Exact Convergence Rates for Particle Distributions in a Non-Lattice Branching Random Walk

Abstract

Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let $Z_n(\cdot )$ be the counting measure which counts the number of particles of generation n in a given set. Révész (1994) studied the convergence rates in the central and local limit theorems for $Z_n(\cdot )$ in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for $Z_n(\cdot )$ in the case the underlying motion is governed by a general non-lattice random walk on ${\mathbb {R}}^d$ with the characteristic function of the motion law satisfying the weak Cramér condition.

1 Introduction and main results

In branching random walks, an attractive theme is the central limit theorem for the counting measure, which describes the configuration of the particle system. The study on this theme has a long history. Since Harris(1963, [18]) raised the related conjecture precisely, the topic has been widely explored (see, e.g., [6, 16, 17, 19, 21, 25, 26] and references therein). As further extension, Révész [23] started the research on convergence speeds in the central limit theorems, for two special cases where the moving mechanism of particles is governed by simple random walk or Wiener process. Later, Chen [9] confirmed the strengthened version of Révész’s conjectures on the exact convergence rates in central limit theorems for the two special models. Afterward, around the generalization of Chen’s results, several works [11,12,13,14,15, 17, 20, 24] were developed much further. But the study is still incomplete, since the works mentioned above didn’t cover the case where the underlying motion is governed by a general non-lattice random walk on ${\mathbb {R}}^d$.

The main objective of this paper is to complete the picture. Precisely, we derive the exact convergence rates in the central limit theorem and the local one for the unsettled case mentioned above. In addition, the moment conditions assumed in our main results are weaker than previous ones, since we make use of some new techniques in the proofs.

Before we state our main results, we first described the model in detail. The process starts with an initial particle $\varnothing$ located at $S_\varnothing =0$. At time 1, the initial particle $\varnothing$ dies and reproduces $N= N_\varnothing$ new particles $\varnothing i = i$ of generation 1, with displacement $L_{\varnothing i}= L_i$, so that each particle $\varnothing i= i$ is situated at the position $S_{\varnothing i} = S_{\varnothing }+ L_{\varnothing i}$, that is, $S_i = L_i, 1\le i \le N$. In general, at time $n+1$, each particle u of generation n, which is denoted by a sequence of positive integers of length n, i.e., $u=u_1u_2\cdots u_n$, dies and reproduces $N_u$ new particles of generation $n+1$, with displacements $L_{u1}, L_{u2}, \cdots , L_{uN_u}$, so that each particle ui is located at $S_{ui}= S_u+L_{ui}$, $1\le i\le N_u$. All the random elements $N_u, L_u$, indexed by finite sequences of integers u, are independent copies of the generic random elements N, L, respectively. Throughout this paper, we shall assume that both N and L are independent random variables, with N having the law $\{ p_i\}_ {i \in {\mathbb {N}}}$ (${\mathbb {N}}=\{0,1,2,\cdots \}$), and L taking values in ${\mathbb {R}}^d$ and having zero mean and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$.

Let ${\mathbb {T}}$ be the genealogical tree with $\{N_u\}$ as defining elements. By definition, we have: (a) $\varnothing \in {\mathbb {T}}$; (b) $ui \in {\mathbb {T}}$ implies $u\in {\mathbb {T}}$; (c) if $u\in {\mathbb {T}}$, then $ui\in {\mathbb {T}}$ if and only if $1\le i\le N_u$. Let

$$\begin{aligned} {\mathbb {T}}_n =\{u\in {\mathbb {T}} :|u|=n\} \end{aligned}$$

be the set of particles of generation n, where |u| denotes the length of the sequence u.

As indicated in abstract, we denote by $Z_n(\cdot )$ the counting measure which counts the number of particles of generation n situated in a given set: for $B \subset {\mathbb {R}}$,

$$\begin{aligned} Z_n(B)= \sum _{u\in {\mathbb {T}}_n} {\mathbf {1}}_{ B}(S_u). \end{aligned}$$

Then, $\{Z_n({\mathbb {R}}^d) \}$ is a Galton–Watson process. When $m = \sum _{i=0}^\infty i p_i>1$, the process is supercritical, i.e., ${\mathbb {P}}(Z_n\rightarrow \infty )>0 .$ Under the additional assumption ${\mathbb {E}}N\log N <\infty$, the martingale $\{ W_n= Z_n({\mathbb {R}}^d) /{ m^{n}} \}$ converges to a nonzero random variable W with ${\mathbb {E}}W=1$ (see, e.g., [3]).

In addition to this well-known martingale $\{W_n\}$, we also need another two martingales $\{N_{1,n}\}$ and $\{N_{2,n}\}$ concerning with the BRW, which are defined through the following

$$\begin{aligned} N_{1,n}= \frac{1}{m^n} \sum _{u\in {\mathbb {T}}_n} S_u, \quad N_{2,n} = \frac{1}{m^n} \sum _{u\in {\mathbb {T}}_n} \Big ( \left\langle {S_u},{\Gamma ^{-1}S_u}\right\rangle -dn\Big ), \end{aligned}$$

(1)

where $\left\langle {\cdot },{\cdot }\right\rangle$ is the inner product in ${\mathbb {R}}^d$ and $\Gamma ^{-1}$ is the inverse of $\Gamma$. These two sequences are convergent a.s. under some suitable moment conditions. Precisely, Proposition 2.1 shows that if ${\mathbb {E}}N (\ln ^+N)^2<\infty$ and ${\mathbb {E}}{\left\| L \right\| }^4<\infty$ , then the following limits exist a.s.

$$\begin{aligned} \mathcal {V}_1:= \displaystyle \lim _{n\rightarrow \infty }N_{1,n} \in {\mathbb {R}}^d \quad \text{ and } \quad \mathcal {V}_2:= \displaystyle \lim _{n\rightarrow \infty }N_{2,n} \in {\mathbb {R}}. \end{aligned}$$

where the notion ${\left\| \cdot \right\| }$ is the norm in ${\mathbb {R}}^d$, i.e., ${\left\| \cdot \right\| } = \sqrt{\left\langle {\cdot },{\cdot }\right\rangle }$.

With the help of the quantities $W, \mathcal {V}_1,\mathcal {V}_2$ mentioned above, we can describe the convergence rates in the central limit theorem as well as local one for $Z_n(\cdot )$.

To state our main results, we need some assumptions.

Throughout the paper, we suppose that the underlying branching process is supercritical and satisfies

$$\begin{aligned} \sum _{k=1}^\infty k (\ln k)^{1+\lambda } p_k <\infty , \end{aligned}$$

(2)

where the value of $\lambda >0$ is to be specified later.

We shall always assume that L satisfies the weak Cramér condition with exponent $b>0$,

$$\begin{aligned} \left|{\mathbb {E}}e^{i\left\langle {t},{L}\right\rangle } \right| \le 1- \frac{C}{{\left\| t \right\| }^b} \quad \text{ for } \text{ all } {\left\| t \right\| } > R, \end{aligned}$$

(3)

where $R>0$ is a large number and C is a constant. This condition was studied in [8] and [1], where different limit theorems were derived under the condition, respectively, generalizing results in [4].

It is easy to see that the classical Cramér condition on the characteristic function (i.e., $\limsup _{{\left\| t \right\| }\rightarrow \infty } \left|{\mathbb {E}}e^{i\left\langle {t},{L}\right\rangle } \right| <1$) implies that (3) holds for each $b>0$.

We set, for each vector $\nu =(\nu _1,\nu _2,\cdots ,\nu _d)\in {\mathbb {N}}^d$,

$$\begin{aligned} \nu ! = \nu _1 !\, \nu _2! \cdots \nu _d !, \quad |\nu |= \sum _{j=1}^d \nu _j, \quad D^{\nu }= \Big (\frac{\partial }{\partial x_1}\Big )^{\nu _1}\cdots \Big (\frac{\partial }{\partial x_d}\Big )^{\nu _d}, \end{aligned}$$

and denote by $\chi _\nu$ the $\nu$-cumulant of the random vector L, i.e.,

$$\begin{aligned} \chi _\nu = i^{-|\nu | } \Big (D^{\nu } \log {\mathbb {E}}e^{i\left\langle {x},{L}\right\rangle }\Big )\Big |_{x={\mathbf {0}}}, \quad {\mathbf {0}}= (0,0,\cdots ,0). \end{aligned}$$

We shall need the vector differential operator $\nabla$ and another two operators defined by

$$\begin{aligned} \nabla= & {} \Big (\frac{\partial }{\partial x_1}, \frac{\partial }{\partial x_2},\cdots , \frac{\partial }{\partial x_d}\Big ), \quad \mathcal {P}_1(D) = \sum _{|\nu |=3} \frac{\chi _\nu }{\nu !}D^{\nu } ,\nonumber \\&\quad \mathcal {P}_2(D)= \sum _{|\nu |=4} \frac{\chi _\nu }{\nu !} D^{\nu } +\frac{1}{2} \Big (\sum _{|\nu |=3} \frac{\chi _\nu }{\nu !} D^{\nu } \Big )^2. \end{aligned}$$

