)-\mathbb E[\ell_d(A_n)]}{\sqrt{\mathbb{V}ar(\ell_d(A_n))}},\quad n\geq 1.\end{equation}

The following corollary holds.

Corollary 2.5. If A is compact and convex then $\{F_n\}_{n\geq 1}$ defined by (12) satisfies the ASCLT.

2.3. An ASCLT on the Poisson space

As already mentioned in the introduction, in this section we present an abstract ASCLT for the Poisson space, which provides the theoretical foundation for the proof of Theorem 2.2. To state such a general result we need to introduce some additional notation.

By $L_{\eta}^r$, $r\in\mathbb R_+$, we denote the space of all random variables $F\in L^r(\mathbb P)$ such that $F=f(\eta)$$\mathbb P$-almost surely for some measurable function $f\,:\,\textbf{N}_{\mathbb X}\to\mathbb R$, where $L^r(\mathbb P)$ is the set of random variables $X\,:\,\Omega\to\mathbb R$ such that $\mathbb{E}[|X|^r]<\infty$. For $F\in L_{\eta}^2$, $F=f(\eta)$, and $x_1,x_2\in\mathbb X$ we define the Malliavin derivative operators

\begin{align*}D_{x_1}F&\,:\!=f(\eta+\varepsilon_{x_1})-f(\eta),\\ D_{x_1,x_2}^{2}F&\,:\!=D_{x_2}(D_{x_1}F)=f(\eta+\varepsilon_{x_1}+\varepsilon_{x_2})-f(\eta+\varepsilon_{x_2})-f(\eta+\varepsilon_{x_1})+f(\eta).\end{align*}

For $H_i\in L_\eta^2$ with $\mathbb{V}ar(H_i)>0$, $i=1,2,3,4$, and positive constants $p,q\in\mathbb R_+$ (defined in the statement of Theorem 2.6), we put

\begin{align}\boldsymbol{\Gamma}(H_1,H_2,H_3,H_4)^2&\,:\!=\boldsymbol{\Lambda}(H_1,H_2,H_3,H_4)^2\nonumber\\&\quad +\frac{1}{\mathbb{V}ar(H_1)\mathbb{V}ar(H_2)}\int_{\mathbb{X}}\psi_x(H_1,q/(2(4+q)))\psi_x(H_2,q/(2(4+q)))\mu(\mathrm d x),\nonumber\end{align}

where

\begin{equation*}\psi_x(G,\beta)\,:\!=\int_{\mathbb{X}}\mathbb{P}(D_{x_1,x}^{2}G\neq 0)^{\beta}\mu(\mathrm d x_1)\end{equation*}

for $x\in\mathbb X$, $G\in L_\eta^2$, $\beta\in\mathbb{R}_+$, and

\begin{align}\boldsymbol{\Lambda}(H_1,H_2,H_3,H_4)^2\,:\!=\frac{1}{\prod_{i=1}^{4}\mathbb{V}ar(H_i)^{1/2}}\int_{\mathbb{X}}\psi_x(H_1,q/(4(4+q)))\psi_x(H_2,q/(4(4+q)))\mu(\mathrm d x).\nonumber\end{align}

We also define

\begin{align*}\boldsymbol{\Theta}(H_1,H_2)&\,:\!=\frac{1}{\prod_{i=1}^{2}\mathbb{V}ar(H_i)^{1/2}}\int_{\mathbb X}\mathbb{P}(D_x H_1\neq 0)^{p/(4(4+p))}\mathbb{P}(D_x H_2\neq 0)^{p/(4(4+p))}\mu(\mathrm{d}x),\\[4pt] \boldsymbol{\Gamma}_1(H_1)^2&\,:\!=\boldsymbol{\Gamma}(H_1,H_1,H_1,H_1)^2,\\[4pt] \boldsymbol{\Gamma}_2(H_1)&\,:\!=\frac{1}{\mathbb{V}ar(H_1)^{3/2}}\int_{\mathbb{X}}\mathbb{P}(D_x H_1\neq 0)^{(1+p)/(4+p)}\mu(\mathrm d x).\end{align*}

The following theorem holds, where we denote by $a\vee b$ the maximum of $a,b\in\mathbb R$.

Theorem 2.6. Assume that $\{H_n\}_{n\geq 1}\subset L_\eta^2$ and $\mathbb{V}ar(H_n)>0$, $n\geq 1$, and that there exist constants $c,p,q\in\mathbb{R}_+$ such that

(13)\begin{equation}\sup_{n\geq 1}\mathbb{E}[|D_{x_1}H_n|^{4+p}]\leq c,\quad\text{$\mu$-a.e. $x_1\in\mathbb X$},\end{equation}

and

(14)\begin{equation}\sup_{n\geq 1}\mathbb{E}\big[|D_{x_1,x_2}^{2}H_n|^{4+q}\big]\}\leq c,\quad\text{$\mu^{\otimes 2}$-a.e. $(x_1,x_2)\in\mathbb X^{2}$.}\end{equation}

Moreover, suppose

(15)\begin{equation}\lim_{n\to\infty}\boldsymbol{\Gamma}_1(H_n)=\lim_{n\to\infty}\boldsymbol{\Gamma}_2(H_n)=0,\end{equation}

(16)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^2}\sum_{k=1}^{n}\frac{\boldsymbol{\Gamma}_i(H_k)}{k}<\infty,\quad\text{$i=1,2$},\end{equation}

(17)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\boldsymbol{\Theta}(H_k,H_l)}{kl}<\infty,\end{equation}

(18)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\boldsymbol{\Gamma}(H_k,H_l,H_k,H_l)}{kl}<\infty,\end{equation}

and

(19)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\boldsymbol{\Lambda}_{max}(H_k,H_l)}{kl}<\infty,\end{equation}

where

\begin{align*}\boldsymbol{\Lambda}_{max}(H_k,H_l)&\,:\!=\boldsymbol{\Lambda}(H_k,H_k,H_k,H_l)\vee\boldsymbol{\Lambda}(H_l,H_l,H_l,H_k)\vee \boldsymbol{\Lambda}(H_k,H_k,H_l,H_l) \nonumber\\&\quad\vee\boldsymbol{\Lambda}(H_l,H_l,H_k,H_k)\vee\boldsymbol{\Lambda}(H_l,H_k,H_k,H_k)\vee\boldsymbol{\Lambda}(H_k,H_l,H_l,H_l)\\&\quad\vee\boldsymbol{\Lambda}(H_k,H_l,H_l,H_k).\end{align*}

Then $\{F_n\}_{n\geq 1}$, $F_n\,:\!=(H_n-\mathbb{E}[H_n])/\sqrt{\mathbb{V}ar(H_n)}$, satisfies the ASCLT.

3.1. Preliminary lemmas

The following lemma is an immediate consequence of Lemma 5.5 in [Reference Lachièze-Rey, Schulte and Yukich8].

Lemma 3.1. If the functionals $H_n$, $n\geq 1$, defined by (3) are stabilizing, then for any fixed ${p''}\in (0,p')$ there exists a constant $c({p''})\in\mathbb{R}_+$ (depending only on the constants $C_{stab}$, $c_{stab}$, $\alpha_{stab}$, pʹ, and $C_{p'}$ appearing in the definition of stabilizing functional) such that

\begin{equation*}\sup_{n\geq 1}\sup_{y\in\mathbb Y}\mathbb{E}\Big[|\mathcal{D}_y H_n|^{4+{p''}}\Big]\leq c({p''})\quad\text{and}\quad\sup_{n\geq 1}\sup_{y_1,y_2\in\mathbb{Y}}\mathbb{E}\Big[|\mathcal{D}_{y_1,y_2}^{2,+}H_n|^{4+{p''}}\Big]\leq c({p''}).\end{equation*}

The following lemma is an immediate consequence of Lemma 5.10 in [Reference Lachièze-Rey, Schulte and Yukich8] (see the inequalities $(5.6)$ and $(5.8)$ therein).

Lemma 3.2. If the functionals $H_n$, $n\geq 1$, defined by (3) are stabilizing, then for any fixed $\beta\in\mathbb{R}_+$ there exists a constant $\tilde{C}_\beta\in\mathbb R_+$ (not depending on n) such that

\begin{equation*}n\int_{\mathbb{Y}}\Biggl(n\int_{\mathbb Y}\mathbb{P}(\mathcal{D}_{y_1,y_2}^2H_n \neq 0)^\beta\,\mathrm d y_1\Biggr)^2\,\mathrm d y_2\leq\tilde{C}_\beta n\int_{\mathbb Y}\exp\left(-\frac{c_{p'}\beta}{2^{2\alpha+1}}(n^{1/d}\mathrm d (x,K))^\alpha\right)\,\mathrm d x\end{equation*}

and

\begin{equation*}n\int_{\mathbb Y}\mathbb{P}(\mathcal{D}_y H_n\neq 0)^\beta\,\mathrm d y\nonumber\leq\tilde{C}_\beta n\int_{\mathbb Y}\exp\left(-\frac{c_{p'}\beta}{2^{\alpha+1}}(n^{1/d}\mathrm d (x,K))^\alpha\right)\,\mathrm d x,\end{equation*}

where the constants $c_{p'}$ and $\alpha$ are defined in (9).

We conclude this subsection with a lemma which relates the operators D and $D^2$ to the operators $\mathcal{D}$ and $\mathcal{D}^2$, respectively.

Lemma 3.3. If the functionals $H_n$, $n\geq 1$, defined by (3) are stabilizing and $I_{K,n}<\infty$, then $H_n\in L^2(\mathbb P)$, and for any $x_1,x_2\in\mathbb{R}^d$,

(20)\begin{align}D_{x_1}H_n=\textbf{1}\{n^{-1/d}x_1\in\mathbb Y\}\mathcal{D}_{n^{-1/d}x_1}H_n\end{align}

and

(21)\begin{align}D_{x_1,x_2}^2 H_n&=\textbf{1}\{n^{-1/d}x_1,n^{-1/d}x_2\in\mathbb Y\}\mathcal{D}_{n^{-1/d}x_1,n^{-1/d}x_2}^2H_n.\end{align}

Proof. We divide the proof into two steps. In the first step we check the square-integrability of $H_n$; in the second step we verify the relations among the operators.