(4)

For the vectors $x=(x_1,x_2,\cdots ,x_d), y=(y_1,y_2,\cdots ,y_d)\in {\mathbb {R}}^d$, the partial order ${x}\le {y}$ means that $x_i\le y_i$ for $1\le i\le d$. Recall that the inner product of x and y is defined by

$$\begin{aligned} \left\langle {x},{y}\right\rangle = x_1y_1+x_2y_2+\cdots +x_dy_d, \end{aligned}$$

and the norm of the vector x by ${\left\| x \right\| }= \sqrt{\left\langle {x},{x}\right\rangle }$. The multiplication of a vector $x\in {\mathbb {R}}^d$ by a real number $\lambda \in {\mathbb {R}}$ is defined by $\lambda {x}=(\lambda x_1, \lambda x_2,\cdots , \lambda x_d).$ As usual, put

$$\begin{aligned} (-\infty , x]=(-\infty ,x_1]\times (-\infty ,x_2]\times \cdots (-\infty ,x_d]. \end{aligned}$$

Then for $t=(t_1,t_2,\cdots , t_d)\in {\mathbb {R}}^d$ and $n\in {\mathbb {N}}$, we have

$$\begin{aligned} (-\infty , \sqrt{n}t]=(-\infty ,\sqrt{n}t_1]\times (-\infty ,\sqrt{n}t_2]\times \cdots (-\infty ,\sqrt{n}t_d]. \end{aligned}$$

Given a measurable $A\subset {\mathbb {R}}^d$, denote by |A| its Lebesgue measure, and set

$$\begin{aligned} {\mathfrak {x}}_A= {|A|} ^{-1} \int _A z\mathrm {d}z. \end{aligned}$$

Recall that ${\mathbb {E}}L=0$ and $\mathrm { Cov}(L)=\Gamma$. By convention, the notation $\det \Gamma$ denotes the determinant of a matrix $\Gamma$, $\Gamma ^{-1}$ is the inverse of $\Gamma$, and set

$$\begin{aligned} \varphi _{\Gamma }(x) = (2\pi )^{-d/2} (\det \Gamma )^{-1/2} \exp \Big \{-\frac{1}{2} \left\langle {x},{\Gamma ^{-1}x}\right\rangle \Big \}, \quad \Phi _{\Gamma } (t)= \int _{(-\infty ,t] } \varphi _{\Gamma }(z) \mathrm {d}z. \end{aligned}$$

With the above notation and preliminary results, we can state our main results for $Z_n(\cdot )$, which give exact convergence rates of the central limit theorem (CLT) and the local limit theorem (LLT).

Theorem 1.1

(Convergence rate in CLT) Assume that L has zero mean ${\mathbb {E}}L=0$ and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$ and satisfies (3) with $b>0$. If (2) holds for some $\lambda >2$ and ${\mathbb {E}}||L||^3<\infty$, then a.s. for each $t\in {\mathbb {R}}^d$,

$$\begin{aligned} \sqrt{n}\left[ \frac{1}{m^n}{Z_n\Big (( -\infty ,\sqrt{n}t]\Big )}-\Phi _{\Gamma }(t)\, W \right] \xrightarrow []{n \longrightarrow \infty } -\left\langle {\nabla \Phi _{\Gamma }(t)},{\mathcal {V}_1}\right\rangle + \mathcal {P}_1 (D) \Phi _\Gamma (t) W . \end{aligned}$$

(5)

Remark 1.2

It is pointed out by the referee that this result holds for non-lattice L without condition (3). In fact, this condition (3) is only used in Theorem E. While by Esseen [10], Theorem E holds for non-lattice L in the one-dimensional case and the close variations of it, that one can find in Battacharya and Rao [4], can be adapted to the multi-dimensional case .

Theorem 1.3

(Convergence rate in LLT) Assume that L has zero mean ${\mathbb {E}}L=0$ and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$ and satisfies (3) with a positive exponent $b<2/(d+1)$. If (2) holds for some $\lambda > d+3$ and ${\mathbb {E}}||L||^{d+4}<\infty$, then for each bounded convex set $A \subset {\mathbb {R}}^d$,

$$\begin{aligned}&{n^{1+d/2}}\left[ \frac{1}{m^n}{Z_n(A)}-\frac{1}{n^{d/2}}\int _A \varphi _{\Gamma } \Big (\frac{z}{\sqrt{n}}\Big )\mathrm {d}z \, W\right] \xrightarrow [a.s.]{n \longrightarrow \infty } { { \dfrac{1 }{(2\pi )^{d/2}\sqrt{\det \Gamma }}}|A|} \cdot \Bigg \{-\frac{1}{2}\mathcal {V}_2 \nonumber \\&\quad +\left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)+\Gamma ^{-1}{\mathfrak {x}}_A },{\mathcal {V}_1}\right\rangle + \left[ \bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0)- \left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)},{{\mathfrak {x}}_A}\right\rangle \right] W\Bigg \}.\nonumber \\ \end{aligned}$$

(6)

Next we give another version of local limit theorem by considering a smooth function with compact support $f \in C^{\infty }_0({\mathbb {R}}^d)$. In this case, we don’t need to assume the condition $b<2/(d+1)$.

Theorem 1.4

(Convergence rate in LLT) Assume that L has zero mean ${\mathbb {E}}L=0$ and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$ and satisfies (3) for $b>0$. If (2) holds for some $\lambda > d+3$ and ${\mathbb {E}}||L||^{d+2}<\infty$, then for each smooth function $f\in C^{\infty }_0({\mathbb {R}}^d)$,

$$\begin{aligned}&{n^{1+d/2}}\left[ \frac{1}{m^n}{ Z_n[f] }-\frac{1}{n^{d/2}}\int _{{\mathbb {R}}^d} \varphi _{\Gamma } \Big (\frac{z}{\sqrt{n}}\Big ) f(z)\mathrm {d}z \, W\right] \xrightarrow [a.s.]{n \longrightarrow \infty } \nonumber \\&{ \dfrac{1 }{(2\pi )^{d/2}\sqrt{\det \Gamma }}} \cdot \Bigg \{-\frac{1}{2}m(f)\mathcal {V}_2 + \left\langle {m(f)\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)+\Gamma ^{-1} \int _{{\mathbb {R}}^d} zf(z) \mathrm {d} z},{\mathcal {V}_1}\right\rangle \nonumber \\&\quad + \left[ m(f)\bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0)- \left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)},{ \int _{{\mathbb {R}}^d} zf(z) \mathrm {d} z}\right\rangle \right] W\Bigg \}, \end{aligned}$$

(7)

where

$$\begin{aligned} Z_n[f]=\sum _{u\in {\mathbb {T}}_{n}}f(S_u),\quad m(f)=\int _{{\mathbb {R}}^d} f(z)\mathrm {d} z. \end{aligned}$$

Remark 1.5

These theorems generalize Theorems 3.1 and 3.2 of Chen [9], which gave the convergence rates in the central limit theorem and local one for branching Wiener processes under the second moment condition on the offspring distribution.

Remark 1.6

We use new techniques in the proofs, which lead to the less restrictive moment conditions on the offspring distribution and the moving laws than before (see, e.g., [11, 13]). For example, in the convergence rate of LLT for the model considered in [13], we have assumed that $\lambda >2d+8$, instead of the condition $\lambda >d+3$ here.

2 Sketch of the proofs

In this section, we outline the proofs of Theorems 1.1 and 1.3. The detailed proofs of some technical lemmas are transferred to Sect. 3.

First, we note that the quantities $\mathcal {V}_1$ and $\mathcal {V}_2$ that appeared in the theorems are well defined by the following proposition:

Proposition 2.1

([13]) The sequences $\{N_{1,n}\}$ and $\{N_{2,n}\}$ defined by (1) are martingales with respect to the filtration $\{ {\mathscr {D}}_n\}$ defined by

$$\begin{aligned} {\mathscr {D}}_0=\{\emptyset ,\Omega \}, \quad {\mathscr {D}}_n = \sigma ( N_u, L_{ui}: i\ge 1, |u| < n) \text{ for } n\ge 1. \end{aligned}$$

Moreover, if ${\mathbb {E}}N (\ln ^+N)^2<\infty$ and ${\mathbb {E}}{\left\| L \right\| }^4<\infty$, then the following limits exist a.s.

$$\begin{aligned} \mathcal {V}_1:= \displaystyle \lim _{n\rightarrow \infty }N_{1,n} \in {\mathbb {R}}^d \quad \text{ and } \quad \mathcal {V}_2:= \displaystyle \lim _{n\rightarrow \infty }N_{2,n} \in {\mathbb {R}}. \end{aligned}$$

This proposition can be proved by slightly modifying the demonstration of Proposition 2.1 in [13], and we omit the proof here.

To sketch the proofs of main theorems, we still need some notation.

For $u\in ({\mathbb {N}}^*)^k (k\ge 0)$ and $n\ge 1$, let $Z_n(u, B )$ be the number of the $n^{th}$ generation descendants from u located in the set $B+S_u\subset {\mathbb {R}}^d$. More precisely,

$$\begin{aligned} Z_{n}(u,B)= \# \{ v \in {\mathbb {T}}_n(u): S_{uv}-S_u\in B\}, \end{aligned}$$

where ${\mathbb {T}}_n(u)=\{v\in {\mathbb {T}}(u): |v|=n\}$ and ${\mathbb {T}}(u)$ is the shifted tree of ${\mathbb {T}}$ at u with defining elements $\{N_{uv}\}$: 1) $\varnothing \in {\mathbb {T}}(u)$, 2) $vi\in {\mathbb {T}}(u)\Rightarrow v\in {\mathbb {T}}(u)$ and 3) if $v\in {\mathbb {T}}(u)$, then $vi\in {\mathbb {T}}(u)$ if and only if $1\le i\le N_{uv}$.

Set $1_n = \underbrace{1 \cdots 1}_{n} \text{ for } n\ge 1 \text{ and } 1_0= \varnothing .$ Write ${\widehat{S}}_{n}= S_{1_n}$ for $n\ge 0$. Then ${\widehat{S}}_{n}$ is an ordinary random walk with generic jump L.

Observe that for each set $B \subset {\mathbb {R}}^d$, for $k\le n$,

$$\begin{aligned} Z_n(B)=\sum _{u\in {\mathbb {T}}_k} Z_{n-k}(u,B-S_u), \end{aligned}$$

(8)

and

$$\begin{aligned} {{\mathbb {E}}_{{\mathscr {D}}_{k}}}\Big ( \frac{Z_{n-k}(u, B-S_u)}{m^{n-k}} \Big | S_u\Big )={\mathbb {P}}\Big ({\widehat{S}}_{n-k} +y\in B \Big )\Big |_{y=S_u}. \end{aligned}$$

(9)

Let $k_n$ be an integer much smaller than n and related to n in an appropriate way, which will be made precise later. Then, we have the following decomposition:

$$\begin{aligned} \frac{1}{m^n} Z_n(B)= & {} \frac{1}{m^{k_n }} \sum _{u\in {\mathbb {T}}_{k_n}} \left[ \frac{Z_{n-k_n}(u,B-S_u)}{m^{n-k_n}}- {\mathbb {P}}\Big ({\widehat{S}}_{n-k_n}\in B-y\Big )\Big |_{y=S_u}\right] \nonumber \\&+ \frac{1}{m^{k_n }} \sum _{u\in {\mathbb {T}}_{k_n}} {\mathbb {P}}\Big ({\widehat{S}}_{n-k_n}\in B-y\Big )\Big |_{y=S_u}=: {\mathbb {A}}_n(B) + {\mathbb {B}}_n(B). \end{aligned}$$

(10)

Similarly, for a function f,

$$\begin{aligned}&\frac{1}{m^n} Z_n[f]=\frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} \bigg [\frac{ \sum _{v\in {\mathbb {T}}_{n-k_n}(u) } f(y+ S_{uv}-S_u)}{m^{n-k_n} } - {\mathbb {E}}f(y+ {\widehat{S}}_{n-k_n}) \bigg ]_{y=S_u}\nonumber \\&\quad + \frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} \Big [ {\mathbb {E}}f(y+ {\widehat{S}}_{n-k_n}) \Big ]_{y=S_u}={\mathbb {A}}_n[f] + {\mathbb {B}}_n[f]. \end{aligned}$$

(11)

Now we can give the proofs of Theorems 1.1 and  1.3.

Proof of Theorem 1.1

We begin with the explicit definition of $k_n$.

Let $\kappa > \frac{9}{ 2(\lambda -2)}$ be an integer large enough and put $\mu =2\kappa +3$. For each $n\in {\mathbb {N}}$, set

$$\begin{aligned} k_n= \big (\lfloor n^{1/\mu }\rfloor \big ) ^{ \kappa }, \end{aligned}$$

where $\lfloor x\rfloor$ is the largest integer not bigger than x.

By using decomposition (10), we deduce the desired (5) from the following lemmas:

Lemma 2.2

Under the conditions of Theorem  1.1,

$$\begin{aligned} \sqrt{n} {\mathbb {A}}_n ((-\infty , \sqrt{n}t])\xrightarrow [a.s.] {n\rightarrow \infty }0. \end{aligned}$$

(12)

Lemma 2.3

Under the conditions of Theorem 1.1,

$$\begin{aligned} \sqrt{n}\bigg [ {\mathbb {B}}_n\big ((-\infty ,\sqrt{n}t ]\big )-W\Phi _{\Gamma }(t) \bigg ] \xrightarrow [a.s.]{n \longrightarrow \infty } -\left\langle {\nabla \Phi _{\Gamma }(t) },{\mathcal {V}_1}\right\rangle + \mathcal {P}_1 (D) \Phi _\Gamma (t) W. \end{aligned}$$

(13)

The proofs of Lemmas 2.2 and 2.3 will be given in the next section. $\square$

Proof of Theorem 1.3

Let $\kappa$ be an large integer bigger than $(d+4)/ (\lambda - d-3)$ and $\mu =2\kappa +2$. For each $n\in {\mathbb {N}}$, set $k_n= \big (\lfloor n^{1/\mu }\rfloor \big ) ^{ \kappa }$.

On the basis of (10), we deduce the desired (6) from the following lemmas.

Lemma 2.4

Under the conditions of Theorem 1.3,

$$\begin{aligned} n^{1+d/2} {\mathbb {A}}_n (A)\xrightarrow [a.s.] {n\rightarrow \infty }0. \end{aligned}$$

(14)

Lemma 2.5

Under the conditions of Theorem 1.3,

$$\begin{aligned}&{n^{1+d/2}}\left[ {\mathbb {B}}_n\big (A\big )-\frac{1}{n^{d/2}}\int _A \varphi _{\Gamma } \Big (\frac{z}{\sqrt{n}}\Big )\mathrm {d}z \, W\right] \xrightarrow [a.s.]{n \longrightarrow \infty } { \varphi _{\Gamma }(0)|A|} \cdot \Bigg \{-\frac{1}{2}\mathcal {V}_2 \nonumber \\&\quad +\left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)+\Gamma ^{-1}{\mathfrak {x}}_A },{\mathcal {V}_1}\right\rangle + \left[ \bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0)- \left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)},{{\mathfrak {x}}_A}\right\rangle \right] W\Bigg \}.\nonumber \\ \end{aligned}$$

(15)

The proofs of Lemmas 2.4 and 2.5 will be given in the next section. $\square$

Proof of Theorem 1.4

Let $\kappa$ be an large integer bigger than $(d+4)/ (\lambda - d-3)$ and $\mu =2\kappa +2$. For each $n\in {\mathbb {N}}$, set $k_n= \big (\lfloor n^{1/\mu }\rfloor \big ) ^{ \kappa }$. On the basis of (11), we deduce the desired (7) from the following lemmas.

Lemma 2.6

Under the conditions of Theorem 1.3,

$$\begin{aligned} n^{1+d/2} {\mathbb {A}}_n (f)\xrightarrow [a.s.] {n\rightarrow \infty }0. \end{aligned}$$

(16)

Lemma 2.7

Under the conditions of Theorem 1.3,

$$\begin{aligned}&{n^{1+d/2}}\left[ {\mathbb {B}}_n\big (f\big )-\frac{1}{n^{d/2}}\int _A \varphi _{\Gamma } \Big (\frac{z}{\sqrt{n}}\Big )\mathrm {d}z \, W\right] \xrightarrow [a.s.]{n \longrightarrow \infty }\nonumber \\&\quad { \dfrac{1 }{(2\pi )^{d/2}\sqrt{\det \Gamma }}} \cdot \Bigg \{-\frac{1}{2}m(f)\mathcal {V}_2 + \left\langle {m(f)\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)+\Gamma ^{-1} \int _{{\mathbb {R}}^d} zf(z) \mathrm {d} z},{\mathcal {V}_1}\right\rangle \nonumber \\&\quad + \left[ m(f)\bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0)- \left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)},{ \int _{{\mathbb {R}}^d} zf(z) \mathrm {d} z}\right\rangle \right] W\Bigg \}. \end{aligned}$$

(17)

The proofs of Lemmas 2.6 and 2.7 will be given in the next section. $\square$

3 Proofs of Lemmas

In this section, we shall give the proofs of Lemmas  2.2$\sim$2.7. Note that throughout the proofs, K denotes a generic positive constant, and thus, its value may vary even in a single inequality.