Step 1: Checking $H_n\in L^2(\mathbb P)$, $n\geq 1$. Throughout this proof, ${p''}\in (0,p')$ is the constant appearing in the definition of $I_{K,n}$. We shall check later on that for any $0<\gamma\leq 2^{2\alpha+3}(4+{p''})$ and $x\in\mathbb Y$,

(22)\begin{equation}\exp(-c_K n^{\alpha_K/d}\mathrm{d}(x,K)^{\alpha_K}/\gamma)\leq\hat{C}\exp(-{p''}\,c_{p'}n^{\alpha/d}\mathrm{d}(x,K)^{\alpha}/(2^{2\alpha+3}(4+{p''})))\end{equation}

for a suitable constant $\hat C\in\mathbb R_+$ not depending on n, where the constants $c_{p'}$ and $\alpha$ are defined in (9). Setting $M\,:\!=\sum_{x\in\eta_n\cap\mathbb Y}\textbf{1}(\xi_n(x,\eta_n\cap\mathbb Y)\neq 0)$, by Jensen’s inequality we have

\begin{align}H_n^2&=M^2\left(\sum_{x\in\eta_n\cap\mathbb Y}\frac{\textbf{1}\{\xi_n(x,\eta_n\cap\mathbb Y)\neq 0\}}{M}\xi_n(x,\eta_n\cap\mathbb Y)\right)^2\nonumber\\&\leq M\sum_{x\in\eta_n\cap\mathbb Y}\textbf{1}\{\xi_n(x,\eta_n\cap\mathbb Y)\neq 0\}\xi_n(x,\eta_n\cap\mathbb Y)^2\nonumber\\&=\sum_{x,x'\in\eta_n\cap\mathbb Y}\textbf{1}\{\xi_n(x',\eta_n\cap\mathbb Y)\neq 0,\,\xi_n(x,\eta_n\cap\mathbb Y)\neq 0\}\xi_n(x,\eta_n\cap\mathbb Y)^2\nonumber\\&=\sum_{x\in\eta_n\cap\mathbb Y}\textbf{1}\{\xi_n(x,\eta_n\cap\mathbb Y)\neq 0\}\xi_n(x,\eta_n\cap\mathbb Y)^2\nonumber\\&\quad+\sum_{x,x'\in\eta_n\cap\mathbb Y:\,x\neq x'}\textbf{1}\{\xi_n(x',\eta_n\cap\mathbb Y)\neq 0,\,\xi_n(x,\eta_n\cap\mathbb Y)\neq 0\}\xi_n(x,\eta_n\cap\mathbb Y)^2.\nonumber\end{align}

By the (multivariate) Mecke formula (see e.g. [10, Chapter 4]), the Cauchy–Schwarz inequality, Hölder’s inequality, (6), (7), and (22), we get

\begin{align}\mathbb{E}\big[H_n^2\big]&\leq n\int_{\mathbb Y}\mathbb{E}[\textbf{1}\{\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})\neq 0\}\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})^2]\,\mathrm d x\nonumber\\&\qquad+n^2\int_{\mathbb{Y}^2}\mathbb{E}[\textbf{1}\{\xi_n(x',(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0,\,\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0\}\nonumber\\&\qquad\times\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})^2]\,\mathrm d x\mathrm d x'\nonumber\\&\leq n\int_{\mathbb Y}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})\neq 0)^{1/2}\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})^4]^{1/2}\,\mathrm d x\nonumber\\&\qquad+n^2\int_{\mathbb{Y}^2}\mathbb{E}[\textbf{1}\{\xi_n(x',(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0,\,\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0\}]^{1/2}\nonumber\\&\qquad\times\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})^4]^{1/2}\,\mathrm d x\mathrm d x'\nonumber\\&\leq n\int_{\mathbb Y}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})\neq 0)^{1/2}\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})^4]^{1/2}\,\mathrm d x\nonumber\\&\qquad+n^2\int_{\mathbb{Y}^2}\mathbb{P}(\xi_n(x',(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\nonumber\\&\qquad\times\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})^4]^{1/2}\,\mathrm d x\mathrm d x'\nonumber\\&\leq n\int_{\mathbb Y}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})\neq 0)^{1/2}\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})^{4+p'}]^{2/(4+p')}\,\mathrm d x\nonumber\\&\qquad+n^2\int_{\mathbb{Y}^2}\mathbb{P}(\xi_n(x',(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\nonumber\\&\qquad\times\mathbb{E}[\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})^{4+p'}]^{2/(4+p')}\,\mathrm d x\mathrm d x'\nonumber\\&\leq C_{p'}^{2/(4+p')}n\int_{\mathbb Y}\mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x\})\neq 0)^{1/2}\,\mathrm d x\nonumber\\&\qquad+C_{p'}^{2/(4+p')}n^2\int_{\mathbb{Y}^2}\mathbb{P}(\xi_n(x',(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\nonumber\\& \qquad \times \mathbb{P}(\xi_n(x,(\eta_n\cap\mathbb Y)\cup\{x,x'\})\neq 0)^{1/4}\,\mathrm d x\mathrm d x'\nonumber\\&\leq C_K^{1/2}C_{p'}^{2/(4+p')}\left[n\int_{\mathbb Y}\exp(-c_K n^{\alpha_K/d}\mathrm{d}(x,K)^{\alpha_K}/2)\,\mathrm d x \right. \nonumber \\& \qquad \left. +\left(n\int_{\mathbb{Y}}\exp({-}c_K n^{\alpha_K/d}\mathrm{d}(x,K)^{\alpha_K}/4)\,\mathrm d x\right)^2\right]\nonumber\\&\leq C'[I_{K,n}+(I_{K,n})^2]<\infty,\nonumber\end{align}

for a suitable constant $C'\in\mathbb{R}_+$. It remains to check (22). Since $\gamma\leq 2^{2\alpha+3}(4+{p''})$, we have

\begin{align}&\exp(-c_K n^{\alpha_K/d}\mathrm{d}(x,K)^{\alpha_K}/\gamma)\nonumber\\&\qquad\qquad\leq\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha_K}\right)\nonumber\\&\qquad\qquad=\textbf{1}\{n^{1/d}\mathrm{d}(x,K)\leq 1\}\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha_K}\right)\nonumber\displaybreak\\&\qquad\qquad\quad+\textbf{1}\{n^{1/d}\mathrm{d}(x,K)>1\}\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha_K}\right)\nonumber\\&\qquad\qquad\leq\textbf{1}\{n^{1/d}\mathrm{d}(x,K)\leq 1\}\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha}\right)\nonumber\\&\qquad\qquad\quad\times\exp\left(\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}[(n^{1/d}\mathrm{d}(x,K))^{\alpha}-(n^{1/d}\mathrm{d}(x,K))^{\alpha_K}]\right)\nonumber\\&\qquad\qquad\quad+\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha}\right)\nonumber\\&\qquad\qquad\leq\left(\textbf{1}\{n^{1/d}\mathrm{d}(x,K)\leq 1\}\exp\left(\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}\right)+1\right)\nonumber \\& \qquad\qquad \quad \times \exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha}\right)\nonumber\\&\qquad\qquad\leq\hat{C}\exp\left(-\frac{{p''}\,c_{p'}}{2^{2\alpha+3}(4+{p''})}(n^{1/d}\mathrm{d}(x,K))^{\alpha}\right)\!,\nonumber\end{align}

where we used that, for $a>1$, the function $x\mapsto a^x$ is non-decreasing on $[0,\infty)$.

Step 2: Checking the relations among the operators. For any $x_1\in\mathbb{R}^d$, we have

\begin{align}D_{x_1}H_n&=\textbf{1}\{n^{-1/d}x_1\in\mathbb Y\}\Biggl(\xi_n\big(n^{-1/d}x_1,\big(\eta_n\cup\{n^{-1/d}x_1\}\big)\cap\mathbb Y\big)\nonumber\\& \quad+\sum_{z\in\eta_n\cap\mathbb Y}\xi_n\big(z,\big(\eta_n\cup\big\{n^{-1/d}x_1\big\}\big)\cap\mathbb Y\big)-\sum_{z\in\eta_n\cap\mathbb Y}\xi_n(z,\eta_n\cap\mathbb Y)\Biggr)\nonumber\\&=\textbf{1}\{n^{-1/d}x_1\in\mathbb Y\}\mathcal{D}_{n^{-1/d}x_1}H_n.\nonumber\end{align}

The relation (21) can be verified by a similar computation.

3.2. Proof of Theorem 2.2

We apply Theorem 2.6 and start by noticing that the square-integrability of the functionals $H_n$, $n\geq 1$, follows from Lemma 3.3. In the next steps we check all the other conditions of Theorem 2.6. Hereafter, ${p''}\in (0,p')$ is the constant involved in the definition of $I_{K,n}$.

Step 1: Checking the conditions (13) and (14). By (20) and Lemma 3.1, we have

\begin{align}\sup_{n\geq 1}\sup_{x\in\mathbb R^d}\mathbb{E}\Big[|D_{x}H_n|^{4+{p''}}\Big]\leq c({p''}).\nonumber\end{align}

By (21), the inequality $|a+b|^r\leq 2^{r-1}(|a|^r+|b|^r)$, $r\geq 1$, and again Lemma 3.1, we have

\begin{align}\sup_{n\geq 1}\sup_{x_1,x_2\in\mathbb{R}^d}\mathbb{E}\Big[|D_{x_1,x_2}^2 H_n|^{4+{p''}}\Big]&\leq\sup_{n\geq 1}\sup_{y_1,y_2\in\mathbb{Y}}\mathbb{E}\Big[|\mathcal{D}_{y_1,y_2}^{2}H_n|^{4+{p''}}\Big]\nonumber\\ &\leq 2^{3+{p''}}\sup_{n\geq 1}\sup_{y_1,y_2\in\mathbb{Y}}\Big(\mathbb{E}\Big[|\mathcal{D}_{y_1,y_2}^{2,+}H_n|^{4+{p''}}\Big]+\mathbb{E}\Big[|\mathcal{D}_{y_1}H_n|^{4+{p''}}\Big ]\Big)\nonumber\\&\leq 2^{4+{p''}}c({p''}).\nonumber\end{align}

So the conditions (13) and (14) are verified with $p=q={p''}$ and $c=2^{4+{p''}}c({p''})$. To conclude the proof it remains to check (15), (16), (17), (18), and (19), where the quantities $\boldsymbol\Lambda$, $\boldsymbol\Gamma$, $\boldsymbol{\Theta}$, and $\boldsymbol\Gamma_2$ involve pʹʹ in place of p and q. This task is accomplished in Steps 2–6.

Step 2: Two preliminary inequalities. Let $\beta\in\mathbb{R}_+$ be fixed. By (21) and Lemma 3.2, we have

(23)\begin{align}\int_{\mathbb R^d}\psi_x(H_n,\beta)^2\,\mathrm d x&=\int_{\mathbb{R}^d}\left(\int_{\mathbb R^d}\mathbb{P}(D_{x_1,x}^2H_n\neq 0)^\beta\,\mathrm d x_1\right)^2\,\mathrm d x\nonumber\\&=\int_{\mathbb{R}^d}\Biggl(\int_{\mathbb R^d}\mathbb{P}\Biggl(\textbf{1}\{n^{-1/d}x_1,n^{-1/d}x\in\mathbb Y\}\mathcal{D}_{n^{-1/d}x_1,n^{-1/d}x}^2H_n\neq 0\Biggr)^\beta\,\mathrm d x_1\Biggr)^2\,\mathrm d x\nonumber\\&=n\int_{\mathbb{Y}}\Biggl(n\int_{\mathbb Y}\mathbb{P}(\mathcal{D}_{x_1,x}^2H_n \neq 0)^\beta\,\mathrm d x_1\Biggr)^2\,\mathrm d x\nonumber\\&\leq\tilde{C}_\beta n\int_{\mathbb Y}\exp\left(-\frac{c_{p'}\beta}{2^{2\alpha+1}}(n^{1/d}\mathrm d (x,K))^\alpha\right)\,\mathrm d x\end{align}

and

(24)\begin{align}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_n\neq 0)^\beta\,\mathrm d x&=\int_{\mathbb R^d}\textbf{1}\{n^{-1/d}x\in\mathbb Y\}\mathbb{P}(\mathcal{D}_{n^{-1/d}x}H_n\neq 0)^\beta\,\mathrm d x\nonumber\\&=n\int_{\mathbb Y}\mathbb{P}(\mathcal{D}_y H_n\neq 0)^\beta\,\mathrm d y\nonumber\\&\leq\tilde{C}_\beta n\int_{\mathbb Y}\exp\left(-\frac{c_{p'}\beta}{2^{\alpha+1}}(n^{1/d}\mathrm d (x,K))^\alpha\right)\,\mathrm d x\nonumber\\&\leq\tilde{C}_\beta n\int_{\mathbb Y}\exp\left(-\frac{c_{p'}\beta}{2^{2\alpha+1}}(n^{1/d}\mathrm d (x,K))^\alpha\right)\,\mathrm d x.\end{align}

Since $\psi_x(G,\beta)$ is a non-increasing function of $\beta$, from the relation (23) we get

(25)\begin{align}&\int_{\mathbb R^d}\psi_x(H_n,{p''}/(4(4+{p''})))^2\,\mathrm d x\bigvee\int_{\mathbb R^d}\psi_x(H_n,{p''}/(2(4+{p''})))^2\,\mathrm d x\nonumber\\&\quad=\int_{\mathbb R^d}\psi_x(H_n,{p''}/(4(4+{p''})))^2\,\mathrm d x\leq\tilde{C}_{{p''}/(4(4+{p''}))}\,I_{K,n}.\end{align}

Similarly, from (24), since $\mathbb{P}(D_x H_n\neq 0)^{\beta}$ is non-increasing in $\beta$ we obtain

(26)\begin{align}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_n\neq 0)^{(1+{p''})/(4+{p''})}\,\mathrm d x&\leq\int_{\mathbb{R}^d}\mathbb{P}(D_x H_n\neq 0)^{{p''}/(2(4+{p''}))}\,\mathrm d x\nonumber\\&\leq\int_{\mathbb{R}^d}\mathbb{P}(D_x H_n\neq 0)^{{p''}/(4(4+{p''}))}\,\mathrm d x\nonumber \\& \leq\tilde{C}_{{p''}/(4(4+{p''}))}\,I_{K,n}.\end{align}

Step 3: Checking the conditions (15) and (16). From now on in this proof, $C>0$ denotes a generic positive constant (not depending on n) which may vary from line to line. Note that from the assumptions it follows that

(27)\begin{equation}\mathbb{V}ar(H_n)\geq C n^\tau \quad\text{for all \textit{n} large enough}\end{equation}

and

(28)\begin{equation}I_{K,n}\leq C n^\tau \quad\text{for all \textit{n} large enough.}\end{equation}

By (27), (25), and (28), we have

(29)\begin{align}\boldsymbol{\Gamma}_1(H_n)&=\frac{1}{\mathbb{V}ar(H_n)}\sqrt{\int_{\mathbb{R}^d}\psi_x(H_n,{p''}/(4(4+{p''})))^2\,\mathrm d x+\int_{\mathbb{R}^d}\psi_x(H_n,{p''}/(2(4+{p''})))^2\,\mathrm d x}\end{align}

(30)\begin{align}&\leq\frac{C}{n^\tau}\sqrt{\int_{\mathbb R^d}\psi_x(H_n,{p''}/(4(4+{p''})))^2\,\mathrm d x+\int_{\mathbb R^d}\psi_x(H_n,{p''}/(2(4+{p''})))^2\,\mathrm d x}\nonumber\\&\leq Cn^{-\tau/2},\quad\text{for all \textit{n} large enough.}\\[11pt]\nonumber\end{align}

Similarly, by (27), (26), and (28), we have

(31)\begin{align}\boldsymbol{\Gamma}_2(H_n)&=\frac{1}{\mathbb{V}ar(H_n)^{3/2}}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_n\neq 0)^{(1+{p''})/(4+{p''})}\,\mathrm d x\end{align}

(32)\begin{align}&\leq\frac{C}{n^{3\tau/2}}\int_{\mathbb R^d}\mathbb{P}(D_x H_n\neq 0)^{(1+{p''})/(4+{p''})}\,\mathrm d x\nonumber\\&\leq C n^{-\tau/2},\quad\text{for all \textit{n} large enough.}\\[11pt]\nonumber\end{align}

The relation (15) follows immediately by (30) and (32). By (30) and (32) we also have, for some $\bar n\geq 2$ large enough,

\begin{align}\sum_{n\geq\bar n}\frac{1}{n(\log n)^2}\sum_{k=\bar n}^{n}\frac{\boldsymbol{\Gamma}_i(H_k)}{k}&\leq C\sum_{n\geq\bar n}\frac{1}{n(\log n)^2}\sum_{k\geq\bar n}\frac{1}{k^{1+\tau/2}}<\infty,\quad\text{$i=1,2$}.\nonumber\end{align}

Therefore (16) follows since $\boldsymbol{\Gamma}_i(H_n)<\infty$, for any $i=1,2$ and $n\geq 1$, as a consequence of the following assumptions: $\mathbb{V}ar(H_n)\in\mathbb R_+$ and $I_{K,n}\in\mathbb R_+$ for $n\geq 1$, (29), (31), (25), and (26) (which hold for any $n\geq 1$).

Step 4: Checking the condition (17). By (27), (26), and (28), we have

(33)\begin{align}\boldsymbol{\Theta}(H_k,H_l)&=\frac{1}{\sqrt{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_k\neq 0)^{{p''}/(4(4+{p''}))}\mathbb{P}(D_x H_l\neq 0)^{{p''}/(4(4+{p''}))}\,\mathrm{d}x\nonumber\\&\leq\frac{1}{\sqrt{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_k\neq 0)^{{p''}/(4(4+{p''}))}\,\mathrm{d}x\end{align}

(34)\begin{align}&\leq\frac{C}{\sqrt{k^\tau}\sqrt{l^\tau}}\int_{\mathbb{R}^d}\mathbb{P}(D_x H_k\neq 0)^{{p''}/(4(4+{p''}))}\,\mathrm{d}x\nonumber\\&\leq C\left(\frac{k}{l}\right)^{\tau/2},\quad\text{for all \textit{k},\ \textit{l} large enough.}\\[11pt]\nonumber\end{align}

Therefore, for some $\bar k\geq 2$ large enough,

(35)\begin{align}\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=\bar k}^{l}\frac{\boldsymbol{\Theta}(H_k,H_l)}{kl}&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/2}}\sum_{k=\bar k}^{l}k^{\tau/2-1}\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/2}}\sum_{k=2}^{l}\int_{k-1}^{k}k^{\tau/2-1}\,\mathrm d x\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/2}}\int_{1}^{l}x^{\tau/2-1}\,\mathrm d x\end{align}

(36)\begin{align}&=C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/2}}(l^{\tau/2}-1)\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l}\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^2}<\infty,\\[11pt]\nonumber\end{align}

where in (35) we used that $0<\tau<2$. By (27), the Cauchy–Schwarz inequality, (26), and (28), for $\bar k\geq 2$ large enough and $n\geq\bar k$,

(37)\begin{align}&\sum_{l=\bar k}^{n}\sum_{k=1}^{\bar k-1}\frac{\boldsymbol{\Theta}(H_k,H_l)}{kl}\nonumber\\&\quad\leq C\sum_{l=\bar k}^{n}\sum_{k=1}^{\bar k-1}\frac{1}{k\mathbb{V}ar(H_k)^{1/2}l^{1+\tau/2}}\left(\int_{\mathbb{R}^d}\mathbb{P}(D_x H_k\neq 0)^{{p''}/(2(4+{p''}))}\,\mathrm d x\right)^{1/2}\nonumber\\& \qquad \times\left(\int_{\mathbb R^d}\mathbb{P}(D_x H_l\neq 0)^{{p''}/(2(4+{p''}))}\,\mathrm{d}x\right)^{1/2}\nonumber\\&\quad\leq C\sum_{k=1}^{\bar k-1}\frac{\sqrt{I_{K,k}}}{k\mathbb{V}ar(H_k)^{1/2}}\sum_{l=\bar k}^{n}\frac{1}{l}\nonumber\\&\quad\leq C\log n.\end{align}

The relations (36) and (37) yield, for $\bar k\geq 2$ large enough,

\begin{equation*}\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=1}^{l}\frac{\boldsymbol{\Theta}(H_k,H_l)}{kl}<\infty.\end{equation*}

This relation implies (17), since $\boldsymbol{\Theta}(H_k,H_l)<\infty$ for any $k,l\geq 1$, as a consequence of the following assumptions: $\mathbb{V}ar(H_n)\in\mathbb R_+$ and $I_{K,n}\in\mathbb R_+$ for any $n\geq 1$, (33), and (26) (which hold for any $k,l\geq 1$).

Step 5: Checking the condition (18). For $k,l,m,n\geq 1$, using the fact that $\psi_x(G,\beta)$ is non-increasing in $\beta$ and the Cauchy–Schwarz inequality, we have

(38)\begin{align}&\boldsymbol{\Gamma}(H_k,H_l,H_m,H_n)^2\nonumber\\&\quad\leq\frac{1}{\sqrt{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)\mathbb{V}ar(H_m)\mathbb{V}ar(H_n)}}\end{align}

(39)\begin{align}& \qquad \times \int_{\mathbb{R}^d}\psi_x(H_k,{p''}/(4(4+{p''})))\psi_x(H_l,{p''}/(4(4+{p''})))\,\mathrm d x\nonumber\\&\qquad+\frac{1}{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}\int_{\mathbb{R}^d}\psi_x(H_k,{p''}/(2(4+{p''})))\psi_x(H_l,{p''}/(2(4+{p''})))\,\mathrm d x\nonumber\\&\quad\leq\left(\frac{1}{\sqrt{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)\mathbb{V}ar(H_m)\mathbb{V}ar(H_n)}}+\frac{1}{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}\right)\nonumber\\&\qquad\times\int_{\mathbb{R}^d}\psi_x(H_k,{p''}/(4(4+{p''})))\psi_x(H_l,{p''}/(4(4+{p''})))\,\mathrm d x\nonumber\\&\quad\leq\left(\frac{1}{\sqrt{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)\mathbb{V}ar(H_m)\mathbb{V}ar(H_n)}}+\frac{1}{\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}\right)\nonumber\\&\qquad\times\left(\int_{\mathbb{R}^d}\psi_x(H_k,{p''}/(4(4+{p''})))^2\,\mathrm d x\right)^{1/2}\left(\int_{\mathbb{R}^d}\psi_x(H_l,{p''}/(4(4+{p''})))^2\,\mathrm d x\right)^{1/2}.\\[11pt]\nonumber\end{align}

By this relation, (27), (25), and (28), for l, k large enough we have

\begin{align}\boldsymbol{\Gamma}(H_k,H_l,H_k,H_l)\leq\frac{C}{(kl)^{\tau/4}}.\nonumber\end{align}

Therefore, for some $\bar k\geq 2$ large enough,

(40)\begin{equation}\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=\bar k}^{l}\frac{\boldsymbol{\Gamma}(H_k,H_l,H_k,H_l)}{kl}\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/4}}\sum_{k=\bar k}^{l}\frac{1}{k^{1+\tau/4}}<\infty.\end{equation}

For $\bar k\geq 2$ large enough, by (39), (25), (28), and (27) we have

\begin{align}&\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=1}^{\bar k-1}\frac{\boldsymbol{\Gamma}(H_k,H_l,H_k,H_l)}{kl}\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=1}^{\bar k-1}\frac{1}{kl}\sqrt{\frac{I_{K,k}^{1/2}I_{K,l}^{1/2}} {\mathbb{V}ar(H_k)\mathbb{V}ar(H_l)}}\nonumber\\&\leq C \left(\sum_{k=1}^{\bar k-1}\frac{1}{k}\sqrt{\frac{I_{K,k}^{1/2}} {\mathbb{V}ar(H_k)}}\right) \sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\left(\sum_{l=\bar k}^{n} \frac{1}{l}\sqrt{\frac{I_{K,l}^{1/2}} {\mathbb{V}ar(H_l)}}\right)\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l}\sqrt{\frac{I_{K,l}^{1/2}} {\mathbb{V}ar(H_l)}}\nonumber\\&\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l^{1+\tau/4}}<\infty.\nonumber\end{align}

This relation, along with (40), yields (18), since $\boldsymbol{\Gamma}(H_k,H_l,H_k,H_l)<\infty$ for any $k,l\geq 1$ as a consequence of the following assumptions: $\mathbb{V}ar(H_n)\in\mathbb R_+$ and $I_{K,n}\in\mathbb R_+$ for any $n\geq 1$, (39), and (25) (which hold for any $k,l\geq 1$).