The proofs of Lemma 2.2, Lemma 2.4 and Lemma 2.6 are similar to that of Lemma 2.2 in [13]. For the convenience of the reader, we give the proof of Lemma 2.2, as well as sketch the proofs of Lemmas 2.4 and 2.6.

Proof of Lemma 2.2

We start by introducing some notations. For $u\in ({\mathbb {N}}^*)^{k_n}$, set

$$\begin{aligned}&X_{n,u}= \frac{1}{m^{n-k_n}}Z_{n-k_n}\Big (u,(-\infty , \sqrt{n}t-S_u]\Big )- {\mathbb {P}}\Big ({\widehat{S}}_{n-k_n}\in (-\infty , \sqrt{n}t-y]\Big )\Big |_{y=S_u},\\&{\overline{X}}_{n,u}= X_{n,u} {\mathbf {1}}_{\{ |X_{n,u}|\le m^{k_n}\}}, \quad \overline{{\mathbb {A}}}_n = \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {\overline{X}}_{n,u}, \quad {\mathbb {A}}_n= {\mathbb {A}}_n((-\infty , \sqrt{n}t ])= \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {X}_{n,u}. \end{aligned}$$

It is easy to see the following fact:

$$\begin{aligned} | X_{n,u}|\le W_{n-k_n}(u) +1, \quad \text{ with } W_{n-k_n}(u) =m^{-(n-k_n)} Z_{n-k_n}(u,{\mathbb {R}}^d) . \end{aligned}$$

(18)

We remind that $\{W_{n-k_n}(u): u\in ({\mathbb {N}}^*)^{k_n}\}$ are mutually independent and identically distributed as $W_{n-k_n}$.

The lemma will be proved if we can establish the following:

$$\begin{aligned}&\sqrt{n} ({\mathbb {A}}_n-{\overline{{\mathbb {A}}}}_n ) =0 \text{ for } \text{ all } \text{ but } \text{ finitely } \text{ many } n \text{ a.s. } \end{aligned}$$

(19)

$$\begin{aligned}&\sqrt{n} \Big ({\overline{{\mathbb {A}}}}_n -{\mathbb {E}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n \Big )\xrightarrow {n \rightarrow \infty } 0\quad \text{ a.s. } \end{aligned}$$

(20)

$$\begin{aligned}&\sqrt{n} {\mathbb {E}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n \xrightarrow {n \rightarrow \infty } 0\quad \text{ a.s. } \end{aligned}$$

(21)

In proving (19), (20) and (21), we shall need the following moment result on $W^*=\sup _{n\ge 0} W_n$.

Lemma 3.1

Assume $m>1$ and ${\mathbb {E}}N(\log N)^{ 1+\lambda }<\infty$ for some $\lambda >1$. Then

$$\begin{aligned} {{\mathbb {E}}}(W^*+1)\big (\log (W^*+1)\big )^{\lambda } <\infty . \end{aligned}$$

(22)

This lemma is an immediate corollary of [7, Theorem 7] by taking $l_0(t)=(\log t)^{\lambda -1}$ therein.

To prove (19), it suffices to show that

$$\begin{aligned} \sum _{n=1}^\infty {\mathbb {P}}( \overline{{\mathbb {A}}}_n \ne {\mathbb {A}} _n ) <\infty . \end{aligned}$$

(23)

Observe that

$$\begin{aligned} {\mathbb {P}}({\mathbb {A}}_n\ne {\overline{{\mathbb {A}}}}_n )&~\le {\mathbb {E}}\sum _{u\in {\mathbb {T}}_{k_n} } {\mathbb {P}}_{ {\mathscr {D}}_{k_n}}(X_{n,u}\ne {\overline{X}}_{n,u}) ={\mathbb {E}}\sum _{u\in {\mathbb {T}}_{k_n} } {\mathbb {P}}_{ {\mathscr {D}}_{k_n}}(|X_{n,u}|\ge m^{k_n})\\&~ \le _{(18)} {\mathbb {E}}\sum _{u\in {\mathbb {T}}_{k_n} } {\mathbb {P}}( W_{n-k_n} (u)+1 \ge m^{k_n})= m^{k_n} {\mathbb {P}}( W_{n-k_n} +1 \ge m^{k_n}) \\&~\le m^{k_n} {\mathbb {P}}( W^* +1 \ge m^{k_n}) . \end{aligned}$$

Recall that we take $k_n=\lfloor n^{1/\mu }\rfloor ^{\kappa }$ and $\lambda > 2+ 4.5/\kappa$. Then we have

$$\begin{aligned} \sum _{n=1}^\infty {\mathbb {P}}( \overline{{\mathbb {A}}}_n \ne {\mathbb {A}} _n ) \le ~&\sum _{n=1}^\infty m^{k_n} {\mathbb {P}}( W^* +1 \ge m^{k_n}) = {\mathbb {E}}\sum _{n=1}^\infty m^{k_n} {\mathbf {1}}_{\{W^* +1 \ge m^{k_n}\}} \\ \le ~&K {\mathbb {E}}(W^* +1 )\log ^{ \mu /\kappa -1} (W^* +1) = K {\mathbb {E}}(W^* +1 )\log ^{ 1+3/\kappa } (W^* +1)\\<&~{\mathbb {E}}(W^* +1 )\log ^{\lambda } (W^* +1)<\infty , \end{aligned}$$

since $\lambda>2+ 4.5/\kappa > 1+3/\kappa$. Therefore, (19) follows.

Now we turn to the proof of (20). To this end, we will need the following inequality (by [5, P.31  (5.3)]): for $1<\alpha <2$,

$$\begin{aligned}&{\mathbb {P}}_{ {\mathscr {D}}_{k_n}} \left[ \bigg | \sum _{u\in {\mathbb {T}}_{k_n}} {m^{-k_n}}({\overline{X}}_{n,u}- {\mathbb {E}}_{{\mathscr {D}}_{k_n} } {\overline{X}}_{n,u}) \bigg |> \varepsilon /n^{ 1/2}\right] \nonumber \\&\quad \le K \dfrac{n^{ \alpha /2}}{\varepsilon ^{\alpha }} \left\{ m^{ -\alpha k_n} Z_{k_n}({\mathbb {R}}^d) {\mathbb {E}}(W^*+1) ^\alpha {{\mathbf {1}}_{\left\{ |W^*+1| \le m^{k_n} \right\} }} + Z_{k _n}({\mathbb {R}}^d) {\mathbb {E}}{{\mathbf {1}}_{\left\{ |W^*+1| > m^{k_n} \right\} }} \right\} .\nonumber \\ \end{aligned}$$

(24)

Thus by taking expected value of the above, we deduce that

$$\begin{aligned}&\sum _{n=1}^{\infty } {{\mathbb {P}}} (|{\overline{{\mathbb {A}}}}_n-{{\mathbb {E}}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n|>{\varepsilon }{{n}^{-1/2}}) \\&\quad =~\sum _{n=1}^{\infty }{\mathbb {E}}{\mathbb {P}}_{ {\mathscr {D}}_{k_n}} \left[ \bigg | \sum _{u\in {\mathbb {T}}_{k_n}} {m^{-k_n}}({\overline{X}}_{n,u}- {\mathbb {E}}_{{\mathscr {D}}_{k_n} } {\overline{X}}_{n,u}) \bigg |> \varepsilon n^{-1/2}\right] \\&\quad \le ~ \sum _{n=1}^{\infty } K \dfrac{n^{ \alpha /2}}{\varepsilon ^{\alpha }} \left\{ m^{ (1-\alpha ) k_n} {\mathbb {E}}(W^*+1) ^\alpha {{\mathbf {1}}_{\left\{ |W^*+1| \le m^{k_n} \right\} }} + m^{k _n} {\mathbb {E}}{{\mathbf {1}}_{\left\{ |W^*+1|> m^{k_n} \right\} }} \right\} \\&\quad = K {\varepsilon ^{-\alpha }} {\mathbb {E}}\left\{ (W^*+1) ^\alpha \sum _{n=1}^{\infty } {n^{\alpha /2}} m^{ (1-\alpha ) k_n} {{\mathbf {1}}_{\left\{ |W^*+1| \le m^{k_n} \right\} }} + \sum _{n=1}^{\infty } n^{ \alpha /2} m^{k _n} {{\mathbf {1}}_{\left\{ |W^*+1| > m^{k_n} \right\} }}\right\} \\&\quad \le K {\varepsilon ^{-\alpha }} {\mathbb {E}}(W^*+1) \Big (\log (W^*+1) \Big )^{\mu ( \alpha +2)/(2\kappa ) -1}\\&\quad =~ K {\varepsilon ^{-\alpha }} {\mathbb {E}}(W^*+1) \Big (\log (W^*+1) \Big )^{2+ 9/(2\kappa ) +(\alpha -1 ) (1+ 3/(2\kappa )) } , \end{aligned}$$

which is finite, since ${\mathbb {E}}(W^*+1) \big (\log (W^*+1) \big )^{ \lambda } <\infty$ and $2+ 9/(2\kappa ) +(\alpha -1 ) (1+ 3/(2\kappa )) <\lambda$, provided that $\alpha$ is sufficiently near one.