Step 6: Checking the condition (19). By the trivial inequality

\begin{equation*}\boldsymbol{\Lambda}(H_k,H_l,H_m,H_n)\leq\boldsymbol{\Gamma}(H_k,H_l,H_m,H_n),\end{equation*}

(39), the inequality $(a+b)^{1/2}\leq a^{1/2}+b^{1/2}$ for $a,b\geq 0$, and (25), for $k,l\geq 1$ we have

(41)\begin{align}\boldsymbol{\Lambda}_{max}(H_k,H_l)&\leq C\left(\frac{1}{\mathbb{V}ar(H_k)^{3/4}\mathbb{V}ar(H_l)^{1/4}}+\frac{1}{\mathbb{V}ar(H_k)}\right)I_{K,k}^{1/2}\nonumber\\&\qquad\vee\left(\frac{1}{\mathbb{V}ar(H_l)^{3/4}\mathbb{V}ar(H_k)^{1/4}}+\frac{1}{\mathbb{V}ar(H_l)}\right)I_{K,l}^{1/2}\nonumber\\&\qquad\vee\left(\frac{1}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}+\frac{1}{\mathbb{V}ar(H_k)}\right)I_{K,k}^{1/2}\nonumber\\&\qquad\vee\left(\frac{1}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}+\frac{1}{\mathbb{V}ar(H_l)}\right)I_{K,l}^{1/2}\nonumber\\&\qquad\vee\left(\frac{1}{\mathbb{V}ar(H_k)^{3/4}\mathbb{V}ar(H_l)^{1/4}}+\frac{1}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}\right)(I_{K,k}I_{K,l})^{1/4}\nonumber\\&\qquad\vee\left(\frac{1}{\mathbb{V}ar(H_l)^{3/4}\mathbb{V}ar(H_k)^{1/4}}+\frac{1}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}\right)(I_{K,k}I_{K,l})^{1/4}\nonumber\\&\qquad\vee\frac{(I_{K,k}I_{K,l})^{1/4}}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}\nonumber\\&=C\left(\frac{I_{K,k}^{1/2}}{\mathbb{V}ar(H_k)^{3/4}}\frac{1}{\mathbb{V}ar(H_l)^{1/4}}+\frac{I_{K,k}^{1/2}}{\mathbb{V}ar(H_k)}\right)\nonumber\\&\qquad\vee\left(\frac{1}{\mathbb{V}ar(H_k)^{1/4}}\frac{I_{K,l}^{1/2}}{\mathbb{V}ar(H_l)^{3/4}}+\frac{I_{K,l}^{1/2}}{\mathbb{V}ar(H_l)}\right)\nonumber\\&\qquad\vee\left(\frac{I_{K,k}^{1/2}}{(\mathbb{V}ar(H_k))^{1/2}}\frac{1}{(\mathbb{V}ar(H_l))^{1/2}}+\frac{I_{K,k}^{1/2}}{\mathbb{V}ar(H_k)}\right)\nonumber\\&\qquad\vee\left(\frac{1}{(\mathbb{V}ar(H_k))^{1/2}}\frac{I_{K,l}^{1/2}}{(\mathbb{V}ar(H_l))^{1/2}}+\frac{I_{K,l}^{1/2}}{\mathbb{V}ar(H_l)}\right)\nonumber\\&\qquad\vee\left(\frac{(I_{K,k}I_{K,l})^{1/4}}{\mathbb{V}ar(H_k)^{3/4}\mathbb{V}ar(H_l)^{1/4}}+\frac{(I_{K,k}I_{K,l})^{1/4}}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}\right)\nonumber\\&\qquad\vee\left(\frac{(I_{K,k}I_{K,l})^{1/4}}{\mathbb{V}ar(H_l)^{3/4}\mathbb{V}ar(H_k)^{1/4}}+\frac{(I_{K,k}I_{K,l})^{1/4}}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}\right)\nonumber\\&\qquad\vee\frac{(I_{K,k}I_{K,l})^{1/4}}{(\mathbb{V}ar(H_k)\mathbb{V}ar(H_l))^{1/2}}.\end{align}

Consequently, by (28) and (27), for some $\bar k\geq 2$ large enough and $l\geq k\geq\bar k$, we have

\begin{align}\boldsymbol{\Lambda}_{max}(H_k,H_l)\leq\frac{C}{k^{\tau/2}}.\nonumber\end{align}

Therefore, for some $\bar k\geq 2$ large enough, we have

(42)\begin{equation}\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=\bar k}^{l}\frac{\boldsymbol{\Lambda}_{max}(H_k,H_l)}{kl}\leq C\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l}\sum_{k=\bar k}^{l}\frac{1}{k^{1+\tau/2}}<\infty.\end{equation}

Let $\bar k\geq 2$ be large enough and $l\geq\bar k>k$, and let $C_i(k)\in\mathbb{R}_+$, $i=0,1,\ldots,12$, denote suitable constants, depending on k but not on l, which may vary from line to line. By (41), (28), and (27), we have

\begin{align}\boldsymbol{\Lambda}_{max}(H_k,H_l)&\leq\left(C_1(k)\frac{1}{\mathbb{V}ar(H_l)^{1/4}}+C_2(k)\right)\nonumber\\&\vee\left(C_3(k)\frac{I_{K,l}^{1/2}}{\mathbb{V}ar(H_l)^{3/4}}+C_4(k)\frac{1}{\mathbb{V}ar(H_l)}\right)\vee\left(C_5(k)\frac{1}{(\mathbb{V}ar(H_l))^{1/2}}+C_6(k)\right)\nonumber\\&\vee\left(C_7(k)\frac{I_{K,l}^{1/2}}{(\mathbb{V}ar(H_l))^{1/2}}+\frac{I_{K,l}^{1/2}}{\mathbb{V}ar(H_l)}\right)\nonumber\\&\vee\left(C_8(k)\frac{(I_{K,l})^{1/4}}{\mathbb{V}ar(H_l)^{1/4}}+C_9(k)\frac{(I_{K,l})^{1/4}}{(\mathbb{V}ar(H_l))^{1/2}}\right)\nonumber\\&\vee\left(C_{10}(k)\frac{(I_{K,l})^{1/4}}{\mathbb{V}ar(H_l)^{3/4}}+C_{11}(k)\frac{(I_{K,l})^{1/4}}{(\mathbb{V}ar(H_l))^{1/2}}\right)\vee C_{12}(k)\frac{(I_{K,l})^{1/4}}{(\mathbb{V}ar(H_l))^{1/2}}\nonumber\\&\leq\left(C_1(k)l^{-\tau/4}+C_2(k)\right)\vee\left(C_3(k)l^{-\tau/4}+C_4(k)l^{-\tau/2}\right)\nonumber\\&\vee\left(C_5(k)l^{-\tau/2}+C_6(k)\right)\vee\left(C_7(k)+l^{-\tau/2}\right)\nonumber\\&\vee\left(C_8(k)+C_9(k)l^{-\tau/4}\right)\vee\left(C_{10}(k)l^{-\tau/2}+C_{11}(k)l^{-\tau/4}\right)\vee C_{12}(k)l^{-\tau/4}\nonumber\\&\leq C_0(k).\nonumber\end{align}

Therefore, for $\bar k\geq 2$ large enough, we have

\begin{align*}\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\sum_{k=1}^{\bar k-1}\frac{\boldsymbol{\Lambda}_{max}(H_k,H_l)}{kl}&\leq C_0(k)\sum_{n\geq\bar k}\frac{1}{n(\log n)^3}\sum_{l=\bar k}^{n}\frac{1}{l}<\infty.\end{align*}

This relation, along with (42), yields (19), since $\boldsymbol{\Lambda}_{max}(H_k,H_l)<\infty$ for any $k,l\geq 1$ as a consequence of the following assumptions: $\mathbb{V}ar(H_n)\in\mathbb R_+$ and $I_{K,n}\in\mathbb R_+$ for any $n\geq 1$, and (41) (which holds for any $k,l\geq 1$).

4.1. Proof of Corollary 2.3

Set $H_n\,:\!=n^{m/d}L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y)$ and note that for $F_n$ defined in (10) we have

\begin{equation*}F_n=\frac{H_n-\mathbb{E}[H_n]}{\sqrt{\mathbb{V}ar(H_n)}}.\end{equation*}

The claim follows by Theorem 2.2 with $\tau=1$. Indeed: (i) By Theorem 2.1 and Lemma 6.3 in [Reference Penrose and Yukich16] (see also the discussion on the related variance bounds in [Reference Lachièze-Rey, Schulte and Yukich8]) we have $n^{1-2m/d}=O(\mathbb{V}ar(L_{NG_k}^{(m)}(\eta_n\cap\mathbb Y))$, and so $n=O(\mathbb{V}ar(H_n))$. (ii) One can show that the functionals $H_n$, $n\geq 1$, are stabilizing following the proof of Theorem 3.1 in [Reference Lachièze-Rey, Schulte and Yukich8]. We briefly sketch the proof. For $n\geq 1$ and $x\in\eta_n\cap\mathbb Y$, we set $\xi_n(x,\eta_n\cap\mathbb Y)\,:\!=n^{m/d}\xi^{(m)}(x,\eta_n\cap\mathbb Y)$. It turns out that $H_n=\sum_{x\in\eta_n\cap\mathbb Y}\xi_n(x,\eta_n\cap\mathbb Y)$ and that the score functions $\xi_n$, $n\geq 1$, satisfy (5), (6), and (7) with $K\,:\!=\mathbb Y$. (iii) By (8) it easily follows that $I_{K,n}=I_{\mathbb Y,n}=n\ell_d(\mathbb Y)$.

4.2. Proof of Corollary 2.4

We set $H_n\,:\!=\mathcal{C}_k(\eta_n\cap\mathbb Y,rn^{-1/d})$, $n\geq 1$; the claim then follows by Theorem 2.2 with $\tau=1$. Indeed: (i) As noticed in the proof of Theorem 2.5 in Section 7 of [Reference Penrose and Yukich17] (see also Remark (iv) on p. 967 in [Reference Lachièze-Rey, Schulte and Yukich8]), we have $\inf_{n\geq 1}\mathbb{V}ar(H_n)/n>0$, and so $n=O(\mathbb{V}ar(H_n))$. (ii) One can show that the functionals $H_n$, $n\geq 1$, are stabilizing following the proof of Theorem 3.15 in [Reference Lachièze-Rey, Schulte and Yukich8]. We briefly sketch the proof. For $n\geq 1$ and $x\in\eta_n\cap\mathbb Y$, we denote by $\xi_n(x,\eta_n\cap\mathbb Y)$ the number of cliques of order $k+1$ in $G(\eta_n\cap\mathbb Y,rn^{-1/d})$ containing x. It turns out that $H_n=\sum_{x\in\eta_n\cap\mathbb Y}\xi_n(x,\eta_n\cap\mathbb Y)$ and that the score functions $\xi_n$, $n\geq 1$, satisfy (5), (6), and (7) with $K\,:\!=\mathbb Y$. (iii) By (8) it easily follows that $I_{K,n}=I_{\mathbb Y,n}=n\ell_d(\mathbb Y)$.

4.3. Proof of Corollary 2.5

Set $H_n\,:\!=n(\ell_d(A_n)-\ell_d(A))$, $n\geq 1$, and note that for $F_n$ defined in (12) we have

\begin{equation*}F_n=\frac{H_n-\mathbb{E}[H_n]}{\sqrt{\mathbb{V}ar(H_n)}}.\end{equation*}

The claim follows by applying Theorem 2.2 with $\tau\,:\!=1-1/d$. Indeed: (i) By Theorem 1.2 in [Reference Schulte20] (see also Theorem 1.1 in [Reference Thäle and Yukich21]) we have $n^{1-1/d}=O(\mathbb{V}ar(H_n))$. (ii) One can show that the functionals $H_n$, $n\geq 1$, are stabilizing following the proof of Theorem 3.4 in [Reference Lachièze-Rey, Schulte and Yukich8] (and the references cited therein). For the sake of completeness, we briefly sketch the proof. For $n\geq 1$ and $x\in\eta_n\cap\mathbb Y$, we set

\begin{equation*}\xi(x,\eta_n\cap\mathbb Y)\,:\!=\textbf{1}_{A}(x)\ell_d(\mathrm{C}(x,\eta_n\cap\mathbb Y)\cap A^{c})-\textbf{1}_{A^{c}}(x)\ell_d(\mathrm{C}(x,\eta_n\cap\mathbb Y)\cap A)\end{equation*}

and $\xi_n(x,\eta_n\cap\mathbb Y)\,:\!=n\xi(x,\eta_n\cap\mathbb Y)$. It turns out that $H_n=\sum_{x\in\eta_n\cap\mathbb Y}\xi_n(x,\eta_n\cap\mathbb Y)$ and that the score functions $\xi_n$, $n\geq 1$, satisfy (5), (6), and (7) with $K\,:\!=\partial A$. (iii) To prove $I_{K,n}=I_{\partial A,n}=O(n^{1-1/d})$, for the sake of clarity, we reproduce the argument in the proof of Theorem 2.3 in [Reference Lachièze-Rey, Schulte and Yukich8]. For all the notions and results from geometric measure theory considered here we refer the reader to [Reference Ambrosio, Fusco and Pallara1]. We first note that since A is a convex body, it has a $(d-1)$-rectifiable boundary (i.e., its boundary is the Lipschitz image of a bounded set in $\mathbb{R}^{d-1}$). Then the $(d-1)$-dimensional upper Minkowski content of $\partial A$, denoted by $\overline{\mathcal{M}}^{d-1}(\partial A)$, is a scalar multiple of the $(d-1)$-dimensional Hausdorff measure of $\partial A$, denoted by $\mathcal{H}^{d-1}(\partial A)$. Therefore $\overline{\mathcal{M}}^{d-1}(\partial A)<\infty$, and by Lemma 5.12 in [Reference Lachièze-Rey, Schulte and Yukich8], there exists a positive constant $C>0$ such that

(43)\begin{equation}\mathcal{H}^{d-1}(\partial A_r)\leq C\big(1+r^{d-1}\big),\quad r>0,\end{equation}

where $\partial A_r\,:\!=\{x\in\mathbb{R}^d:\,\,\mathrm{d}(x,\partial A)\leq r\}$. Setting $c\,:\!={p''}\,c_{p'}/[2^{2\alpha+3}(4+{p''})]$, we finally have

\begin{align}I_{K,n}&=n\int_{\mathbb Y}\exp(-c n^{\alpha/d}\mathrm{d}(x,\partial A)^\alpha)\,\mathrm d x\nonumber\\&=n\int_{\mathbb Y\setminus\partial A}\exp(-c n^{\alpha/d}\mathrm{d}(x,\partial A)^\alpha)\,\mathrm d x\nonumber\displaybreak\\&=n\int_0^{\infty}\exp(-c n^{\alpha/d}r^\alpha)\mathcal{H}^{d-1}(\partial A_r)\,\mathrm d r\end{align}

(44)\begin{align}&\leq C n\int_0^{\infty}\exp(-c n^{\alpha/d}r^\alpha)\big(1+r^{d-1}\big)\,\mathrm d r\end{align}

(45)\begin{align}& \le Cn\int_0^\infty {\exp ( - c{n^{\alpha /d}}{r^\alpha })(1 + {r^{d - 1}}} \\ \le C{n^{1 - 1/d}}\int_0^\infty {{e^{ - c{u^\alpha }}}(1 + {u^{d - 1}}){\rm{d}}u,}\end{align}

where (44) follows by the coarea formula and (45) is a consequence of (43).

5.1. Elements of the Malliavin calculus on the Poisson space

For $r\in\mathbb{R}_+$ and $n\geq 1$ an integer, we denote by $L^r(\mu^{\otimes n})$ the set of all measurable functions $g\,:\,\mathbb X^n\to\mathbb R$ such that $\int_{\mathbb X^n}|g(x)|^r\mu^{\otimes n}(\mathrm d x)<\infty$. We call a function $g\,:\,\mathbb X^n\to\mathbb R$ symmetric if it is invariant under permutations of its arguments, and denote by $L_s^2(\mu^{\otimes n})$ the set of all symmetric functions $g\in L^2(\mu^{\otimes n})$. For $g_1,g_2\in L^2(\mu^{\otimes n})$, we define $\langle g_1,g_2\rangle_n\,:\!=\int_{\mathbb X^n}g_1(x)g_2(x)\mu^{\otimes n}(\mathrm d x)$ and $\|g_1\|_n\,:\!=\sqrt{\langle g_1,g_1\rangle_n}$.

For $F\in L_{\eta}^2$, $F=f(\eta)$, we extend the definition of the Malliavin operators D and $D^2$ by defining

\begin{equation*}D_{x_1,\ldots,x_n}^n F\,:\!=D_{x_n}\Big(D_{x_1,\ldots,x_{n-1}}^{n-1}F\Big), \qquad x_1,\ldots,x_n\in\mathbb X, \qquad n\geq 3.\end{equation*}

It is well-known that every $F\in L_{\eta}^2$ admits the representation

(46)\begin{equation}F=\mathbb{E}[F]+\sum_{n\geq 1}I_n(g_n),\end{equation}

where $g_n(x_1,\ldots,x_n)=\frac{1}{n!}\mathbb{E}[D_{x_1,\ldots,x_n}^n F]$ and, for $g_n\in L_s^2(\mu^{\otimes n})$, we denote by $I_n(g_n)$ the nth-order Wiener–Itô integral with respect to the centered Poisson measure $\eta(\mathrm d x)-\mu(\mathrm d x)$; see e.g. [Reference Last and Penrose9]. Another operator that we shall consider is the so-called Ornstein–Uhlenbeck generator L. Given $F\in L_{\eta}^2$ of the form (46), we write $F\in\mathrm{Dom}(L)$ if $\sum_{n\geq 1}n^2 n!\|g_n\|_n^2<\infty$. In this case we define

\begin{equation*}L F\,:\!=-\sum_{n\geq 1}n I_n(g_n).\end{equation*}

The (pseudo) inverse of L is given by

\begin{equation*}L^{-1}F\,:\!=-\sum_{n\geq 1}n^{-1}I_n(g_n).\end{equation*}

It can be easily checked that the random variable $L^{-1}F$ is a well-defined element of $L_{\eta}^2$ for every $F\in L_{\eta}^2$. In the following, we write $F\in\mathrm{Dom}(D)$ if $F\in L_{\eta}^2$ and $\int_{\mathbb X}\mathbb{E}[|D_x F|^2]\mu(\mathrm d x)<\infty$.

5.2. Preliminary lemmas

In this subsection we provide some relations among Malliavin operators on the Poisson space, which will be crucial for the proof of Theorem 2.6. The following lemmas hold.

Lemma 5.1. For $F\in L_\eta^2$ and $r\geq 1$, we have

\begin{equation*}\mathbb{E}[|D_x L^{-1}F|^r]\leq\mathbb{E}[|D_xF|^r],\quad\text{$\mu$-a.e. $x\in\mathbb X$.}\end{equation*}

Lemma 5.2. For any $F,G\in\mathrm{Dom}(D)$ such that $\mathbb{E}[F]=0$, we have

\begin{equation*}\mathbb{E}[FG]=\mathbb{E}[\langle DG,-DL^{-1}F\rangle_{1}].\end{equation*}

Lemma 5.3. For $F\in\mathrm{Dom}(D)$ such that $\mathbb{E}[F]=0$, we have

\begin{equation*}\mathbb{V}\mathrm{ar}(\langle DF,-DL^{-1}F\rangle_{1})\leq\gamma_1(F)^2,\end{equation*}

where

(47)\begin{align}\gamma_1(F)^2&\,:\!=4\int_{\mathbb{X}^3}(\mathbb{E}[(D_{x_1,x_3}^{2}F)^2(D_{x_2,x_3}^{2}F)^2])^{1/2}(\mathbb{E}[(D_{x_1}F)^2(D_{x_2}F)^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3)\nonumber\\&\qquad\qquad\qquad+\int_{\mathbb{X}^3}\mathbb{E}[(D_{x_1,x_3}^{2}F)^2(D_{x_2,x_3}^{2}F)^2]\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3).\end{align}

We refer the reader to [Reference Last, Peccati and Schulte11] for the proof of these lemmas (see, therein, Lemma 3.4, the first displayed formula in the proof of Proposition 4.1, and the statement of Proposition 4.1 itself, respectively).

The next lemma gives a Gaussian bound for functionals of the Poisson measure. We refer the reader to Lemma 2.2 in [Reference Bercu, Nourdin and Taqqu3] and Lemma 3.1 in [Reference Zheng25] for analogous inequalities on the Wiener and Rademacher spaces, respectively.