Hence, (20) follows by the Borel–Cantelli lemma.

It remains to prove (21). Since ${\mathbb {E}}_{{\mathscr {D}}_{k_n}} X_{n,u}=0$, we have

$$\begin{aligned}&|n^{1/2}{\mathbb {E}}_{{\mathscr {D}}_{k_n} } {\overline{{\mathbb {A}}}}_{n} |\\&\quad =~ n^{1/2}\bigg |{m^{-k_n}}\sum _{u\in {\mathbb {T}}_{k_n}}{\mathbb {E}}_{{\mathscr {D}}_{k_n} } {\overline{X}}_{n,u} \bigg | =n^{1/2}\bigg |-{m^{-k_n}}\sum _{u\in {\mathbb {T}}_{k_n}}{\mathbb {E}}_{{\mathscr {D}}_{k_n} }X_{n,u} {\mathbf {1}}_{\{ |X_{n,u}|\ge m^{k_n}\}}\bigg | \\&\quad \le ~\frac{ n^{1/2}}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}}{\mathbb {E}}_{{\mathscr {D}}_{k_n} } (W_{n-k_n}(u)+1) {\mathbf {1}}_{\{ |W_{n-k_n}(u)+1|\ge m^{k_n}\}} \\&\quad = ~n^{1/2} W_{k_n} {\mathbb {E}}( W_{n-k_n}+1) {\mathbf {1}}_{\{ |W_{n-k_n}+1|\ge m^{k_n}\}} \\&\quad \le ~ n^{1/2} W^* {\mathbb {E}}(W^*+1) {\mathbf {1}}_{\{ |W^*+1|\ge m^{k_n}\}}\\&\quad \le ~ n^{1/2} k_n^{-\lambda } (\log m) ^{-\lambda } W^* {\mathbb {E}}(W^*+1) \log ^{\lambda } (W^*+1). \end{aligned}$$

Since $\lambda \kappa /\mu - 1/2>0$,

$$\begin{aligned} n^{1/2} k_n^{-\lambda } \sim n^{1/2- \frac{ \lambda \kappa }{\mu }}\xrightarrow {n\rightarrow \infty } 0. \end{aligned}$$

Accordingly (21) follows. The lemma has been proved. $\square$

Proof of Lemma 2.3

We start the proof by the following result, which is a particular case of Corollary 4.4 in [1]:

Theorem E Assume that L has mean zero ${\mathbb {E}}L=0$ and invertible covariance $\mathrm { Cov}(L)=\Gamma$ and satisfies the weak Cramér condition (3) with a positive exponent b. If ${\mathbb {E}}{\left\| L \right\| }^3<\infty$, then

$$\begin{aligned} \sup _{t\in {\mathbb {R}}^d }\left| {\mathbb {P}}({\widehat{S}}_{n}/\sqrt{n} \le t )- \Big (\Phi _{\Gamma }(t) - n^{-1/2}\big (\mathcal {P}_1(D) \Phi _\Gamma \big ) (t) \Big ) \right| = o(n^{-1/2}). \end{aligned}$$

By using Theorem E, we deduce that

$$\begin{aligned}&{\mathbb {P}}\Big ({\widehat{S}}_{n-k_n}\le \sqrt{n}t -y\Big ) \nonumber \\&\quad = \Phi _\Gamma \bigg (\frac{ \sqrt{n}t -y}{ \sqrt{n-k_n}} \bigg ) -\frac{ 1}{(n-k_n)^{ 1/2} } \big (\mathcal {P}_1(D) \Phi _\Gamma \big )\bigg ( { \frac{ \sqrt{n}t -y}{ \sqrt{n-k_n}} } \bigg ) + \frac{ 1}{(n-k_n)^{ 1/2} }\varepsilon _{0,n}(y) ,\nonumber \\ \end{aligned}$$

(25)

where

$$\begin{aligned} \tau _n= \sup \{|\varepsilon _{0,n}(y)|: y\in {\mathbb {R}}^d\} \xrightarrow [ ]{ n\rightarrow \infty } 0. \end{aligned}$$

Here and in all that follows we suppress the dependence on t, as t is fixed.

By using Taylor’s expansion, together with the choice of $k_n$, we have that as n tends to infinity,

$$\begin{aligned}&\Phi _\Gamma \Bigg (\frac{ \sqrt{n}t -y}{ \sqrt{n-k_n}} \Bigg )= \Phi _\Gamma (t)- n^{-1/2}\left\langle {\nabla \Phi _\Gamma (t) },{y}\right\rangle +\varepsilon _{1,n}(y)n^{-1/2},\\&\frac{ 1}{(n-k_n)^{ 1/2} } \big (\mathcal {P}_1(D) \Phi _\Gamma \big )\bigg ( { \frac{ \sqrt{n}t -y}{ \sqrt{n-k_n}} } \bigg )= n^{-1/2} \big (\mathcal {P}_1(D) \Phi _\Gamma \big )( t ) + \varepsilon _{2,n}(y)n^{-1/2}, \end{aligned}$$

where $\varepsilon _{i,n}(y) (i=1,2)$ are infinitesimals satisfying

$$\begin{aligned} \sup _{y\in {\mathbb {R}}^d} \{ |\varepsilon _{i,n}(y) |\} \le K \frac{ {\left\| y \right\| }^2+k_n }{\sqrt{n}}, \quad i=1,2. \end{aligned}$$

Combining the above formulas, we conclude that

$$\begin{aligned}&{\mathbb {B}}_n\big ((-\infty ,\sqrt{n}t ]\big )= -\Phi _\Gamma (t)W_{k_n} - n^{-1/2}\left\langle {\nabla \Phi _\Gamma (t) },{N_{1,k_n} }\right\rangle \nonumber \\&\quad +n^{-1/2} \big (\mathcal {P}_1(D) \Phi _\Gamma \big ) (t) W_{k_n}+ n^{-1/2} {\varepsilon }_{n}, \end{aligned}$$

(26)

where

$$\begin{aligned} {\varepsilon }_{n} = \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} \Big [{\varepsilon }_{0,n}(S_u) +\varepsilon _{1,n}(S_u) +{\varepsilon }_{2,n}(S_u) \Big ] \end{aligned}$$

satisfies

$$\begin{aligned} |{\varepsilon }_{n}|\le K n^{-1/2} \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| }^2+ K W_{k_n} (n^{-1/2}k_n +\tau _n). \end{aligned}$$

(27)

We now prove that

$$\begin{aligned} n^{-1/2} \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| }^2 \xrightarrow [n\rightarrow \infty ]{a.s.} 0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} j^{-\mu /2} \frac{1}{m^{j^{\kappa }}} \sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^2 \xrightarrow [n\rightarrow \infty ]{a.s.} 0, \end{aligned}$$

(28)

where $j= \lfloor n^{1/\mu }\rfloor$, and $k_n= (\lfloor n^{1/\mu }\rfloor )^{\kappa }= j^{\kappa }$.

Recall that $\mu =2\kappa +3$. Observe that

$$\begin{aligned}&{\mathbb {E}}\sum _{j=1}^\infty j^{-\mu /2} \frac{1}{m^{j^{\kappa }}} \sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^2 \\ =&\sum _{j=1}^\infty j^{-\mu /2} {\mathbb {E}}{\left\| {\widehat{S}}_{j^{\kappa }} \right\| }^2 = \sum _{j=1}^\infty j^{-\mu /2} j^{\kappa } {\mathbb {E}}{\left\| L \right\| }^2= \sum _{j=1}^\infty j^{-3/2} {\mathbb {E}}{\left\| L \right\| }^2 <\infty . \end{aligned}$$

This implies $\sum _{j=1}^\infty j^{-\mu /2} \frac{1}{m^{j^{\kappa }}} \sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^2 <\infty$ a. s. and hence (28) follows.

It is easy to see that

$$\begin{aligned} K W_{k_n} (n^{-1/2}k_n+\tau _n) \rightarrow 0 \qquad \text{ a.s. } \end{aligned}$$

Combining this with (27) and (28), we have $\varepsilon _n \rightarrow 0$ a.s. as n tends to infinity.