Lemma 5.4. For $F\in\mathrm{Dom}(D)$ such that $\mathbb{E}[F]=0$, we have

\begin{align}|\mathbb{E}[\mathrm{e}^{\textbf{i}tF}]-\mathrm{e}^{-t^2/2}|\leq |t|^2(|1-\mathbb{E}[F^2]|+\gamma_1(F))+\frac{|t|^3}{\sqrt 2}\gamma_2(F) \quad\text{$\forall$ $t\in\mathbb R$},\nonumber\end{align}

where $\textbf{i}\,:\!=\sqrt{-1}$ and

(48)\begin{equation}\gamma_2(F)\,:\!=\int_{\mathbb{X}}\mathbb{E}[|D_xF|^3]\mu(\mathrm{d}x).\end{equation}

$\textit{Proof.}$ Let $\mathcal{C}_b^{1}(\mathbb R)$ be the family of bounded and differentiable functions from $\mathbb R$ to $\mathbb R$ with a bounded first derivative. For any $g\in\mathcal{C}_b^1(\mathbb R)$ it holds that $g(F)\in\mathrm{Dom}(D)$ (indeed, we clearly have $g(F)\in L_\eta^2$, and by the mean value theorem, $|D_xg(F)|\leq\|g'\|_\infty |D_x F|$ for any $x\in\mathbb X$). Therefore, by Lemma 5.2,

\begin{equation*}\mathbb{E}[Fg(F)]=\mathbb{E}[\langle Dg(F),-DL^{-1}F\rangle_{1}] \quad\text{$\forall$ $g\in\mathcal{C}_b^1(\mathbb R)$.}\end{equation*}

So, for any $t\in\mathbb R$,

(49)\begin{align}\mathbb{E}[F\mathrm{e}^{\textbf{i}tF}]&=\mathbb{E}[F\cos(tF)]+\textbf{i}\mathbb{E}[F\sin(tF)]\nonumber\\&=\mathbb{E}[\langle D\cos(tF),-DL^{-1}F\rangle_{1}]+\textbf{i}\mathbb{E}[\langle D\sin(tF),-DL^{-1}F\rangle_{1}].\end{align}

For $F=f(\eta)$, by Taylor’s formula with integral remainder, for $(x,t)\in\mathbb X\times\mathbb R$,

(50)\begin{equation}D_x\cos(tF)=\cos(tf(\eta+\varepsilon_x))-\cos(t f(\eta))=-t\sin(tF)D_xF+t^2R_{1,t}(x),\end{equation}

and similarly

(51)\begin{equation}D_x\sin(tF)=\sin(tf(\eta+\varepsilon_x))-\sin(t f(\eta))=t\cos(tF)D_xF+t^2R_{2,t}(x),\end{equation}

where

\begin{equation*}R_{1,t}(x)\,:\!=\int_{f(\eta)}^{f(\eta+\varepsilon_x)}(u-f(\eta+\varepsilon_x))\cos tu\,\mathrm{d}u\end{equation*}

and

\begin{equation*}R_{2,t}(x)\,:\!=\int_{f(\eta)}^{f(\eta+\varepsilon_x)}(u-f(\eta+\varepsilon_x))\sin tu\,\mathrm{d}u.\end{equation*}

Note that $|R_{i,t}(x)|\leq\frac{|D_xF|^2}{2}$ for $i=1,2$, and so

(52)\begin{equation}|R_{1,t}(x)+\textbf{i}R_{2,t}(x)|\leq\frac{|D_x F|^2}{\sqrt 2}.\end{equation}

By (49), (50), and (51), we have

(53)\begin{align}\mathbb{E}[F\mathrm{e}^{\textbf{i}tF}]&=t\mathbb{E}[(-\sin(tF)+\textbf{i}\cos(tF))\langle DF,-DL^{-1}F\rangle_{1}]+t^2\mathbb{E}[\langle R_{1,t}+\textbf{i}R_{2,t},-DL^{-1}F\rangle_{1}]\nonumber\\&=\textbf{i}t\mathbb{E}[\mathrm{e}^{\textbf{i}tF}\langle DF,-DL^{-1}F\rangle_{1}]+t^2\mathbb{E}[\langle R_{1,t}+\textbf{i}R_{2,t},-DL^{-1}F\rangle_{1}].\end{align}

We put $\varphi(t)\,:\!=\mathrm{e}^{t^2/2}\mathbb{E}[\mathrm{e}^{\textbf{i}tF}]$, $t\in\mathbb R$. By the mean value theorem, (53), and (52), we have

\begin{align}|\varphi(t)-\varphi(0)|&\leq |t|\sup_{u\in [0,t]}|\varphi'(u)|\nonumber\\[4pt]&=|t|\sup_{u\in [0,t]}|u\mathrm{e}^{u^2/2}\mathbb{E}[\mathrm{e}^{\textbf{i}uF}]+\textbf{i}\mathrm{e}^{u^2/2}\mathbb{E}[F\mathrm{e}^{\textbf{i}uF}]|\nonumber\\[4pt]&=|t|\sup_{u\in [0,t]}|u\mathrm{e}^{u^2/2}\mathbb{E}[\mathrm{e}^{\textbf{i}uF}]-u\mathrm{e}^{u^2/2}\mathbb{E}[\mathrm{e}^{\textbf{i}uF}\langle DF,-DL^{-1}F\rangle_{1}]\nonumber\\[4pt]&\quad+\textbf{i}u^2\mathrm{e}^{u^2/2}\mathbb{E}[\langle R_{1,u}+\textbf{i}R_{2,u},-DL^{-1}F\rangle_{1}]|\nonumber\\[4pt]&\leq|t|^2\mathrm{e}^{t^2/2}\sup_{u\in [0,t]}|\mathbb{E}[\mathrm{e}^{\textbf{i}uF}(1-\langle DF,-DL^{-1}F\rangle_{1})]|\nonumber\\[4pt]&\quad+|t|^3\mathrm{e}^{t^2/2}\sup_{u\in [0,t]}\mathbb{E}[|\langle R_{1,u}+\textbf{i}R_{2,u},-DL^{-1}F\rangle_{1}|]\nonumber\\[4pt]&\leq|t|^2\mathrm{e}^{t^2/2}\mathbb{E}[|1-\langle DF,-DL^{-1}F\rangle_{1}|]\nonumber\\[4pt]&\quad+\frac{|t|^3\mathrm{e}^{t^2/2}}{\sqrt 2}\int_{\mathbb X}\mathbb{E}[|D_x F|^2|D_xL^{-1}F|]\mu(\mathrm{d}x).\nonumber\end{align}

So by Lemma 5.2, the Cauchy–Schwarz inequality, and Lemma 5.3, it follows that

(54)\begin{align}|\mathbb{E}[\mathrm{e}^{\textbf{i}tF}]-\mathrm{e}^{-t^2/2}|&\leq |t|^2|1-\mathbb{E}[F^2]|+|t|^2\mathbb{E}[|\mathbb{E}[F^2]-\langle DF,-DL^{-1}F\rangle_{1}|]\nonumber\\&\qquad\qquad\qquad+\frac{|t|^3}{\sqrt 2}\int_{\mathbb X}\mathbb{E}[|D_x F|^2|DL^{-1}F|]\mu(\mathrm{d}x)\nonumber\\&\leq |t|^2|1-\mathbb{E}[F^2]|+|t|^2\sqrt{\mathbb{V}\mathrm{ar}(\langle DF,-DL^{-1}F\rangle_{1})}\nonumber\\&\qquad\qquad\qquad+\frac{|t|^3}{\sqrt 2}\int_{\mathbb X}\mathbb{E}[|D_x F|^2|D_xL^{-1}F|]\mu(\mathrm{d}x)\nonumber\\&\leq |t|^2(|1-\mathbb{E}[F^2]|+\gamma_1(F))+\frac{|t|^3}{\sqrt 2}\int_{\mathbb X}\mathbb{E}[|D_x F|^2|D_xL^{-1}F|]\mu(\mathrm{d}x).\end{align}

By Hölder’s inequality and Lemma 5.1, we have

\begin{align}\int_X\mathbb{E}[|D_xF|^2|D_x L^{-1}F|]\mu(\mathrm{d}x)&\leq\int_{\mathbb{X}}\mathbb{E}[|D_xF|^3]^{2/3}\mathbb{E}[|D_x L^{-1}F|^3]^{1/3}\mu(\mathrm{d}x)\nonumber\\&\leq\int_{\mathbb{X}}\mathbb{E}[|D_xF|^3]^{2/3}\mathbb{E}[|D_x F|^3]^{1/3}\mu(\mathrm{d}x)\nonumber\\&=\int_{\mathbb{X}}\mathbb{E}[|D_xF|^3]\mu(\mathrm{d}x).\nonumber\end{align}

The claim follows from combining this latter inequality with (54).

We conclude this subsection by recalling a result, due to Ibragimov and Lifshits [Reference Ibragimov and Lifshits7], which provides general sufficient conditions for the ASCLT to hold.

Lemma 5.5. Let $\{X_n\}_{n\geq 1}$ be a sequence of real-valued random variables converging in distribution to a random variable X, and put

(55)\begin{equation}\Delta_n(t)\,:\!=\frac{1}{\log n}\sum_{k=1}^{n}\frac{1}{k}\left(\mathrm{e}^{\textbf{i}t X_k}-\mathbb{E}[\mathrm{e}^{\textbf{i}t X}]\right)\!,\quad t\in\mathbb R.\end{equation}

If, for all $r\in\mathbb{R}_+$,

\begin{equation*}\sup_{|t|\leq r}\sum_{n\geq 2}\frac{\mathbb{E}[|\Delta_n(t)|^2]}{n\log n}<\infty,\end{equation*}

then (1) holds with X in place of Z. If X is distributed according to the standard normal law, then $\{X_n\}_{n\geq 1}$ satisfies the ASCLT.

5.3. Ancillary ASCLTs on the Poisson space and proof of Theorem 2.6

Let $F_1,F_2,F_3,F_4\in L_\eta^2$ be such that $\mathbb{E}[F_i]=0$, $i=1,2,3,4$. We generalize the definition of the quantity $\gamma_1(F)^2$ by setting

\begin{align}\gamma(F_1,F_2,F_3,F_4)^2&\,:\!=4\lambda(F_1,F_2,F_3,F_4)^2+\int_{\mathbb{X}^3}\mathbb{E}[(D_{x_1,x_3}^{2}F_1)^2(D_{x_2,x_3}^{2}F_2)^2]\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3),\nonumber\end{align}

where

\begin{align}\lambda(F_1,F_2,F_3,F_4)^2&\,:\!=\int_{\mathbb{X}^3}(\mathbb{E}[(D_{x_1,x_3}^{2}F_1)^2(D_{x_2,x_3}^{2}F_2)^2])^{1/2}\nonumber\\& \qquad\qquad\qquad \times (\mathbb{E}[(D_{x_1}F_3)^2(D_{x_2}F_4)^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3).\nonumber\end{align}

Note that $\gamma_1(F_1)^2=\gamma(F_1,F_1,F_1,F_1)^2$,

(56)\begin{equation}\lambda(F_1,F_2,F_3,F_4)^2=\lambda(F_2,F_1,F_4,F_3)^2, \quad\text{and}\quad\gamma(F_1,F_2,F_3,F_4)^2=\gamma(F_2,F_1,F_4,F_3)^2.\end{equation}

We also put

\begin{equation*}\theta(F_1,F_2)\,:\!=\int_{\mathbb X}(\mathbb{E}[|D_x F_1|^2])^{1/2}(\mathbb{E}[|D_x F_2|^2])^{1/2}\mu(\mathrm{d}x).\end{equation*}

The following ASCLT will be proved at the end of this subsection.

Proposition 5.6. Assume $\{F_n\}_{n\geq 1}\subset L_\eta^2$ and $\mathbb{E}[F_n]=0$, $\mathbb{E}[F_n^2]=1$, for any $n\geq 1$. Moreover, suppose

(57)\begin{equation}\lim_{n\to\infty}\gamma_1(F_n)=\lim_{n\to\infty}\gamma_2(F_n)=0,\end{equation}

(58)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^2}\sum_{k=1}^{n}\frac{\gamma_i(F_k)}{k}<\infty,\quad\text{$i=1,2$},\end{equation}

(59)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\theta(F_k,F_l)}{kl}<\infty,\end{equation}

(60)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\gamma(F_k,F_l,F_k,F_l)}{kl}<\infty,\end{equation}

and

(61)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\lambda_{max}(F_k,F_l)}{kl}<\infty,\end{equation}

where

\begin{align}\lambda_{max}(F_k,F_l)&\,:\!=\lambda(F_k,F_k,F_k,F_l)\vee\lambda(F_l,F_l,F_l,F_k)\vee\lambda(F_k,F_k,F_l,F_l)\vee\lambda(F_l,F_l,F_k,F_k)\nonumber\\&\quad\vee\lambda(F_l,F_k,F_k,F_k)\vee\lambda(F_k,F_l,F_l,F_l)\vee\lambda(F_k,F_l,F_l,F_k).\nonumber\end{align}

Then $\{F_n\}_{n\geq 1}$ satisfies the ASCLT.