By [2, Theorem 2], we see that under the condition ${\mathbb {E}}N(\log N)^{1+\lambda }<\infty$,

$$\begin{aligned} W_{k_n}-W= o(k_n^{-\lambda } ) =o(1/\sqrt{n}), \end{aligned}$$

(29)

since $\lambda \kappa /\mu >1/2.$ By Proposition 2.1,

$$\begin{aligned} N_{1,k_n}- \mathcal {V}_1= o(1). \end{aligned}$$

(30)

These convergence results, together with (26), yield the desired (13). The lemma is proved. $\square$

Proof of Lemma 2.4

Set for $u\in {({\mathbb {N}}^* ) }^{k_n}$,

$$\begin{aligned}&X_{n,u}= \frac{1}{m^{n-k_n}}Z_{n-k_n}\Big (u, A-S_u \Big )- {\mathbb {P}}\Big ({\widehat{S}}_{n-k_n}\in A-y\Big )\Big |_{y=S_u}, \\&{\overline{X}}_{n,u}= X_{n,u} {\mathbf {1}}_{\{ |X_{n,u}|\le m^{k_n}\}}, \quad \overline{{\mathbb {A}}}_n = \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {\overline{X}}_{n,u}, \quad {\mathbb {A}}_n= {\mathbb {A}}_n(A)= \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {X}_{n,u}. \end{aligned}$$

It is easy to see the following fact:

$$\begin{aligned} | X_{n,u}|\le W_{n-k_n}(u) +1, \quad \text{ with } W_{n-k_n}(u) =m^{-(n-k_n)} Z_{n-k_n}(u,{\mathbb {R}}^d) . \end{aligned}$$

We remind that $\{W_{n-k_n}(u): u\in {\mathbb {T}}_{k_n}\}$ are mutually independent and identically distributed as $W_{n-k_n}$.

The lemma will be proved if we can establish the following:

$$\begin{aligned}&{n}^{1+d/2} ({\mathbb {A}}_n-{\overline{{\mathbb {A}}}}_n ) =0 \text{ for } \text{ all } \text{ but } \text{ finitely } \text{ many } n. \end{aligned}$$

(31)

$$\begin{aligned}&{n}^{1+d/2} \Big ({\overline{{\mathbb {A}}}}_n -{\mathbb {E}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n \Big )\xrightarrow {n \rightarrow \infty } 0\quad \text{ a.s. } \end{aligned}$$

(32)

$$\begin{aligned}&{n}^{1+d/2} {\mathbb {E}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n \xrightarrow {n \rightarrow \infty } 0\quad \text{ a.s. } \end{aligned}$$

(33)

The result (31) follows from that

$$\begin{aligned} \sum _{n=1}^\infty {\mathbb {P}}( \overline{{\mathbb {A}}}_n \ne {\mathbb {A}} _n )&\le \sum _{n=1}^\infty m^{k_n} {\mathbb {P}}( W^* +1 \ge m^{k_n}) = {\mathbb {E}}\sum _{n=1}^\infty m^{k_n} {\mathbf {1}}_{\{W^* +1 \ge m^{k_n}\}}\\&\le K {\mathbb {E}}(W^* +1 )\log ^{\mu /\kappa -1} (W^* +1)=K {\mathbb {E}}(W^* +1 )\log ^{1+2/\kappa } (W^* +1)\\&\le {\mathbb {E}}(W^* +1 )\log ^{\lambda } (W^* +1)<\infty , \end{aligned}$$

since $1+2/\kappa <\lambda$.

The proof of (32) is similar to that of (20). Observe

$$\begin{aligned}&\sum _{n=1}^{\infty } {{\mathbb {P}}} (|{\overline{{\mathbb {A}}}}_n-{{\mathbb {E}}}_{{\mathscr {D}}_{k_n}} {\overline{{\mathbb {A}}}}_n|>{\varepsilon }{{n}^{-(1+d/2)}}) \le K {\varepsilon ^{-\alpha }} {\mathbb {E}}(W^*+1) \Big (\log (W^*+1) \Big )^{\mu ((d+2) \alpha +2)/(2\kappa ) -1}, \end{aligned}$$

which is finite, since ${\mathbb {E}}(W^*+1) \big (\log (W^*+1) \big )^{ \lambda } <\infty$ and $\mu ((d+2) \alpha +2)/(2\kappa ) -1<\lambda$, provided that $\alpha$ is sufficiently near one. Then by the Borel–Cantelli lemma (32) follows.

It remains to prove (33). By using the fact ${\mathbb {E}}_{{\mathscr {D}}_{k_n}} X_{n,u}=0$, we see that

$$\begin{aligned}&|n^{1+d/2}{\mathbb {E}}_{{\mathscr {D}}_{k_n} } {\overline{{\mathbb {A}}}}_{n} | \le n^{1+d/2} k_n^{-\lambda } (\log m) ^{-\lambda } W^* {\mathbb {E}}(W^*+1) \log ^{\lambda } (W^*+1) \xrightarrow [a.s.]{n\rightarrow \infty } 0, \end{aligned}$$

since $\lambda \kappa /\mu >1+d/2$ holds for $\kappa$ large. Accordingly, (33) follows. The lemma has been proved. $\square$

Proof of Lemma 2.5

Recall that we assume

$$\begin{aligned} \kappa >(d+4)/ (\lambda - d-3) , \qquad \mu =2\kappa +2,\qquad k_n= \big (\lfloor n^{1/\mu }\rfloor \big ) ^{ \kappa }, \forall n\in {\mathbb {N}}. \end{aligned}$$

To begin with, we need the following result, which is a particular case of Corollary 4.4 in [1]:

Theorem F   Assume that L has mean zero ${\mathbb {E}}L=0$ and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$ and satisfies the weak Cramér condition (3) with a positive exponent $b < {2}/{(d+1)}$. If ${\mathbb {E}}{\left\| L \right\| }^{d+4}<\infty$, then

$$\begin{aligned}&\sup _{C\in {\mathfrak {C}} }\Bigg |{\mathbb {P}}({\widehat{S}}_{n}/\sqrt{n}\in C )- \int _C \Big [\varphi _{\Gamma }(x) - n^{-1/2}\mathcal {P}_1(D) \varphi _\Gamma (x) + n^{-1}\mathcal {P}_2(D) \varphi _\Gamma (x)\\&\quad +\sum _{k=3}^{d+2}n^{-k/2} \mathcal {P}_k(-D) \varphi _\Gamma (x)\Big ] \mathrm {d}x\Bigg | = o(n^{-(d+2)/2}), \end{aligned}$$

where ${\mathfrak {C}}$ is the class of all Borel-measurable convex subsets of ${\mathbb {R}}^d$, $\mathcal {P}_k(-D)$ are linear combinations of $D^{\nu } (1\le \left|\nu \right| \le 3k)$ with coefficients involving $\chi _\nu (\left|\nu \right| \le k+2 )$.

By using Theorem F, we have the following result:

$$\begin{aligned}&{\mathbb {P}}\Big ({\widehat{S}}_{n-k_n} +y \in A \Big ) = \int _{\frac{ A-y}{\sqrt{n-k_n}}} \varphi _{\Gamma }( x)d x + \int _{\frac{ A-y}{\sqrt{n-k_n}}} \bigg [- (n-k_n)^{-1/2}\mathcal {P}_1(D) \varphi _\Gamma (x) \\&\quad + (n-k_n)^{-1}\mathcal {P}_2(D) \varphi _\Gamma (x) + \sum _{k=3}^{d+2}(n-k_n)^{-k/2} \mathcal {P}_k(-D) \varphi _\Gamma (x)\bigg ] \mathrm {d}x +n^{-(1+d/2)} \varepsilon _0(n,y), \end{aligned}$$

where $\varepsilon _0(n,y)$ is an infinitesimal uniformly bounded for y:

$$\begin{aligned} \varrho _n=\sup _{y\in {\mathbb {R}}^d}|\varepsilon _0(n,y)|\xrightarrow {n\rightarrow \infty } 0. \end{aligned}$$

Observe that

$$\begin{aligned} \frac{ 1}{( n-k_n)^{-d/2}}=\frac{1}{n^{d/2}} + \frac{ dk_n}{2n^{1+d/2}} + O(1) \frac{ k_n^2}{n^{2+d/2}}. \end{aligned}$$