Next, to state a corollary of this proposition, we introduce some more notation. For $F_1,F_2,F_3,F_4\in L_\eta^2$ such that $\mathbb{E}[F_i]=0$, $i=1,2,3,4$, we set

\begin{align}\Gamma_1(F_1)^2&\,:\!=4\int_{\mathbb{X}^3}\mathbb{E}\Big[\big(D_{x_1,x_3}^{2}F_1\big)^4\Big]^{1/4}\mathbb{E}\Big[\big(D_{x_2,x_3}^{2}F_1\big)^4\Big]^{1/4}\mathbb{E}\Big[(D_{x_1}F_1)^4\Big]^{1/4}\nonumber\\ &\quad \times \mathbb{E}\Big[\big(D_{x_2}F_1\big)^4\Big]^{1/4}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3)\nonumber\\ &\quad+\int_{\mathbb{X}^3}\mathbb{E}\Big[\big(D_{x_1,x_3}^{2}F_1\big)^4\Big]^{1/2}\mathbb{E}\Big[\big(D_{x_2,x_3}^{2}F_1\big)^4\Big]^{1/2}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3)\nonumber\\ &=4\int_{\mathbb{X}}\left(\int_{\mathbb{X}}\mathbb{E}\Big[\big(D_{x_1,x_2}^{2}F_1\big)^4\Big]^{1/4}\mathbb{E}\Big[(D_{x_1}F_1)^4\Big]^{1/4}\mu(\mathrm d x_1)\right)^2\mu(\mathrm d x_2)\nonumber\\ &\quad+\int_{\mathbb{X}}\left(\int_{\mathbb X}\mathbb{E}\Big[\big(D_{x_1,x_2}^{2}F_1\big)^4\Big]^{1/2}\mu(\mathrm d x_1)\right)^2\mu(\mathrm d x_2),\nonumber\end{align}

\begin{align}\Gamma(F_1,F_2,F_3,F_4)^2&\,:\!=4\Lambda(F_1,F_2,F_3,F_4)^2+\int_{\mathbb{X}^3}\mathbb{E}\Big[\big(D_{x_1,x_3}^{2}F_1\big)^4\Big]^{1/2}\nonumber\\ & \qquad \mathbb{E}\Big[\big(D_{x_2,x_3}^{2}F_2\big)^4\Big]^{1/2}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3)\nonumber\\ &=4\Lambda(F_1,F_2,F_3,F_4)^2\nonumber\\& \quad+\int_{\mathbb{X}}\left(\int_{\mathbb X}\mathbb{E}\Big[\big(D_{x_1,x_2}^{2}F_1\big)^4\Big]^{1/2}\mu(\mathrm d x_1)\right)\nonumber\\ &\quad \times\left(\int_{\mathbb X}\mathbb{E}[(D_{x_1,x_2}^{2}F_2)^4]^{1/2}\mu(\mathrm d x_1)\right)\mu(\mathrm d x_2),\nonumber\end{align}

\begin{align}&\Lambda(F_1,F_2,F_3,F_4)^2\nonumber\\&\,:\!=\int_{\mathbb{X}^3}\mathbb{E}\Big[\big(D_{x_1,x_3}^{2}F_1\big)^4\Big]^{1/4}\mathbb{E}\Big[\big(D_{x_2,x_3}^{2}F_2\big)^4\Big]^{1/4} \mathbb{E}\Big[(D_{x_1}F_3)^4\Big]^{1/4}\nonumber\\ & \quad \times \mathbb{E}\Big[(D_{x_2}F_4)^4\Big]^{1/4}\mu^{\otimes 3}(\mathrm{d}x_1,\mathrm d x_2,\mathrm d x_3)\nonumber\\ &=\int_{\mathbb{X}}\left(\int_{\mathbb{X}}\mathbb{E}\Big[\big(D_{x_1,x_2}^{2}F_1\big)^4\Big]^{1/4}\mathbb{E}\Big[(D_{x_1}F_3)^4\Big]^{1/4}\mu(\mathrm d x_1)\right) \nonumber \\ & \quad \times\left(\int_{\mathbb{X}}\mathbb{E}\Big[\big(D_{x_1,x_2}^{2}F_2\big)^4\Big]^{1/4} \mathbb{E}\Big[(D_{x_1}F_4)^4\Big]^{1/4}\mu(\mathrm d x_1)\right)\mu(\mathrm d x_2),\nonumber\end{align}

and

\begin{equation*}\Theta(F_1,F_2)\,:\!=\int_{\mathbb X}\mathbb{E}\Big[(D_x F_1)^4\Big]^{1/4}\mathbb{E}\Big[(D_x F_2)^4\Big]^{1/4}\mu(\mathrm{d}x).\end{equation*}

Note that $\Gamma_1(F_1)^2=\Gamma(F_1,F_1,F_1,F_1)^2$,

(62)\begin{equation}\Lambda(F_1,F_2,F_3,F_4)^2=\Lambda(F_2,F_1,F_4,F_3)^2, \quad\text{and}\quad\Gamma(F_1,F_2,F_3,F_4)^2=\Gamma(F_2,F_1,F_4,F_3)^2.\end{equation}

Note also that by the Cauchy–Schwarz inequality

(63)\begin{align}\gamma(F_1,F_2,F_3,F_4)^2\leq\Gamma&(F_1,F_2,F_3,F_4)^2, \quad \lambda(F_1,F_2,F_3,F_4)^2\leq\Lambda(F_1,F_2,F_3,F_4)^2, \nonumber\\ & \qquad \quad \theta(F_1,F_2)\leq\Theta(F_1,F_2)\end{align}

(and so, in particular, $\gamma_1(F_1)\leq\Gamma_1(F_1)$).

The following corollary is an immediate consequence of the inequalities (63) and Proposition 5.6.

Proof of Theorem 2.6. For $n\geq 1$, $x_1,x_2\in\mathbb X$, we have

\begin{equation*}D_{x_1}F_n=\frac{D_{x_1}H_n}{\sqrt{\mathbb{V}ar(H_n)}}\quad\text{and}\quad D_{x_1,x_2}^2 F_n=\frac{D_{x_1,x_2}^2 H_n}{\sqrt{\mathbb{V}ar(H_n)}}.\end{equation*}

So, letting $C\in\mathbb R_+$ denote a positive constant which may vary from line to line, by Hölder’s inequality and the assumptions (13) and (14), for any $n\geq 1$ we have

(64)\begin{align}\mathbb{E}[|D_{x_1}F_n|^4]&\leq\mathbb{P}(D_{x_1}H_n\neq 0)^{p/(4+p)}\frac{\mathbb{E}[|D_{x_1}H_n|^{4+p}]^{4/(4+p)}}{\mathbb{V}ar(H_n)^2}\nonumber\\&\leq C\frac{\mathbb{P}(D_{x_1}H_n\neq 0)^{p/(4+p)}}{\mathbb{V}ar(H_n)^2},\quad\text{$\mu$-a.e. $x_1\in\mathbb X$},\end{align}

(65)\begin{align}\mathbb{E}[|D_{x_1,x_2}^2 F_n|^4]&\leq C\frac{\mathbb{P}(D_{x_1,x_2}^2 H_n\neq 0)^{q/(4+q)}}{\mathbb{V}ar(H_n)^2},\quad\text{$\mu^{\otimes 2}$-a.e. $(x_1,x_2)\in\mathbb X^2$},\\[11pt]\nonumber\end{align}

and

(66)\begin{align}\mathbb{E}[|D_{x_1}F_n|^3]&\leq\mathbb{P}(D_{x_1}H_n\neq 0)^{(1+p)/(4+p)}\frac{\mathbb{E}[|D_{x_1}H_n|^{4+p}]^{3/(4+p)}}{\mathbb{V}ar(H_n)^{3/2}}\nonumber\\&\leq C\frac{\mathbb{P}(D_{x_1}H_n\neq 0)^{(1+p)/(4+p)}}{\mathbb{V}ar(H_n)^{3/2}},\quad\text{$\mu$-a.e. $x_1\in\mathbb X$.}\end{align}

For $k,l,m,n\geq 1$, the inequalities (64), (65), and (66) imply

\begin{align*}\Gamma_1(F_k)^2\leq C\boldsymbol{\Gamma}_1(H_k)^2,\quad\Gamma(F_k,F_l,F_m,F_n)^2\leq C\boldsymbol{\Gamma}(H_k,H_l,H_m,H_n)^2,\end{align*}

\begin{align*}\Lambda(F_k,F_l,F_m,F_n)^2\leq C\boldsymbol{\Lambda}&(H_k,H_l,H_m,H_n)^2,\quad\qquad \quad \Theta(F_k,F_l)^2\leq C\boldsymbol{\Theta}(H_k,H_l)^2, \\ & \gamma_2(F_k)\leq C\boldsymbol{\Gamma}_2(H_k).\end{align*}

The claim follows from these relations and Corollary 5.7.

Proof of Proposition 5.6. We start by noticing that in fact $F_n\in\mathrm{Dom}(D)$, $n\geq 1$. Indeed, by (59) it follows that

\begin{equation*}\int_{\mathbb X}\mathbb{E}[|D_x F|^2]\mu(\mathrm d x)=\theta(F_n,F_n)<\infty,\quad n\geq 1.\end{equation*}

To prove the ASCLT we are going to apply the Ibragimov–Lifshits criterion stated as Lemma 5.5. So, let $\Delta_n(t)$ be given by (55) with $F_k$ in place of $X_k$ and Z in place of X. We have

\begin{align}\mathbb{E}[|\Delta_n(t)|^2]&=\frac{1}{(\log n)^2}\sum_{k,l}^{1,n}\frac{1}{kl}\mathbb{E}\left[\left(\mathrm{e}^{\textbf{i}tF_k}-\mathrm{e}^{-t^2/2}\right)\left(\mathrm{e}^{-\textbf{i}tF_l}-\mathrm{e}^{-t^2/2}\right)\right]\nonumber\\ &=\frac{1}{(\log n)^2}\sum_{k,l}^{1,n}\frac{1}{kl}\Big[\left(\mathbb{E}\big[\mathrm{e}^{\textbf{i}t(F_k-F_l)}\big]-\mathrm{e}^{-t^2}\right)-\mathrm{e}^{-t^2/2}\Big(\mathbb{E}\big[\mathrm{e}^{\textbf{i}tF_k}\big]-\mathrm{e}^{-t^2/2}\Big)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad-\mathrm{e}^{-t^2/2}\left(\mathbb{E}\big[\mathrm{e}^{-\textbf{i}t F_l}\big]-\mathrm{e}^{-t^2/2}\right)\Big].\nonumber\end{align}

Therefore, for $r\in\mathbb R_+$,

(67)\begin{align}\sup_{|t|\leq r}\sum_{n\geq 2}\frac{\mathbb{E}[|\Delta_n(t)|^2]}{n\log n}&\leq\sup_{|t|\leq r}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k,l}^{1,n}\frac{|\mathbb{E}[\mathrm{e}^{\textbf{i}t(F_k-F_l)}]-\mathrm{e}^{-t^2}|}{kl}\end{align}

(68)\begin{align}&\quad+\sup_{|t|\leq r}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\frac{1}{l}\sum_{k=1}^{n}\frac{|\mathbb{E}[\mathrm{e}^{\textbf{i}tF_k}]-\mathrm{e}^{-t^2/2}|}{k}\end{align}