By Taylor’s expansion, we find that

$$\begin{aligned}&\varphi _\Gamma \Big ( \frac{z-y}{\sqrt{n-k_n}}\Big ) = \varphi _\Gamma (0)\exp \Big \{ -\frac{1}{2(n-k_n)} \left\langle { z-y},{\Gamma ^{-1}(z-y)}\right\rangle \Big \} \\ =\&\varphi _\Gamma (0)\bigg [ 1-\frac{1}{2(n-k_n)} \left\langle { z-y},{\Gamma ^{-1}(z-y)}\right\rangle + \frac{O(1)}{8(n-k_n)^2} \left\langle { z-y},{\Gamma ^{-1}(z-y)}\right\rangle ^2 \bigg ] \\ =\&\varphi _\Gamma (0)\bigg [ 1- \frac{1}{2n}\Big ( \left\langle {z},{\Gamma ^{-1}z}\right\rangle -2 \left\langle { z},{\Gamma ^{-1}y}\right\rangle + \left\langle { y},{\Gamma ^{-1}y}\right\rangle \Big )\bigg ]+ \eta _1(n,y), \end{aligned}$$

where $\left| \eta _1(n,y)\right| \le K (k_n^2+ {\left\| z \right\| }^4+{\left\| y \right\| }^4 )/n^2$ . From these observations, we find that

$$\begin{aligned} \int _{\frac{ A-y}{\sqrt{n-k_n}}} \varphi _{\Gamma }( x)d x= & {} (n-k_n)^{-d/2} \int _A \varphi _\Gamma \Big ( \frac{z-y}{\sqrt{n-k_n}}\Big )\mathrm {d}z = \frac{1}{n^{d/2}}\int _A\varphi _\Gamma \Big ( \frac{z }{\sqrt{n }} \Big ) \mathrm {d}z \\&+\frac{ \varphi _{\Gamma }(0) }{2n^{1+d/2} } \left[ 2\left\langle {\int _A z\mathrm {d}z},{\Gamma ^{-1}y}\right\rangle - |A|\left( \left\langle { y},{\Gamma ^{-1} y}\right\rangle -dk_n\right) \right] + \frac{1}{n^{1+d/2}} \varepsilon _1(n,y), \end{aligned}$$

where $|\varepsilon _1(n,y) | \le K \Big ( \frac{ {\left\| y \right\| }^4 }{n} + \frac{ k_n^2}{n} + \frac{ {\left\| y \right\| }^2}{\sqrt{n}}\Big )$. Also arguing as above gives that

$$\begin{aligned} \mathcal {P}_1(D) \varphi _\Gamma \Big ( \frac{z-y}{\sqrt{n-k_n}} \Big ) =&\mathcal {P}_1(D) \varphi _\Gamma (0) + \left\langle {\nabla \big ( \mathcal {P}_1(D) \varphi _\Gamma \big )(0)},{\frac{z-y}{\sqrt{n-k_n}} }\right\rangle +O(1) {\left\| \frac{z-y}{\sqrt{n-k_n}} \right\| }^2 \\ =&\frac{1}{\sqrt{n}}\left\langle {\nabla \big ( \mathcal {P}_1(D) \varphi _\Gamma \big )(0)},{z-y }\right\rangle + \eta _2(n,y) , \end{aligned}$$

where $\left| \eta _2(n,y)\right| \le K ({\left\| z \right\| }^2+{\left\| y \right\| }^2 )/n$. Hence we have that as n tends to infinity,

$$\begin{aligned}&\int _{\frac{ A-y}{\sqrt{n-k_n}}} (n-k_n )^{ -1/2}\mathcal {P}_1(D) \varphi _\Gamma (x)\mathrm {d}x \\&\quad = (n-k_n )^{ -(d+1)/2}\int _A \mathcal {P}_1(D) \varphi _\Gamma \Big ( \frac{z-y}{\sqrt{n-k_n}} \Big ) \mathrm {d}z \\&\quad = n^{-(d+2)/2} \left\langle {\nabla \big ( {\mathcal {P}_1(D) \varphi _\Gamma } \big )(0)},{\int _A z \mathrm {d}z- |A|y}\right\rangle + n^{-(d+2)/2} \varepsilon _2(n,y) \\&\quad = n^{-(d+2)/2} \varphi _\Gamma (0) \left\langle {\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)},{\int _A z \mathrm {d}z- |A|y}\right\rangle + n^{-(d+2)/2} \varepsilon _2(n,y), \end{aligned}$$

where $|\varepsilon _2(n,y) | \le K \Big ( \frac{ {\left\| y \right\| }^4 }{n} + \frac{ k_n^2}{n} + \frac{ {\left\| y \right\| }^2}{\sqrt{n}}\Big ).$

Similarly, we can deduce that

$$\begin{aligned}&\int _{\frac{ A-y}{\sqrt{n-k_n}}} (n-k_n )^{ -(d+2)/2} \mathcal {P}_2(D)\varphi _\Gamma (x) \mathrm {d}x \\&\quad = (n-k_n )^{ -(d+2)/2}\int _A \mathcal {P}_2(D)\varphi _\Gamma \Big ( \frac{z-y}{\sqrt{n-k_n}} \Big ) \mathrm {d}z \\&\quad = n^{-(d+2)/2} \mathcal {P}_2(D) \varphi _\Gamma (0) |A|+ n^{-(d+2)/2} \varepsilon _3(n,y) \\&\quad = n^{-(d+2)/2} \varphi _\Gamma (0) \bigg ( \frac{\mathcal {P}_2(D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0) |A|+ n^{-(d+2)/2} \varepsilon _3(n,y), \end{aligned}$$

and

$$\begin{aligned}&\int _{\frac{ A-y}{\sqrt{n-k_n}}} \bigg [ \sum _{k=3}^{d+2}(n-k_n)^{-k/2} \mathcal {P}_k(-D) \varphi _\Gamma (x)\bigg ] \mathrm {d}x\\&\quad = \int _A \bigg [ \sum _{k=3}^{d+2}(n-k_n)^{-(d+k)/2} \mathcal {P}_k(-D) \varphi _\Gamma \Big (\frac{z-y}{\sqrt{n-k_n}}\Big )\bigg ]\mathrm {d}z = n^{-(d+2)/2} \varepsilon _4(n,y), \end{aligned}$$

where

$$\begin{aligned} |\varepsilon _3(n,y) | \le K \frac{1+{\left\| y \right\| } }{\sqrt{n}} , ~~|\varepsilon _4(n,y) | \le K \frac{1 }{\sqrt{n}} . \end{aligned}$$

Combining these formulas, we get

$$\begin{aligned} B_{n}(A)= & {} \frac{1}{n^ { d/2}} \int _A\varphi _\Gamma \Big ( \frac{z }{\sqrt{n }} \Big ) \mathrm {d}z \, {W}_{k_n}\nonumber \\&+ \frac{ \varphi _{\Gamma }(0) |A|}{n^ { 1+ d/2}} \Bigg \{ \left[ \bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0)- \left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)},{{\mathfrak {x}}_A}\right\rangle \right] {W}_{k_n}\nonumber \\&+\left\langle {\nabla \bigg ( \frac{\mathcal {P}_1 (D) \varphi _\Gamma }{\varphi _\Gamma } \bigg )(0)+\Gamma ^{-1}{\mathfrak {x}}_A },{ {N}_{1,k_n}}\right\rangle -\frac{1}{2} {N}_{2,k_n} \Bigg \} +\frac{1}{n^ {1+ d/2}} \varepsilon _{n}, \end{aligned}$$

(34)

where

$$\begin{aligned} \varepsilon _{n}= \frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}}\sum _{i=0}^4\varepsilon _{i}(n,S_u) \end{aligned}$$

satisfies

$$\begin{aligned} |\varepsilon _{n}|\le K n^{-1}\frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| }^4 +K n^{-1/2} \frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| }+ K n^{-1}k_n^2W_{k_n} + \varrho _n W_{k_n}. \end{aligned}$$

(35)

We go to prove that $\varepsilon _{n}$ tends to zero a.s. as n tends to infinity. As it is easily seen that

$$\begin{aligned} K n^{-1}k_n^2W_{k_n} + \varrho _n W_{k_n} \xrightarrow [a.s. ]{n\rightarrow \infty } 0, \end{aligned}$$

we only need to show that

$$\begin{aligned} n^{-1}\frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| }^4 + n^{-1/2} \frac{1}{m^{k_n}}\sum _{u\in {\mathbb {T}}_{k_n}} {\left\| S_u \right\| } \xrightarrow [a.s. ]{n\rightarrow \infty } 0. \end{aligned}$$

With $j= \lfloor n^{1/\mu }\rfloor$ and $k_n= j^{\kappa }$, this is equivalent to show that

$$\begin{aligned} j^{-\mu } \frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^4 + j^{-\mu /2} \frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| } \xrightarrow [a.s. ]{n\rightarrow \infty } 0. \end{aligned}$$

(36)

By using the moment inequality of sums of independent random variables (see, for instance, [22]), we observe that for $\kappa$ large,

$$\begin{aligned}&{\mathbb {E}}\sum _{j=1}^\infty \bigg [ j^{-\mu }\frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^4 + j^{-\mu /2} \frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| } \bigg ] \\= & {} \sum _{j=1}^\infty \bigg [ j^{-\mu }{\mathbb {E}}{\left\| {\widehat{S}}_{j^{\kappa }} \right\| }^4 + j^{-\mu /2} {\mathbb {E}}{\left\| {\widehat{S}}_{j^{\kappa }} \right\| }\bigg ] \le K \sum _{j=1}^\infty j^{-\mu } j^{2 \kappa }{\mathbb {E}}{\left\| L \right\| }^4 + j^{-\mu /2} j^{\kappa /2} \sqrt{ {\mathbb {E}}{\left\| L \right\| }^2} <\infty . \end{aligned}$$

This implies a. s.

$$\begin{aligned} \sum _{j=1}^\infty \bigg [ j^{-\mu }\frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| }^4 + j^{-\mu /2} \frac{1}{m^{j^{\kappa }}}\sum _{u\in {\mathbb {T}}_{j^{\kappa }}} {\left\| S_u \right\| } \bigg ] <\infty \end{aligned}$$

and hence (36) follows. So by (35), we have that $\varepsilon _{n}$ tends to zero as n tends to infinity.