(69)\begin{align}&\quad+\sup_{|t|\leq r}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k=1}^{n}\frac{1}{k}\sum_{l=1}^{n}\frac{|\mathbb{E}[\mathrm{e}^{-\textbf{i}tF_l}]-\mathrm{e}^{-t^2/2}|}{l}.\\[11pt]\nonumber\end{align}

Let $t\in\mathbb R$ and $r\in\mathbb{R}_+$ be such that $|t|\leq r$. Since $\mathbb{E}[F_n^2]=1$ for each n, by Lemma 5.4 we have

\begin{equation*}|\mathbb{E}[\mathrm{e}^{\textbf{i}F_k t}]-\mathrm{e}^{-t^2/2}|\leq r^2\gamma_1(F_k)+\frac{r^3}{\sqrt 2}\gamma_2(F_k).\end{equation*}

Therefore,

(70)\begin{align}\sup_{|t|\leq r} & \sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\frac{1}{l}\sum_{k=1}^{n}\frac{|\mathbb{E}[\mathrm{e}^{\textbf{i}tF_k}]-\mathrm{e}^{-t^2/2}|}{k}\qquad\qquad\end{align}

\begin{align}& \quad\leq\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\frac{1}{l}\sum_{k=1}^{n}\frac{r^2\gamma_1(F_k)+2^{-1/2}r^3\gamma_2(F_k)}{k}.\nonumber\end{align}

Since $\lim_{n\to\infty}\sum_{l=1}^{n}l^{-1}/\log n=1$, the finiteness of the term in (68) is guaranteed by the assumption (58). Since by Lemma 5.4 we also have

\begin{equation*}|\mathbb{E}[\mathrm{e}^{-\textbf{i}tF_l}]-\mathrm{e}^{-t^2/2}|\leq r^2\gamma_1(F_l)+\frac{r^3}{\sqrt 2}\gamma_2(F_l),\end{equation*}

the same argument guarantees the finiteness of the term in (69). As for the finiteness of the term in the right-hand side of (67), we start by noticing that, again by Lemma 5.4, we get

\begin{align}|\mathbb{E}[\mathrm{e}^{\textbf{i}t(F_k-F_l)}]-\mathrm{e}^{-t^2}|&=|\mathbb{E}\Big[\mathrm{e}^{\textbf{i}(\sqrt{2}t)(F_k-F_l)/\sqrt 2}\Big]-\mathrm{e}^{-(\sqrt{2}t)^2/2}|\nonumber\\&\leq 2|t|^2|\mathbb{E}[F_kF_l]|+2|t|^2\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)+2|t|^3\gamma_2\left(\frac{F_k-F_l}{\sqrt 2}\right)\nonumber\\&\leq 2r^2\mathbb{E}[F_kF_l]|+2r^2\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)+2r^3\gamma_2\left(\frac{F_k-F_l}{\sqrt 2}\right).\nonumber\end{align}

Therefore the term in the right-hand side of (67) is finite if the following hold:

(71)\begin{align}&\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k,l}^{1,n}\frac{|\mathbb{E}[F_k F_l]|}{kl}<\infty,\end{align}

(72)\begin{align}&\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k,l}^{1,n}\frac{\gamma_2\left(\frac{F_k-F_l}{\sqrt 2}\right)}{kl}<\infty,\end{align}

(73)\begin{align}&\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k,l}^{1,n}\frac{\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)}{kl}<\infty.\\[11pt]\nonumber\end{align}

Proof of (71). Due to the symmetry (with respect to k and l) of the mapping $(k,l)\mapsto\frac{|\mathbb{E}[F_k F_l]|}{kl}$, it holds that

\begin{align}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k,l}^{1,n}\frac{|\mathbb{E}[F_k F_l]|}{kl}&=\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{k=1}^{n}\frac{1}{k^2}+2\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=2}^{n}\sum_{k=1}^{l-1}\frac{|\mathbb{E}[F_k F_l]|}{kl}.\nonumber\end{align}

Since $\sum_{n\geq 2}\frac{1}{n(\log n)^\beta}<\infty$, $\beta>1$, and $\sum_{n\geq 2}\frac{1}{n^2}<\infty$, the infinite sum (71) converges if and only if

(74)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{|\mathbb{E}[F_k F_l]|}{kl}<\infty.\end{equation}

By Lemmas 5.2 and 5.1 and the Cauchy–Schwarz inequality, we get

\begin{align}|\mathbb{E}[F_k F_l]|&\leq\mathbb{E}[|\langle DF_k,-DL^{-1}F_l\rangle_1|]\nonumber\\&\leq\int_{\mathbb X}\mathbb{E}[|D_x F_k||D_x L^{-1}F_l|]\mu(\mathrm d x)\leq\theta(F_k,F_l),\nonumber\end{align}

and therefore (74) is a consequence of the assumption (59).

Proof of (72). By the inequality $(a+b)^3\leq 4(a^3+b^3)$, $a,b\geq 0$, we have

\begin{align}\gamma_2\left(\frac{F_k-F_l}{\sqrt 2}\right)&=2^{-3/2}\int_{\mathbb{X}}\mathbb{E}[|D_xF_k-D_xF_l|^3]\mu(\mathrm{d}x)\nonumber\\&\leq\sqrt{2}(\gamma_2(F_k)+\gamma_2(F_l))\nonumber.\end{align}

Therefore, (72) follows by the assumption (58) (with $i=2$).

Proof of (73). Due to the symmetry (with respect to k and l) of the mapping

\begin{equation*}(k,l)\mapsto\frac{\gamma_{1}\left(\frac{F_k-F_l}{\sqrt 2}\right)}{kl},\end{equation*}

the infinite sum (73) converges if and only if

(75)\begin{equation}\sum_{n\geq 2}\frac{1}{n(\log n)^3}\sum_{l=1}^{n}\sum_{k=1}^{l}\frac{\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)}{kl}<\infty.\end{equation}

Using the inequalities $(a+b)^2\leq 2a^2+2b^2$ and $(a+b)^{1/2}\leq a^{1/2}+b^{1/2}$, $a,b\geq 0$, we now bound the quantity $\gamma_1((F_k-F_l)/\sqrt 2)$. From here on, for simplicity of notation we put $\mu^{\otimes 3}(\mathrm{d}\textbf{x})\,:\!=\mu^{\otimes 3}(\mathrm d x_1,\mathrm d x_2, \mathrm d x_3)$. We have

\begin{align}&\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)\leq\Biggl(4\int_{\mathbb{X}^3}(\mathbb{E}[(|D_{x_1,x_3}^{2}F_k|^2+|D_{x_1,x_3}^{2}F_l|^2)(|D_{x_2,x_3}^{2}F_k|^2+|D_{x_2,x_3}^{2}F_l|^2)])^{1/2}\nonumber\\&\quad \times(\mathbb{E}[(|D_{x_1}F_k|^2+|D_{x_1}F_l|^2)(|D_{x_2}F_k|^2+|D_{x_2}F_l|^2)])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad+\int_{\mathbb{X}^3}\mathbb{E}[(|D_{x_1,x_3}^{2}F_k|^2+|D_{x_1,x_3}^{2}F_l|^2)(|D_{x_2,x_3}^{2}F_k|^2+|D_{x_2,x_3}^{2}F_l|^2)]\mu^{\otimes 3}(\mathrm{d}\textbf{x})\Biggr)^{1/2}\nonumber\\&=\Biggl(4\int_{\mathbb{X}^3}\Biggl((\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\nonumber\\&\quad +\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])\nonumber\displaybreak\\&\quad \times(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2]+\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2]\nonumber\\&\quad +\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2]+\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2])\Biggr)^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad+\int_{\mathbb{X}^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\nonumber\\&\quad +\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\Biggr)^{1/2},\nonumber\end{align}

from which it follows that

\begin{align}\gamma_1&\left(\frac{F_k-F_l}{\sqrt 2}\right) \nonumber\\[2pt]&\le \Biggl( 4\int_{\mathbb{X}^3}\mu^{\otimes 3}(\mathrm{d}\textbf{x}) \Biggl(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad +\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad +\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2]\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2]\Biggr)^{1/2}\nonumber\\[2pt]&\qquad\qquad\qquad+\int_{\mathbb{X}^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2]\nonumber\\[2pt]&\qquad\qquad\qquad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2]+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\Biggr)^{1/2},\nonumber\end{align}

and therefore

(76)\begin{align}\gamma_1&\left(\frac{F_k-F_l}{\sqrt 2}\right)\nonumber\\&\leq\Biggl[\int_{\mathbb{X}^3}(4(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2])^{1/2}\nonumber\\&\quad +\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +\int_{\mathbb{X}^3}(4(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2])^{1/2}\nonumber\\&\quad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +\int_{\mathbb{X}^3}(4(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2])^{1/2})\nonumber\\&\quad+\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +\int_{\mathbb{X}^3}(4(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2])^{1/2}\nonumber\\&\quad+\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2])\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_k|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_k|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_k|^2|D_{x_2}F_l|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\nonumber\\&\quad +4\int_{\mathbb X^3}(\mathbb{E}[|D_{x_1,x_3}^{2}F_l|^2|D_{x_2,x_3}^{2}F_l|^2])^{1/2}(\mathbb{E}[|D_{x_1}F_l|^2|D_{x_2}F_k|^2])^{1/2}\mu^{\otimes 3}(\mathrm{d}\textbf{x})\Biggr]^{1/2}.\end{align}

Finally, by (76), using firstly again the inequality $(a+b)^{1/2}\leq a^{1/2}+b^{1/2}$, $a,b\geq 0$, and secondly the identities (56), we have

\begin{align}\gamma_1\left(\frac{F_k-F_l}{\sqrt 2}\right)&\leq\gamma_1(F_k)+\gamma_1(F_l)+\gamma(F_k,F_l,F_k,F_l)+\gamma(F_l,F_k,F_l,F_k)\nonumber\\&\quad+\lambda(F_k,F_k,F_k,F_l)+\lambda(F_k,F_k,F_l,F_k)+\lambda(F_k,F_k,F_l,F_l)+\lambda(F_k,F_l,F_k,F_k)\nonumber\\&\quad+\lambda(F_k,F_l,F_l,F_k)+\lambda(F_k,F_l,F_l,F_l)+\lambda(F_l,F_k,F_k,F_k)+\lambda(F_l,F_k,F_k,F_l)\nonumber\\&\quad+\lambda(F_l,F_k,F_l,F_l)+\lambda(F_l,F_l,F_k,F_k)+\lambda(F_l,F_l,F_k,F_l)+\lambda(F_l,F_l,F_l,F_k)\nonumber\\&=\gamma_1(F_k)+\gamma_1(F_l)+2\gamma(F_k,F_l,F_k,F_l)+2\lambda(F_k,F_k,F_k,F_l)\nonumber\\&\quad+\lambda(F_k,F_k,F_l,F_l)+2\lambda(F_l,F_k,F_k,F_k)+2\lambda(F_k,F_l,F_l,F_k)\nonumber\\&\quad+2\lambda(F_k,F_l,F_l,F_l)+\lambda(F_l,F_l,F_k,F_k)+2\lambda(F_l,F_l,F_l,F_k).\nonumber\end{align}

Therefore (75) follows by the assumptions (58) (with $i=1$), (60), and (61).

By Lemma 5.4, the assumption (57), and Lévy’s continuity theorem, the sequence $\{F_n\}_{n\geq 1}$ converges in law to Z. At last, the claim follows by the Ibragimov–Lifshits criterion.