By [2, Theorem 2], under the condition ${\mathbb {E}}N(\log N)^{1+\lambda }<\infty$ for $\lambda >d+3$, we see that

$$\begin{aligned} W_{k_n}-W= o(k_n^{-\lambda } ) =o(1/n). \end{aligned}$$

(37)

Proposition 2.1 implies that

$$\begin{aligned} {N}_{1, k_n}-\mathcal {V}_1=o(1), \quad {N}_{2, k_n}-\mathcal {V}_2=o(1). \end{aligned}$$

(38)

Substituting (37) and (38) into (34) gives the desired (15). The lemma is proved. $\square$

Proof of Lemma 2.6

Set for $u\in {({\mathbb {N}}^* ) }^{k_n}$,

$$\begin{aligned}&X_{n,u}=\bigg [\frac{ \sum _{v\in {\mathbb {T}}_{n-k_n}(u) } f(y+ S_{uv}-S_u)}{m^{n-k_n} } - {\mathbb {E}}f(y+ {\widehat{S}}_{n-k_n}) \bigg ]_{y=S_u},\\&{\overline{X}}_{n,u}= X_{n,u} {\mathbf {1}}_{\{ |X_{n,u}|\le m^{k_n}\}}, \quad \overline{{\mathbb {A}}}_n (f)= \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {\overline{X}}_{n,u}, \quad {\mathbb {A}}_n(f)= \frac{1}{m^{k_n}} \sum _{u\in {\mathbb {T}}_{k_n}} {X}_{n,u}. \end{aligned}$$

As f is bounded by some constant $K_f>0$, it is easy to see the following fact:

$$\begin{aligned} | X_{n,u}|\le K_f(W_{n-k_n}(u) +1) . \end{aligned}$$

A similar argument as in Lemma 2.2 allows us to finish the proof of the lemma. $\square$

Proof of Lemma 2.7

Recall that we assume

$$\begin{aligned} \kappa >(d+4)/ (\lambda - d-3) , \qquad \mu =2\kappa +2,\qquad k_n= \big (\lfloor n^{1/\mu }\rfloor \big ) ^{ \kappa }, \forall n\in {\mathbb {N}}. \end{aligned}$$

To begin with, we need the following result, which is a particular case of Theorem 7.1 in [8]:

Theorem G  Assume that L has mean zero ${\mathbb {E}}L=0$ and invertible covariance matrix $\mathrm { Cov}(L)=\Gamma$ and satisfies the weak Cramér condition (3) with $b >0$. If ${\mathbb {E}}{\left\| L \right\| }^{d+2}<\infty$, then

$$\begin{aligned} {\mathbb {E}}(f({\widehat{S}}_n) )= & {} \int _{{\mathbb {R}}^d} f(x\sqrt{n} )\Big [\varphi _{\Gamma }(x) - n^{-1/2}\mathcal {P}_1(D) \varphi _\Gamma (x) + n^{-1}\mathcal {P}_2(D) \varphi _\Gamma (x) \Big ] \mathrm {d}x \\&+ K_f \cdot o\Big (\frac{1}{n^{(2+d)/2}}\Big ). \end{aligned}$$

From this theorem, we see that

$$\begin{aligned}&{\mathbb {E}}f(y+{\widehat{S}}_{n-k_n}) = \int _{{\mathbb {R}}^d} f(y+x\sqrt{n-k_n} ) \Big [\varphi _{\Gamma }(x) - (n-k_n)^{-1/2}\mathcal {P}_1(D) \varphi _\Gamma (x)\\&\qquad \qquad + (n-k_n)^{-1}\mathcal {P}_2(D) \varphi _\Gamma (x) \Big ] \mathrm {d}x + K_f \cdot o\Big (\frac{1}{(n-k_n)^{(2+d)/2}}\Big ) \\&\quad = \int _{{\mathbb {R}}^d} f(z) \bigg [\frac{1}{(n-k_n)^{d/2}} \varphi _{\Gamma }\Big ( \frac{z-y }{\sqrt{n-k_n}} \Big ) - \frac{1}{(n-k_n)^{(d+1)/2}}\mathcal {P}_1(D) \varphi _\Gamma \Big ( \frac{z-y }{\sqrt{n-k_n}} \Big ) \\&\qquad \qquad + \frac{1}{(n-k_n)^{(d+2)/2}}\mathcal {P}_2(D) \varphi _\Gamma \Big ( \frac{z-y }{\sqrt{n-k_n}} \Big )\bigg ] \mathrm {d}x + K_f \cdot o\Big (\frac{1}{(n-k_n)^{(2+d)/2}}\Big ), \end{aligned}$$

where $K_f$ is a constant only depending on f. Arguing as the proof of Lemma 2.5, we get

$$\begin{aligned}&{\mathbb {E}}f(y+{\widehat{S}}_{n-k_n}) = \frac{1}{n^{d/2} }\int _{{\mathbb {R}}^d} f(z) \bigg \{ \varphi _{\Gamma }\Big ( \frac{z}{\sqrt{n}}\Big ) +\frac{1}{n} \varphi _{\Gamma }(0) \bigg [ - \frac{1}{2} (\left\langle {y},{\Gamma ^{-1} y}\right\rangle -dk_n) \\&\quad + \left\langle {\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)+ \Gamma ^{-1}z},{ y}\right\rangle - \left\langle {\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)},{z}\right\rangle +\bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0) \bigg ]\bigg \} \mathrm {d}z\\&\quad + K_f \cdot o\Big (\frac{1}{n^{(2+d)/2}}\Big ). \end{aligned}$$

Hence,

$$\begin{aligned}&{\mathbb {E}}f(y+{\widehat{S}}_{n-k_n}) = \frac{1}{n^{d/2} } \int _{{\mathbb {R}}^d} f(z)\varphi _{\Gamma }\Big ( \frac{z}{\sqrt{n}}\Big ) \mathrm {d}z +\frac{1}{n^{1+d/2}} \varphi _{\Gamma }(0) \bigg [ - \frac{1}{2} m(f)(\left\langle {y},{\Gamma ^{-1} y}\right\rangle -dk_n) \\&\quad + \left\langle {m(f)\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)+ \Gamma ^{ -1}\int _{{\mathbb {R}}^d} f(z)z \mathrm {d}z},{ y}\right\rangle \\&\quad - \left\langle {\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)},{\int _{{\mathbb {R}}^d} f(z)z \mathrm {d}z}\right\rangle +m(f)\bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0) \bigg ]\\&\quad + K_f \cdot o\Big (\frac{1}{n^{(2+d)/2}}\Big ). \end{aligned}$$

This yields that

$$\begin{aligned} {\mathbb {B}}_n(f)= & {} \frac{1}{n^{d/2} }\int _{{\mathbb {R}}^d} f(z) \varphi _{\Gamma }\Big ( \frac{z}{\sqrt{n}}\Big ) \mathrm {d}z W_{k_n}+ \frac{1}{n^{1+d/2} }\varphi _{\Gamma }(0)\bigg [ - \frac{1}{2} m(f) N_{2,k_n} \\&+ \left\langle {m(f)\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)+ \Gamma ^{ -1}\int _{{\mathbb {R}}^d} f(z)z \mathrm {d}z},{ N_{1,k_n}}\right\rangle \\&- \left\langle {\nabla \big ( \frac{\mathcal {P}_1(D) \varphi _\Gamma }{ \varphi _\Gamma } \big )(0)},{\int _{{\mathbb {R}}^d} f(z)z \mathrm {d}z}\right\rangle W_{k_n}+m(f)\bigg ( \frac{ \mathcal {P}_2 (D) \varphi _\Gamma }{\varphi _\Gamma }\bigg ) (0) W_{k_n}\bigg ] + o(\frac{1}{n^{(2+d)/2}} ) \end{aligned}$$

Substituting(37) and (38) in the last estimation, we find that (17) holds. $\square$